# On a degenerate boundary value problem to relativistic magnetohydrodynamics
with a general pressure law

Rahul Barthwal  T. Raja Sekhar  Department of Mathematics, Indian Institute of
Technology Kharagpur, Kharagpur, India

###### Abstract

This work is concerned with establishing the existence and uniqueness of the
solution to a mixed-type degenerate boundary value problem for a relativistic
magnetohydrodynamics system. We first consider a full relativistic
magnetohydrodynamics system and reduce it to a simplified form under the
assumption that the magnetic field vector is orthogonal to the velocity
vector. We consider a boundary value problem for the steady part of this
reduced system where the boundary data is prescribed on a sonic boundary and a
characteristic curve. Here the main difficulty is the consideration of a
relativistic system, with a general equation of state while considering the
magnetic field effects as well, which we believe has never been analyzed
before in the context of the analytical study of sonic-supersonic flows. Also,
the degeneracy of the governing equations along the sonic curve is a crucial
challenge. However, we employ the iteration method used in the work of Li and
Hu [1] to prove the existence and uniqueness of a local classical supersonic
solution in the partial hodograph plane first and finally, we recover a local
smooth solution to the boundary value problem in the physical plane by
applying an inverse transformation.

###### keywords:

Relativistic magnetohydrodynamics, Characteristic decomposition, Sonic-
supersonic solutions, Degenerate boundary value problem

###### MSC:

[] 35L65; 35L80; 76H05; 76N15

##  1 Introduction

One of the most significant issues in the mathematical analysis of gas
dynamical systems is the analysis of transonic flow which is present in a
variety of significant physical phenomena. The analysis of sonic-supersonic
patches is crucial in the mathematical study of transonic flow problems.
According to Courant and Friedrichs [2], if the Mach number of the flow
passing through an obstacle or duct is not much below one, the flow can become
supersonic near the duct’s surface due to the convexity of the duct and
reverts back to being purely subsonic. Many scientific and aerospace
applications naturally include similar circumstances, examples including the
flow via a nozzle or a flow over a symmetric or asymmetric airfoil. A global
solution to these transonic flow problems has been the subject of numerous
substantial contributions over the past century, but it is still an open
mathematical problem even now. Because a transonic structure is made up of
supersonic and subsonic components that are separated by either a transonic
shock or a sonic boundary, transonic flows are much more complex to analyze in
contrast to a purely supersonic or subsonic flow. Also, the governing
equations of the flow are generally linearly degenerate on a sonic boundary
and may change their nature from hyperbolic (supersonic) to elliptic
(subsonic) across the degenerate boundary; see [3, 4, 5, 6, 7, 8].

According to Morawetz’s research on transonic flow in channels and ducts (see
Figure 1) a smooth flow is absent in general. This suggests that in the
downstream flow, there could be a transonic shock [9]. The nonexistence of a
smooth transonic flow around a plane blunt airfoil was proved by Morawetz [10]
using the compensated-compactness framework. The same framework was utilized
to prove the existence of weak solutions to transonic flow problems by
Morawetz [11] and Chen et al. [12]. Recent years have seen the development of
much significant research for the subsonic-sonic and sonic-supersonic parts of
the transonic flow for two-dimensional gas dynamics systems and other related
mixed-type systems. For the subsonic part, the global existence of solutions
has been developed extensively in the last few decades. For a subsonic-sonic
part of a long nozzle, the global existence of a solution was proved by Xie
and Xin [13, 14]. Further, they proved the well-posedness of the sonic-
subsonic and subsonic flows with critical mass flux for isentropic Euler
equations. Their work was extended to the full Euler equations by Chen et al.
[15]. Chen et al. [16] developed the sonic-subsonic limit of approximate
solutions for the multi-dimensional full Euler equations. Recently, Wang and
Xin studied Meyer-type transonic flows in de Laval nozzles and established the
existence and uniqueness of a smooth solution. Also, they proved the well-
posedness of global subsonic-sonic flow problems [17]. As far as the sonic-
supersonic part is concerned, the relevant results are still very limited.
Here we give a brief overview of some significant developments in the context
of supersonic-sonic flows over the last few years. Under the assumption that
the flow is irrotational, a local sonic-supersonic solution was developed by
Zhang and Zheng [18] for isentropic Euler equations in two dimensions. Their
results were extended to the case of two-dimensional steady and unsteady full
Euler equations by Hu and Li [19, 20]. Further, they also developed a global
smooth sonic-supersonic solution and discussed its behaviour near the sonic
boundary in [21]. When tackling sonic-supersonic boundary value problems, the
partial hodograph mapping employed in the work of Li and Hu has been proven to
be extremely crucial; see [1, 19, 21] and references cited therein for more
details. We also refer interested readers to [22, 23, 24, 25, 26, 1, 12] for
more such results on subsonic-sonic and sonic-supersonic patch problems.

Figure 1: Transonic flow in a duct with a supersonic bubble

The velocity of fluid particles in the stellar to galactic scales (active
galactic nuclei, core-collapse supernovae, superluminal jets, coalescing
neutron stars, gamma-ray bursts and the formation of black holes etc.) in the
areas of plasma physics, astrophysics, and nuclear physics is typically very
high and frequently very close to the speed of light, necessitating the
consideration of relativistic effects and rendering obsolete the classical
Euler equations. The governing system in question for such a high-speed flow
is known as the relativistic Euler system. Recently, a number of sonic-
supersonic boundary value problems for a two-dimensional relativistic Euler
system and axisymmetric relativistic Euler system has been considered; see
viz. [27, 28]. However, in the analytical study of the mathematical theory of
transonic flows, magnetic effects for such complicated systems have not been
considered before to the best of the authors’ knowledge. Throughout celestial
objects as well as the entire cosmos, magnetic fields are present everywhere.
For the treatment of fluids and gases coupled to an electromagnetic field,
magnetohydrodynamics is usually an effective approximation. Relativistic
magnetohydrodynamics is needed to analyze in detail to understand several
astrophysical phenomena, including accretion disks, plasma winds,
magnetospheres, and astrophysical jets near compact objects where the
relativistic effects are considerable. This motivated us to discuss a sonic-
supersonic degenerate boundary value problem for a relativistic
magnetohydrodynamics system in this article, where the boundary data is
prescribed on a characteristic curve and a sonic boundary. We choose a general
convex pressure which makes the problem under consideration even more
complicated and extensive.

Let us consider the motion of a relativistic fluid in an electromagnetic field
in the Minkowski space-time governed by relativistic magnetohydrodynamics
equations. In the covariant form, these equations can be written as [29]

$\displaystyle\begin{cases}\partial_{j}(nu^{j})=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{(Conservation\leavevmode\nobreak\
of\leavevmode\nobreak\ mass),}\\\
\partial_{j}(\mathbf{T}^{jk})=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{(Conservation\leavevmode\nobreak\ of\leavevmode\nobreak\ energy-
momentum),}\\\ \partial_{j}(\mathbf{\Psi}^{jk})=0,\leavevmode\nobreak\
\leavevmode\nobreak\ j,k=0,1,2,3\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{(Maxwell\leavevmode\nobreak\ equations),}\\\
\end{cases}$ (1.1)

where the Einstein summation convention has been used, $n$ denotes the average
rest-mass density of baryon particles, $u^{j}$ denotes the four-velocity
vector and $\partial_{j}$ denote the covariant derivative operator with
respect to space-time coordinates $(t,x^{1},x^{2},x^{3})$. Further, the tensor
$\mathbf{\Psi}^{jk}$ is given by $\mathbf{\Psi}^{jk}=u^{j}h^{k}-u^{k}h^{j}$
and the energy-momentum tensor $\mathbf{T}^{jk}$ is decomposed into the fluid
part $\mathbf{T}_{f}^{jk}$ and electromagnetic part $\mathbf{T}_{m}^{jk}$ such
that $\mathbf{T}^{jk}=\mathbf{T}_{f}^{jk}+\mathbf{T}_{m}^{jk}$ with

$\displaystyle\begin{cases}\mathbf{T}_{f}^{jk}=(p+\rho)u^{j}u^{k}+pg^{jk},\\\
\mathbf{T}_{m}^{jk}=|h|^{2}(u^{j}u^{k}+g^{jk}/2)-h^{j}h^{k},\end{cases}$

where $i=p+\rho$ represents the enthalpy per unit volume such that $p$ is the
gas pressure, $\rho=mn+e$ defines the total mass-energy density with $e$ being
the specific internal energy and $m$ being the average rest mass. Further, we
take $g^{jk}$ as the Minkowski tensor, i.e.
$g^{jk}=\pm\mathrm{diag}\\{-1,1,1,1\\}$ and normalize the speed of light to be
one. Also, relations between the four vectors $u^{j}$ and $h^{j}$ and the
spatial components of the velocity $\vec{\mathbf{U}}=(u_{1},u_{2},u_{3})$ and
the laboratory magnetic field $\vec{\mathbf{H}}=(H_{1},H_{2},H_{3})$ are
$u^{j}=\gamma(1,\vec{\mathbf{U}}),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
h^{j}=\gamma\left(\vec{\mathbf{U}}.\vec{\mathbf{H}},\dfrac{\vec{\mathbf{H}}}{\gamma^{2}}+\vec{\mathbf{U}}(\vec{\mathbf{U}}.\vec{\mathbf{H}})\right),$
where
$\gamma=\dfrac{1}{\sqrt{1-u_{1}^{2}-u_{2}^{2}-u_{3}^{2}}}\leavevmode\nobreak\
(0<u_{1}^{2}+u_{2}^{2}+u_{3}^{2}<1)$ is the Lorentz factor. For simplicity of
notations and computations to be followed, we rewrite (1.1) into the
conservative form given by

$\begin{cases}\begin{aligned} &(n\gamma)_{t}+\mathrm{div}\leavevmode\nobreak\
(n\gamma\vec{\mathbf{U}})=0,\\\
&\left((i\gamma^{2}+|\vec{\mathbf{H}}|^{2})\vec{\mathbf{U}}-(\vec{\mathbf{U}}.\vec{\mathbf{H}})\vec{\mathbf{H}}\right)_{t}+\mathrm{div}\left(\left((i\gamma^{2}+|\vec{\mathbf{H}}|^{2})\vec{\mathbf{U}}-(\vec{\mathbf{U}}.\vec{\mathbf{H}})\vec{\mathbf{H}}\right)\otimes\vec{\mathbf{U}}-\vec{\mathbf{H}}\left(\dfrac{\vec{\mathbf{H}}}{\gamma^{2}}+(\vec{\mathbf{U}}.\vec{\mathbf{H}})\vec{\mathbf{U}}\right)+p^{tot}\mathbf{I}\right)=0,\\\
&\left(i\gamma^{2}-p^{tot}+|\vec{\mathbf{H}}|^{2}\right)_{t}+\mathrm{div}\left((i\gamma^{2}+|\vec{\mathbf{H}}|^{2})\vec{\mathbf{U}}-(\vec{\mathbf{U}}.\vec{\mathbf{H}})\vec{\mathbf{H}}\right)=0,\\\
&\vec{\mathbf{H}}_{t}-\mathrm{rot}\leavevmode\nobreak\
(\vec{\mathbf{U}}\times\vec{\mathbf{H}})=0,\\\
&\mathrm{div}\leavevmode\nobreak\ \vec{\mathbf{H}}=0,\end{aligned}\end{cases}$
(1.2)

where $\mathbf{I}$ is a $3\times 3$ identity matrix and $p^{tot}=p+p_{m}$ is
the total pressure containing the gas pressure $p$ and the magnetic pressure
$p_{m}$ given by
$p_{m}=\dfrac{1}{2}\left(\dfrac{|\vec{\mathbf{H}}|^{2}}{\gamma^{2}}+(\vec{\mathbf{U}}.\vec{\mathbf{H}})^{2}\right)$.

Note that
$\vec{\mathbf{U}}\times\vec{\mathbf{H}}=(u_{2}H_{3}-u_{3}H_{2},u_{3}H_{1}-u_{1}H_{3},u_{1}H_{2}-u_{2}H_{1})$,
which implies that the magnetic field equations can be rewritten as

$\displaystyle\begin{cases}(H_{1})_{t}-(u_{1}H_{2}-u_{2}H_{1})_{y}-(u_{3}H_{1}-u_{1}H_{3})_{z}=0,\\\
(H_{2})_{t}-(u_{2}H_{3}-u_{3}H_{2})_{z}-(u_{1}H_{2}-u_{2}H_{1})_{x}=0,\\\
(H_{3})_{t}-(u_{3}H_{1}-u_{1}H_{3})_{x}-(u_{2}H_{3}-u_{3}H_{2})_{y}=0.\end{cases}$

Therefore, if we assume that the magnetic field is orthogonal to the velocity
field such that $\vec{\mathbf{H}}=(0,0,H)$ and $\vec{\mathbf{U}}=(u,v,0)$
satisfying $\vec{\mathbf{U}}.\vec{\mathbf{H}}=0$ and the variables
$(u,v,p,\rho,H)$ are independent of the space variable $z$, then the system
(1.2) is reduced to

$\displaystyle\begin{cases}(n\gamma)_{t}+(n\gamma u)_{x}+(n\gamma v)_{y}=0,\\\
\left((i\gamma^{2}+H^{2})u\right)_{t}+\left((i\gamma^{2}+H^{2})u^{2}+p+\dfrac{H^{2}}{2\gamma^{2}}\right)_{x}+\left((i\gamma^{2}+H^{2})uv\right)_{y}=0,\\\
\left((i\gamma^{2}+H^{2})v\right)_{t}+\left((i\gamma^{2}+H^{2})uv\right)_{x}+\left((i\gamma^{2}+H^{2})v^{2}+p+\dfrac{H^{2}}{2\gamma^{2}}\right)_{y}=0,\\\
\left(i\gamma^{2}-p-\dfrac{H^{2}}{2\gamma^{2}}+H^{2}\right)_{t}+\left((i\gamma^{2}+H^{2})u\right)_{x}+\left((i\gamma^{2}+H^{2})v\right)_{y}=0,\\\
H_{t}+(Hu)_{x}+(Hv)_{y}=0.\end{cases}$ (1.3)

From the first and last equation, it is trivial to see that

$\displaystyle\left(\dfrac{H}{n\gamma}\right)_{t}+u\left(\dfrac{H}{n\gamma}\right)_{x}+v\left(\dfrac{H}{n\gamma}\right)_{y}=0,$
(1.4)

which implies that $\dfrac{H}{n\gamma}$ is a constant along each stream line.
The relation (1.4) is essentially an expression of the frozen-in law. To put
it another way, all magnetic field lines move with the particles on this line
and it appears like magnetic field lines have “frozen” on the fluid moving
with them, see [30]. For simplicity, we assume that
$\dfrac{H}{n\gamma}=\kappa_{0}$ where $\kappa_{0}$ is a constant.

Also, the Rankine-Hugoniot conditions for system (1.3) are given by

$\displaystyle\begin{cases}\alpha_{t}[n\gamma]=\alpha_{x}[n\gamma
u]+\alpha_{y}[n\gamma v],\\\
\alpha_{t}[(i\gamma^{2}+H^{2})u]=\alpha_{x}\bigg{[}(i\gamma^{2}+H^{2})u^{2}+p+\dfrac{H^{2}}{2\gamma^{2}}\bigg{]}+\alpha_{y}[(i\gamma^{2}+H^{2})uv],\\\
\alpha_{t}[(i\gamma^{2}+H^{2})v]=\alpha_{x}[(i\gamma^{2}+H^{2})uv]+\alpha_{y}\bigg{[}(i\gamma^{2}+H^{2})v^{2}+p+\dfrac{H^{2}}{2\gamma^{2}}\bigg{]},\\\
\alpha_{t}\bigg{[}i\gamma^{2}-p-\dfrac{H^{2}}{2\gamma^{2}}+H^{2}\bigg{]}=\alpha_{x}[(i\gamma^{2}+H^{2})u]+\alpha_{y}[(i\gamma^{2}+H^{2})v],\\\
\alpha_{t}[H]=\alpha_{x}[Hu]+\alpha_{y}[Hv],\\\ \end{cases}$ (1.5)

where $[(.)]$ denotes the jump in the variable $(.)$ across a discontinuity
surface and $(-\alpha_{t},\alpha_{x},\alpha_{y})$ denotes the normal to the
discontinuity surface.

Now one can use the first and fifth equations of (1.5) to obtain

$\displaystyle\bigg{[}\dfrac{H}{n\gamma}\bigg{]}=0,$ (1.6)

which means that $\dfrac{H}{n\gamma}$ remains unchanged across a shock wave.
Therefore, in view of (1.4) and (1.6), we consider the following reduced
system

$\displaystyle\begin{cases}(n\gamma)_{t}+(n\gamma u)_{x}+(n\gamma v)_{y}=0,\\\
\bigg{[}((i+\kappa_{0}^{2}n^{2})\gamma^{2})u\bigg{]}_{t}+\left((i+\kappa_{0}^{2}n^{2})\gamma^{2})u^{2}+p+\dfrac{1}{2}\kappa_{0}^{2}n^{2}\right)_{x}+\bigg{[}(i+\kappa_{0}^{2}n^{2})\gamma^{2})uv\bigg{]}_{y}=0,\\\
\bigg{[}((i+\kappa_{0}^{2}n^{2})\gamma^{2})v\bigg{]}_{t}+\bigg{[}(i+\kappa_{0}^{2}n^{2})\gamma^{2})uv\bigg{]}_{x}+\left((i+\kappa_{0}^{2}n^{2})\gamma^{2})v^{2}+p+\dfrac{1}{2}\kappa_{0}^{2}n^{2}\right)_{y}=0,\\\
\bigg{[}((i+\kappa_{0}^{2}n^{2})\gamma^{2}-p-\dfrac{1}{2}\kappa_{0}^{2}n^{2})v\bigg{]}_{t}+\bigg{[}(i+\kappa_{0}^{2}n^{2})\gamma^{2}u\bigg{]}_{x}+\bigg{[}(i+\kappa_{0}^{2}n^{2})\gamma^{2}v\bigg{]}_{y}=0.\end{cases}$

In this article, we consider that the flow is isentropic and steady which
leads to the following system

$\displaystyle\begin{cases}(n\gamma u)_{x}+(n\gamma v)_{y}=0,\\\
\bigg{[}(\rho+p+\kappa_{0}^{2}n^{2})\gamma^{2})u^{2}+p^{tot}\bigg{]}_{x}+\bigg{[}((\rho+p+\kappa_{0}^{2}n^{2})\gamma^{2})uv\bigg{]}_{y}=0,\\\
\bigg{[}(\rho+p+\kappa_{0}^{2}n^{2})\gamma^{2})uv\bigg{]}_{x}+\bigg{[}(\rho+p+\kappa_{0}^{2}n^{2})\gamma^{2})v^{2}+p^{tot}\bigg{]}_{y}=0,\end{cases}$
(1.7)

where $0<q=\sqrt{u^{2}+v^{2}}<1$ is the flow velocity,
$\gamma=(1-q^{2})^{-1/2}$ is the normalized Lorentz factor, and
$p^{tot}=p+\dfrac{1}{2}\kappa_{0}^{2}n^{2}$ is the total pressure. Moreover,
we suppose that the density $\rho$ and gas pressure $p=p(\rho)$ satisfy the
following [31, 32]

$\displaystyle 0<\rho<\rho_{\max}<\infty,\leavevmode\nobreak\
\leavevmode\nobreak\ 0<p(\rho),\leavevmode\nobreak\ \leavevmode\nobreak\
0<p^{\prime}(\rho)+\dfrac{\kappa_{0}^{2}n^{2}}{p+\rho}(\rho)<1,\leavevmode\nobreak\
\leavevmode\nobreak\ p^{\prime\prime}(\rho)>0.$ (1.8)

Additionally, we assume that for any value of $\rho$, $p^{\prime\prime}(\rho)$
does not become unbounded, which is true in general for all equations of
states of physical importance.

In view of Figure 1, we describe the degenerate Cauchy-Goursat boundary value
problem for relativistic magnetohydrodynamics precisely as follows:

A Cauchy-Goursat mixed-type boundary value problem:

If $\widehat{BA}$ and $\widehat{BC}$ are two given smooth boundaries of the
degenerate mixed-type boundary value problem such that $\widehat{BC}$ is a
sonic curve and $\widehat{BA}$ is a positive characteristic curve of the flow.
Then develop a local classical supersonic solution to the relativistic
magnetohydrodynamics system in the angular domain $ABC$ near the point $B$.

Such a degenerate Cauchy-Goursat problem for two-dimensional steady Euler
equations was studied by Li and Hu [1]. They proved the convergence of a
sequence of iterations generated by a system of integral equations using the
ideas of characteristic decomposition to establish the existence and
uniqueness of the solution. Their work was extended to the relativistic Euler
equations for polytopic gas by Fan et al. [27] recently. We try to extend
these results to a more complicated relativistic magnetohydrodynamics system
with a general convex pressure. Note that due to the presence of a general
pressure law and magnetic effects, the process of proving the convergence is
not straightforward and brings more complexities. The approach to prove the
existence and uniqueness of the solution in this article is inspired by the
work of Li and Hu [1]. We first establish the existence and uniqueness of the
solution in a partial hodograph region by proving the uniform convergence of
an iterated sequence generated by a system of integral equations and then
obtain the solution to the original problem in the physical plane by
transforming the solution obtained in partial hodograph plane via an inverse
transformation.

The rest of the article can be arranged as follows. We analyze the basic
aspects of the steady isentropic rotational relativistic magnetohydrodynamic
equations (1.7) and derive the characteristic decompositions of angle
variables in section 2. In section 3, we precisely define the main problem and
specify the boundary data on the characteristic curve and sonic curve,
respectively. The purpose of section 4 is to establish the existence and
uniqueness of solutions in the partial hodograph plane. We use an inverse
transformation to convert the solution obtained in the partial hodograph plane
back into the physical plane and verify that they satisfy our main problem in
section 5.

##  2 Basic aspects of steady isentropic irrotational relativistic
magnetohydrodynamics

Assuming that the relativistic flow is irrotational, i.e., $u_{y}=v_{x}$ and
using first equation of (1.7) and
$\gamma_{x}=\gamma^{3}(uu_{x}+vv_{x}),\leavevmode\nobreak\
\leavevmode\nobreak\ \gamma_{y}=\gamma^{3}(uu_{y}+vv_{y}),$ we can write the
second equation of system (1.7) as follows

$\dfrac{n}{\gamma}\bigg{\\{}\dfrac{i+\frac{H^{2}}{\gamma^{2}}}{n}\gamma_{x}+\dfrac{\gamma}{n}p_{x}^{tot}+\dfrac{\gamma
u}{n}\bigg{[}n\gamma
u\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{x}+n\gamma
v\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{y}\bigg{]}\bigg{\\}}=0.$
(2.1)

Further, by the first law of thermodynamics, we have

$\displaystyle d\left(\dfrac{i}{n}\right)=\dfrac{1}{n}dp+TdS,$ (2.2)

where $S$ is the entropy of the flow and $T$ denotes absolute temperature.
Since the entropy is assumed to be constant, in view of (2.2), we have

$\displaystyle d\left(\dfrac{i}{n}\right)=\dfrac{1}{n}dp$ (2.3)

or equivalently

$\displaystyle\dfrac{dn}{n}=\dfrac{d\rho}{p+\rho},$

i.e.,

$\displaystyle
n=n(\rho)=n_{0}e^{\displaystyle\int_{\rho_{0}}^{\rho}\dfrac{ds}{s+p(s)}},$
(2.4)

where $n_{0}$ is the proper number density of baryon particles when the mass-
energy density of the fluid is $\rho_{0}>0$. It is straightforward to observe
that

$\displaystyle\begin{cases}n^{\prime}(\rho)=\dfrac{n(\rho)}{p(\rho)+\rho}>0,\\\
n^{\prime\prime}(\rho)=-\dfrac{p^{\prime}(\rho)n(\rho)}{(p(\rho)+\rho)^{2}}<0.\end{cases}$
(2.5)

Also, noting that $H=\kappa_{0}n\gamma$, one has

$\displaystyle d\left(\dfrac{H^{2}}{n\gamma^{2}}\right)=\dfrac{dp_{m}}{n},$
(2.6)

where $p_{m}$ is the magnetic pressure defined by
$p_{m}=\dfrac{H^{2}}{2\gamma^{2}}$. Clearly, one can use (2.3) and (2.6) to
obtain

$\displaystyle
d\left(\dfrac{i+\frac{H^{2}}{\gamma^{2}}}{n}\right)=\dfrac{dp^{tot}}{n}.$
(2.7)

Thus, exploiting (2.1) we have

$\displaystyle\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{x}+\dfrac{\gamma
u}{n}\bigg{[}n\gamma
u\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{x}+n\gamma
v\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{y}\bigg{]}=0.$
(2.8)

In a similar fashion, using the last equation of system (1.7), one can get

$\displaystyle\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{y}+\dfrac{\gamma
v}{n}\bigg{[}n\gamma
u\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{x}+n\gamma
v\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{y}\bigg{]}=0.$
(2.9)

Now one can easily observe that the matrix
$\begin{bmatrix}1+\gamma^{2}u^{2}&\gamma^{2}uv\\\
\gamma^{2}uv&1+\gamma^{2}v^{2}\end{bmatrix}$ is nonsingular which implies that
$\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{x}=\left(\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}\right)_{y}=0$
and yields the Bernoulli’s law for two-dimensional steady relativistic
magnetohydrodynamics of the form

$\displaystyle\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}=\mathrm{constant.}$
(2.10)

Without loss of generality, we rewrite (2.10) as:

$\displaystyle\dfrac{\gamma\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}=m\hat{\gamma},$
(2.11)

where $\hat{\gamma}^{-1}=\sqrt{1-\hat{q}^{2}}$ is a constant.

######  Lemma 2.1.

If for all $n>0$, $\dfrac{\partial p}{\partial n}>0$ then there exists a
constant $\hat{q}\leavevmode\nobreak\ (0<\hat{q}<1)$ usually referred as the
limit speed of the flow such that $q<\hat{q}$, where $q$ is the flow speed.
Further, $q$ approaches $\hat{q}$ when $n$ approaches $0$.

###### Proof.

We use (2.7) to obtain

$\dfrac{d}{dn}\left(\dfrac{i+\frac{H^{2}}{\gamma^{2}}}{n}\right)=\dfrac{d\left(\dfrac{i+\frac{H^{2}}{\gamma^{2}}}{n}\right)}{dp^{tot}}.\dfrac{dp^{tot}}{dn}=\dfrac{1}{n}.\dfrac{dp^{tot}}{dn}>0$

for $n>0$.

In view of (2.11) and the fact that
$\dfrac{i+\dfrac{H^{2}}{\gamma^{2}}}{n}=\dfrac{mn+e+p+\kappa_{0}^{2}n^{2}}{n}\geq
m$ for $n\geq 0$, it is straightforward to observe that $q<\hat{q}<1$ and
$q\longrightarrow\hat{q}$ as $n\longrightarrow 0$. ∎

Again by employing (2.11), system (1.7) can be converted into

$\displaystyle\begin{cases}\gamma\Bigg{[}\left(\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma
u\right)_{x}+\left(\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma
v\right)_{y}\Bigg{]}+\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma[u\gamma_{x}+v\gamma_{y}]=0,\\\
\gamma u\Bigg{[}\left(\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma
u\right)_{x}+\left(\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma
v\right)_{y}\Bigg{]}+\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma[u(\gamma
u)_{x}+v(\gamma u)_{y}]+p^{tot}_{x}=0,\\\ \gamma
v\Bigg{[}\left(\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma
u\right)_{x}+\left(\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma
v\right)_{y}\Bigg{]}+\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma[u(\gamma
v)_{x}+v(\gamma v)_{y}]+p^{tot}_{y}=0.\end{cases}$ (2.12)

Further, by considering scalar product of $(\gamma,-\gamma u,-\gamma v)$ with
(2.12) leads to

$\displaystyle\left(i+\frac{H^{2}}{\gamma^{2}}\right)(\gamma
u)_{x}+\left(i+\frac{H^{2}}{\gamma^{2}}\right)(\gamma v)_{y}=-\gamma
u\rho_{x}-\gamma v\rho_{y}.$ (2.13)

Moreover, from the second and third equation of (1.7), one can obtain

$\displaystyle\begin{cases}\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma^{2}uu_{x}+\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma^{2}vu_{y}+p_{x}^{tot}=0,\\\
\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma^{2}uv_{x}+\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma^{2}vv_{y}+p_{y}^{tot}=0.\end{cases}$
(2.14)

Again by taking the scalar product of $(\gamma u,\gamma v)$ with (2.14), one
can obtain

$\displaystyle\left(i+\frac{H^{2}}{\gamma^{2}}\right)\gamma^{3}[u^{2}u_{x}+uv(u_{y}+v_{x})+v^{2}v_{y}]=-w^{2}(\gamma
u\rho_{x}+\gamma v\rho_{y}),$ (2.15)

where $w=\sqrt{p^{\prime}(\rho)+\dfrac{\kappa_{0}^{2}n^{2}}{p+\rho}}$ is the
magneto-acoustic sound.

Therefore, we combine (2.13) and (2.15) to convert the governing system (1.7)
in the form:

$\displaystyle\begin{cases}(M_{1}^{2}-1)u_{x}+M_{1}M_{2}(v_{x}+u_{y})+(M_{2}^{2}-1)v_{y}=0,\\\
u_{y}-v_{x}=0,\end{cases}$ (2.16)

where $M_{1}=\dfrac{\gamma u}{w\gamma_{w}},\leavevmode\nobreak\
\leavevmode\nobreak\ M_{2}=\dfrac{\gamma v}{w\gamma_{w}},$ and
$\gamma_{w}=\dfrac{1}{\sqrt{1-w^{2}}}$.

For smooth solutions, (2.16) can be rewritten in the matrix form as

$\displaystyle\begin{bmatrix}M_{1}^{2}-1&M_{1}M_{2}\\\
0&-1\end{bmatrix}\begin{bmatrix}u\\\
v\end{bmatrix}_{x}+\begin{bmatrix}M_{1}M_{2}&M_{2}^{2}-1\\\
1&0\end{bmatrix}\begin{bmatrix}u\\\ v\end{bmatrix}_{y}=\begin{bmatrix}0\\\
0\end{bmatrix}.$ (2.17)

The eigenvalues of the system (2.17) are
$\Lambda_{\pm}=\dfrac{M_{1}M_{2}\pm\sqrt{M_{1}^{2}+M_{2}^{2}-1}}{M_{1}^{2}-1}$
and the left eigenvectors are $l_{\pm}=(1,\mp\sqrt{M^{2}-1})$, such that
$M=\sqrt{M_{1}^{2}+M_{2}^{2}}=\dfrac{\gamma q}{\gamma_{w}w}$ is the proper
Mach number. Clearly, (2.17) is a mixed-type system and may change its
behavior across the sonic curve ($M=1$). The system (2.17) is hyperbolic for
$M>1$ while for $M<1$ system (2.17) changes its nature to elliptic. Moreover,
one can define the families of wave characteristics as follows

$\dfrac{dy}{dx}=\Lambda_{\pm}.$ (2.18)

Furthermore, one can easily obtain the characteristic equations of the form

$\displaystyle\begin{cases}{\partial}_{+}u+\Lambda_{-}{\partial}_{+}v=0,\\\
{\partial}_{-}u+\Lambda_{+}{\partial}_{-}v=0,\end{cases}$ (2.19)

where ${\partial}_{\pm}=\partial_{x}+\Lambda_{\pm}\partial_{y}$.

###  2.1 Characteristic decompositions of the flow variables

Differentiating Bernoulli’s law (2.11) with respect to $q$ gives

$\displaystyle\dfrac{\gamma^{3}q\left(i+\frac{H^{2}}{\gamma^{2}}\right)}{n}+\gamma\dfrac{d(\left(i+\frac{H^{2}}{\gamma^{2}}\right)/n)}{dp^{tot}}\dfrac{dp^{tot}}{dw}\dfrac{dw}{dq}=0.$
(2.20)

Then by noting that
$w^{2}=p^{\prime}(\rho)+\dfrac{\kappa_{0}^{2}n^{2}}{p+\rho}$ we have
$\dfrac{dp^{tot}}{dw}=\dfrac{2i^{2}w^{3}}{i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))}$,
which is used in (2.20) along with (2.7) to yield

$\displaystyle\dfrac{dw}{dq}=-\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)q\gamma^{2}\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}}{2i^{2}w^{3}}<0.$
(2.21)

Also, by $M=\dfrac{\gamma q}{w\gamma_{w}}$, it is easy to see that

$\displaystyle\dfrac{dM}{dq}=M\left(\dfrac{1}{q(1-q^{2})}-\dfrac{1}{w(1-w^{2})}\dfrac{dw}{dq}\right).$

Then noting that $\dfrac{dw}{dq}<0$ and $0<w^{2},q^{2}<1$ we have
$\dfrac{dM}{dq}>0$.

Furthermore, we employ the following expressions of velocity [31]

$u=\dfrac{w\gamma_{w}}{\gamma}\dfrac{\cos\theta}{\sin\omega},\leavevmode\nobreak\
\leavevmode\nobreak\
v=\dfrac{w\gamma_{w}}{\gamma}\dfrac{\sin\theta}{\sin\omega},$ (2.22)

and employ the normalized directional derivatives of the form

$\displaystyle\begin{cases}&\bar{\partial}_{+}=\cos\alpha\dfrac{\partial}{\partial
x}+\sin\alpha\dfrac{\partial}{\partial y},\leavevmode\nobreak\
\leavevmode\nobreak\ \bar{\partial}_{-}=\cos\beta\dfrac{\partial}{\partial
x}+\sin\beta\dfrac{\partial}{\partial y},\vspace{0.2 cm}\\\
&\bar{\partial}_{0}=\cos\theta\dfrac{\partial}{\partial
x}+\sin\theta\dfrac{\partial}{\partial y},\leavevmode\nobreak\
\leavevmode\nobreak\
\bar{\partial}_{\perp}=-\sin\theta\dfrac{\partial}{\partial
x}+\cos\theta\dfrac{\partial}{\partial y}.\end{cases}$ (2.23)

From (2.23), we have

$\displaystyle\dfrac{\partial}{\partial
x}=-\dfrac{\sin\beta\bar{\partial}_{+}-\sin\alpha\bar{\partial}_{-}}{\sin(2\omega)},\leavevmode\nobreak\
\leavevmode\nobreak\ \dfrac{\partial}{\partial
y}=\dfrac{\cos\beta\bar{\partial}_{+}-\cos\alpha\bar{\partial}_{-}}{\sin(2\omega)},\leavevmode\nobreak\
\leavevmode\nobreak\
\bar{\partial}_{0}=\dfrac{\bar{\partial}_{+}+\bar{\partial}_{-}}{2\cos\omega},\leavevmode\nobreak\
\leavevmode\nobreak\
\bar{\partial}_{\perp}=\dfrac{\bar{\partial}_{+}-\bar{\partial}_{-}}{2\sin\omega}.$
(2.24)

Therefore, a direct computation leads to the following decompositions of $u$
and $v$

$\displaystyle\begin{cases}\bar{\partial}_{\pm}u=\dfrac{\gamma_{w}}{\gamma\sin^{2}\omega}\bigg{[}f(w)\sin\omega\cos\theta\bar{\partial}_{\pm}w-w\sin\omega\sin\theta\bar{\partial}_{\pm}\theta-w\cos\theta\cos\omega\bar{\partial}_{\pm}\omega\bigg{]},\vspace{0.2
cm}\\\
\bar{\partial}_{\pm}v=\dfrac{\gamma_{w}}{\gamma\sin^{2}\omega}\bigg{[}f(w)\sin\omega\sin\theta\bar{\partial}_{\pm}w+w\sin\omega\cos\theta\bar{\partial}_{\pm}\theta-w\sin\theta\cos\omega\bar{\partial}_{\pm}\omega\bigg{]},\end{cases}$
(2.25)

where
$0<f(w)=\left(\gamma_{w}^{2}+\dfrac{2i^{2}w^{4}}{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}}\right)<\infty$.

Utilizing (2.25) and performing direct calculations leads us to

$\displaystyle\begin{cases}\bar{\partial}_{+}\omega&=\dfrac{wF_{1}(w,\omega)}{2\left(i+\frac{H^{2}}{\gamma^{2}}\right)q\gamma\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\cos\omega\gamma_{w}}\dfrac{\bar{\partial}_{+}w}{w},\\\
\bar{\partial}_{-}\omega&=\dfrac{wF_{1}(w,\omega)}{2\left(i+\frac{H^{2}}{\gamma^{2}}\right)q\gamma\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\cos\omega\gamma_{w}}\dfrac{\bar{\partial}_{-}w}{w},\\\
\bar{\partial}_{+}\theta&=\dfrac{(F_{2}(w,\omega)-wF_{1}(w,\omega)\cos^{2}\omega)}{2\left(i+\frac{H^{2}}{\gamma^{2}}\right)q\gamma\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\cos\omega\sin^{2}\omega\gamma_{w}}\dfrac{\bar{\partial}_{+}w}{w},\\\
\bar{\partial}_{-}\theta&=\dfrac{(wF_{1}(w,\omega)\cos^{2}\omega-
F_{2}(w,\omega))}{2\left(i+\frac{H^{2}}{\gamma^{2}}\right)q\gamma\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\cos\omega\sin^{2}\omega\gamma_{w}}\dfrac{\bar{\partial}_{-}w}{w},\end{cases}$
(2.26)

where

$\displaystyle\begin{cases}F_{1}(a,\omega)=2\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)+4i^{2}w^{2}\sin^{2}\omega>0,\\\
F_{2}(a,\omega)=\left(i+\frac{H^{2}}{\gamma^{2}}\right)q\gamma\gamma_{w}f(w)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\cos\omega\sin
2\omega.\end{cases}$ (2.27)

Noting the expressions of velocity components from (2.22) and employing (2.25)
and (2.21), it is easy to see that $\omega=\omega(\rho)$. Therefore, one can
use
$\dfrac{d\omega}{dw}=\dfrac{d\omega}{dM}\dfrac{dM}{dq}\dfrac{dq}{dw}\dfrac{dw}{d\rho}$
and $\dfrac{dw}{dq}<0,\leavevmode\nobreak\ \dfrac{dM}{dq}>0$ along with
$\sin\omega=\dfrac{1}{M}$ to obtain $\dfrac{d\omega}{dw}>0$. Further, we
denote $\sin\omega=\dfrac{1}{M}:=\varpi,\omega\in[\varpi_{0},\pi/2],$
($\varpi_{0}>0$ is a constant). Thus by utilizing the inverse function
theorem, one can observe that $\rho=\rho(\omega)=\rho(\sin^{-1}\varpi)$ or in
other words $w=w(\omega)=w(\sin^{-1}\varpi)$. Thus we denote
$F_{1}(w,\omega)=:F_{1}(\varpi)$ and $F_{2}(w,\omega)=:F_{2}(\varpi)$ to get

$\displaystyle\begin{cases}\bar{\partial}_{+}\theta+\dfrac{4i^{2}(\varpi)w^{2}(\varpi)\cos\omega}{F_{1}(\varpi)}\bar{\partial}_{+}\varpi=0,\\\
\bar{\partial}_{-}\theta-\dfrac{4i^{2}(\varpi)w^{2}(\varpi)\cos\omega}{F_{1}(\varpi)}\bar{\partial}_{-}\varpi=0.\end{cases}$
(2.28)

In view of the continuity of
$\dfrac{2i^{2}(s)w^{2}(s)}{sF_{1}(s)},s\in[\sin\varpi_{0},1]$, we set

$\displaystyle I(\varpi)=\int_{\sin
k_{0}}^{\varpi}\dfrac{2i^{2}(s)w^{2}(s)}{sF_{1}(s)}ds,\leavevmode\nobreak\
\leavevmode\nobreak\ \varpi\in[\sin\varpi_{0},1].$

Thus, we denote $I:=I(\varpi)$ to reduce (2.28) as

$\displaystyle\begin{cases}\bar{\partial}_{+}\theta+\sin
2\omega\bar{\partial}_{+}I=0,\\\ \bar{\partial}_{-}\theta-\sin
2\omega\bar{\partial}_{-}I=0\end{cases}$ (2.29)

with

$\displaystyle\bar{\partial}_{i}\varpi=\dfrac{\varpi
F_{1}(\varpi)}{2i^{2}(\varpi)w^{2}(\varpi)}\bar{\partial}_{i}I,\leavevmode\nobreak\
\leavevmode\nobreak\ i=0,\pm.$ (2.30)

Moreover, we make use of the commutator relation given by [1]

$\displaystyle\bar{\partial}_{-}\bar{\partial}_{+}-\bar{\partial}_{+}\bar{\partial}_{-}$
$\displaystyle=\dfrac{1}{\sin 2\omega}\bigg{[}(\cos
2\omega\bar{\partial}_{-}\alpha-\bar{\partial}_{+}\beta)\bar{\partial}_{+}-(\bar{\partial}_{-}\alpha-\cos
2\omega\bar{\partial}_{+}\beta)\bar{\partial}_{-}\bigg{]},$

which provides the second-order decomposition of $I$ and $\theta$ of the form

$\displaystyle\begin{cases}\bar{\partial}_{-}\bar{\partial}_{+}I-\bar{\partial}_{+}\bar{\partial}_{-}I&=\dfrac{1}{\sin
2\omega}\bigg{[}(\cos
2\omega\bar{\partial}_{-}\alpha-\bar{\partial}_{+}\beta)\bar{\partial}_{+}I-(\bar{\partial}_{-}\alpha-\cos
2\omega\bar{\partial}_{+}\beta)\bar{\partial}_{-}I\bigg{]},\vspace{0.2 cm}\\\
\bar{\partial}_{-}\bar{\partial}_{+}\theta-\bar{\partial}_{+}\bar{\partial}_{-}\theta&=\dfrac{1}{\sin
2\omega}\bigg{[}(\cos
2\omega\bar{\partial}_{-}\alpha-\bar{\partial}_{+}\beta)\bar{\partial}_{+}\theta-(\bar{\partial}_{-}\alpha-\cos
2\omega\bar{\partial}_{+}\beta)\bar{\partial}_{-}\theta\bigg{]}.\end{cases}$

For subsequent discussions in the article, we define $f(w):=f(w(\varpi))$,
$W:=\bar{\partial}_{+}I$, $Z:=\bar{\partial}_{-}I$, and employ
$F_{1}(\varpi)=2\left(i+\frac{H^{2}}{\gamma^{2}}\right)\left(i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\right)\gamma_{w(\varpi)}^{2}f(w(\varpi))+4i^{2}(\varpi)w^{2}(\varpi)\varpi^{2}$
to derive the following characteristic decompositions of $W$ and $Z$:

$\displaystyle\begin{cases}&\bar{\partial}_{-}W=W\Bigg{[}W+Z\cos
2\omega+\dfrac{F_{1}(\varpi)}{4i^{2}(\varpi)w^{2}(\varpi)\cos^{2}\omega}(W-Z\cos
2\omega)\Bigg{]},\vspace{0.2 cm}\\\ &\bar{\partial}_{+}Z=Z\Bigg{[}Z+W\cos
2\omega+\dfrac{F_{1}(\varpi)}{4i^{2}(\varpi)w^{2}(\varpi)\cos^{2}\omega}(Z-W\cos
2\omega)\Bigg{]},\end{cases}$ (2.31)

or

$\displaystyle\begin{cases}&\bar{\partial}_{-}W=\dfrac{W}{\cos^{2}\omega}\Bigg{[}W+Z\cos^{2}2\omega+\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\varpi)w^{2}(\varpi)}(W-Z\cos
2\omega)\Bigg{]},\vspace{0.2 cm}\\\
&\bar{\partial}_{+}Z=\dfrac{Z}{\cos^{2}\omega}\Bigg{[}Z+W\cos^{2}2\omega+\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\varpi)w^{2}(\varpi)}(Z-W\cos
2\omega)\Bigg{]}.\end{cases}$ (2.32)

##  3 The boundary data and main result

We now provide the boundary data for our main problem on the boundaries
$\widehat{BC}$ and $\widehat{AB}$. Assume that
$\widehat{AB}:x=\psi(y),\leavevmode\nobreak\ y\in[y_{A},y_{B}]$ and
$\widehat{BC}:y=\varphi(x),x\in[x_{B},x_{C}]$ are two smooth curves and
$(\theta,\varpi)|_{\widehat{BC}}=(\hat{\theta},\hat{\varpi})(x)$ and
$(\theta,\varpi)|_{\widehat{AB}}=(\tilde{\theta},\tilde{\varpi})(y)$ are given
boundary data on the curves $\widehat{BC}$ and $\widehat{AB}$ which satisfy

$\displaystyle\varphi(x)\in C^{3}([x_{B},x_{C})),\leavevmode\nobreak\
\leavevmode\nobreak\ \hat{\theta}(x)\in
C^{3}([x_{B},x_{C})),\leavevmode\nobreak\ \leavevmode\nobreak\
\hat{\varpi}(x)=1\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ \leavevmode\nobreak\ x\in[x_{B},x_{C}]$ (3.1)

and

$\displaystyle\begin{cases}x_{B}=\psi(y_{B}),\leavevmode\nobreak\
\leavevmode\nobreak\ \psi(y)\in C^{4}((y_{A},y_{B}]),\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{\theta}(y)=\mathrm{\cot^{-1}}\psi^{\prime}(y)-\sin^{-1}\tilde{\varpi}(y),\leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{\varpi}(y_{B})=1,\vspace{0.3 cm}\\\
\dfrac{1}{\sqrt{1-\tilde{\varpi}^{2}(y_{A})}}-\dfrac{4i^{2}(\tilde{\varpi})w^{2}(\tilde{\varpi})\sqrt{1-\tilde{\varpi}^{2}(y_{A})}}{F_{1}(\tilde{\varpi}(y_{A}))}>0,\vspace{0.3
cm}\\\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sin^{-1}\tilde{\varpi}(y)-Q(\tilde{\varpi}(y))=\mathrm{\cot^{-1}}\psi^{\prime}(y)-\big{[}\hat{\theta}(x_{B})+Q(\hat{\varpi}(x_{B}))\big{]},\end{cases}$
(3.2)

where

$Q(\varpi)=\displaystyle\int\dfrac{4i^{2}(\varpi)w^{2}(\varpi)\sqrt{1-\varpi^{2}}}{F_{1}(\varpi)}d\varpi.$

The last equality of (3.2) indeed holds. From the first equation of (2.28) it
is clear that
$[\tilde{\theta}+Q(\tilde{\varpi})]|_{\widehat{AB}}=\hat{\theta}(x_{B})+Q(\hat{\varpi}(x_{B}))$.
Consequently $(\tilde{\theta},\tilde{\varpi})(y)$ satisfy the last equality of
(3.2).

###  3.1 Boundary data of $(W,Z)$

Now we proceed to provide the boundary data of $W$ and $Z$ on the curves
$\widehat{AB}$ and $\widehat{BC}$ using the boundary information of
$(\theta,\varpi)$.

Using (3.2), we obtain

$\displaystyle\tilde{\theta}(B)=\mathrm{\cot^{-1}}\psi^{\prime}(B)-\sin^{-1}\tilde{\varpi}(B)=\hat{\theta}(B)+Q(\hat{\varpi}(B))-Q(\tilde{\varpi}(B))=\hat{\theta}(B),$

which together with $\hat{\varpi}(B)=\tilde{\varpi}(B)=1$ gives
$(\hat{\theta},\hat{\varpi})(x_{B})=(\tilde{\theta},\tilde{\varpi})(y_{B})$.
In order to check that the compatibility conditions hold true at point $B$, we
need to verify that the equations in (2.28) are satisfied at point $B$.
Moreover, one can easily check that the second equation of (2.28) is satisfied
at point $B$. For this, we just need to check that
$\bar{\partial}_{-}\theta|_{B}=0$ as $\cos\omega|_{B}=0$.

Now we use (3.2) to obtain

$\displaystyle\tilde{\theta}^{\prime}(y)=-\dfrac{4i^{2}(\tilde{\varpi})w^{2}(\tilde{\varpi})\tilde{\varpi}^{\prime}(y)\sqrt{1-\tilde{\varpi}^{2}(y)}}{F_{1}(\tilde{\varpi}(y))},$
(3.3)

which implies that $\tilde{\theta}^{\prime}(B)=0$ or in other words
$\bar{\partial}_{+}\theta=0$ at $B$. Further, since $\cos\omega|_{B}=0$,
therefore noting the expression of $\bar{\partial}_{0}\theta$ we must have
$\bar{\partial}_{-}\theta(B)=-\bar{\partial}_{+}\theta(B)=0$, which proves
that the compatibility condition is satisfied at $B$.

Now we discuss the boundary values of $W$ and $Z$ on the curves $\widehat{AB}$
and $\widehat{BC}$. Noting that $\widehat{AB}:x=\psi(y)$ is a positive
characteristic, we must have

$\displaystyle
W|_{\widehat{AB}}=\dfrac{2i^{2}(\tilde{\varpi}(y))w^{2}(\tilde{\varpi}(y))\tilde{\varpi}^{\prime}(y)}{\tilde{\varpi}(y)F_{1}(\tilde{\varpi}(y))\sqrt{1+(\psi^{\prime}(y))^{2}}}=:\tilde{b}_{0}(y)\in
C^{2}((y_{A},y_{B}]).$ (3.4)

Again since $\cos\omega|_{\widehat{BC}}=0$, we must have
$W+Z|_{\widehat{BC}}=2\cos\omega\bar{\partial}_{0}I=0$. Also, by exploiting
(2.29) we obtain

$\displaystyle
W-Z=-\dfrac{\bar{\partial}_{+}\theta+\bar{\partial}_{-}\theta}{\sin(2\omega)}=-\dfrac{\bar{\partial}_{0}\theta}{\sin\omega},$

which yields

$\displaystyle
W|_{\widehat{BC}}=-Z|_{\widehat{BC}}=-\dfrac{\bar{\partial}_{0}\theta}{2}\bigg{|}_{\widehat{BC}}.$
(3.5)

Therefore, in order to find the boundary values of $W$ and $Z$, we must derive
the boundary information of $\bar{\partial}_{0}\theta$ on the curve
$\widehat{BC}$. Again from (2.28), one can observe that
$\bar{\partial}_{\perp}\theta=-(W+Z)\cos\omega$, which implies
$\bar{\partial}_{\perp}\theta|_{\widehat{BC}}=0$. Therefore, in view of
$\theta(x,y)|_{\widehat{BC}}=\theta(x,\varphi(x))=\hat{\theta}(x)$, we must
have

$\displaystyle\hat{\theta}^{\prime}(x)=\theta_{x}(x,\varphi(x))+\varphi^{\prime}(x)\theta_{y}(x,\varphi(x)),$

which together with $\bar{\partial}_{\perp}\theta=0$ implies

$\displaystyle\theta_{x}(x,\varphi(x))=\dfrac{\hat{\theta}^{\prime}\cos\hat{\theta}^{\prime}}{\cos\hat{\theta}^{\prime}+\varphi^{\prime}\sin\hat{\theta}^{\prime}},\leavevmode\nobreak\
\leavevmode\nobreak\
\theta_{y}(x,\varphi(x))=\dfrac{\hat{\theta}^{\prime}\sin\hat{\theta}^{\prime}}{\cos\hat{\theta}^{\prime}+\varphi^{\prime}\sin\hat{\theta}^{\prime}},$

or in other words

$\displaystyle\bar{\partial}_{0}\theta|_{\widehat{BC}}=\dfrac{\hat{\theta}^{\prime}}{\cos\hat{\theta}^{\prime}+\varphi^{\prime}\sin\hat{\theta}^{\prime}}.$

which implies that

$\displaystyle
W|_{\widehat{BC}}=-Z|_{\widehat{BC}}=-\dfrac{\hat{\theta}^{\prime}}{2(\varphi^{\prime}\sin\hat{\theta}^{\prime}+\cos\hat{\theta}^{\prime})}=:-\hat{a}_{0}(x)\in
C^{2}([x_{B},x_{C})).$ (3.6)

Furthermore, the above derivation process clearly shows that
$-\hat{a}_{0}(x_{B})=\tilde{b}_{0}(y_{B})$. Considering the continuity of the
functions $\hat{a}_{0}(x_{B})$ and $\tilde{b}_{0}(y_{B})$ there must exist two
sufficiently small positive constants $\epsilon_{0},\delta_{0}$ such that
$\tilde{b}_{0}(y)\geq\epsilon_{0}$ for arbitrary
$y\in[y_{B}-\delta_{0},y_{B}]$ and $-\hat{a}_{0}(x_{B})\geq\epsilon_{0}$ for
arbitrary $x\in[x_{B},x_{B}+\delta_{0}]$.

One can take into account the following assumptions; given the solutions are
being considered in the neighbourhood of point $B$.

$\displaystyle\hat{\theta}^{\prime}(x)\leq-\epsilon_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \hat{a}_{0}(x)\leq-\epsilon_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\
x\in[x_{B},x_{C}),\leavevmode\nobreak\ \leavevmode\nobreak\
\psi^{\prime\prime}(y)\leq\epsilon_{0},\leavevmode\nobreak\
\tilde{b}_{0}(y)\geq\epsilon_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ y\in(y_{A},y_{B}].$ (3.7)

##  4 Solution in a partial hodograph plane

In this section, we develop the existence and uniqueness of a solution to the
nonlinear system (2.32) with the boundary data (3.4) and (3.6) under the
assumptions (3.7) in the neighbourhood of the point $B$ by utilizing a partial
hodograph mapping. We convert the system (2.32) into a system of linear
singular equations by a change of variables and then prove the convergence of
a sequence of iterations generated by a set of integral equations.

###  4.1 Formulation of the main problem in $(t,r)$ plane

We now reformulate the main problem by using a change of variables. First we
define a partial hodograph mapping $(x,y)\longrightarrow(t,r)$ of the form

$\displaystyle t=\cos\omega(x,y),\leavevmode\nobreak\ \leavevmode\nobreak\
r=\theta(x,y).$ (4.1)

It is a simple observation to see that $\varpi=\sqrt{1-t^{2}}$. Thus, we
denote $F_{1}(\varpi):=\hat{F}_{1}(t)$ where

$\displaystyle\hat{F}_{1}(t)=4i^{2}(t)w^{2}(t)(1-t^{2})+2\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}(t)p^{\prime\prime}(\rho(t))+\kappa_{0}^{2}n^{2}(t)(1-p^{\prime}(\rho(t)))\bigg{]}\gamma_{w}^{2}(t)f(w(t))>0.$
(4.2)

Further, by exploiting (2.29) and (2.30), we obtain

$\displaystyle J$
$\displaystyle:=\dfrac{\partial(t,r)}{\partial(x,y)}=\begin{vmatrix}\dfrac{\partial
t}{\partial x}&\dfrac{\partial t}{\partial y}\vspace{0.2 cm}\\\
\dfrac{\partial r}{\partial x}&\dfrac{\partial r}{\partial y}\end{vmatrix}$
$\displaystyle=\dfrac{(1-t^{2})\hat{F}_{1}(t)}{i^{2}(t)w^{2}(t)t}WZ\neq
0\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
0\leq t<1.$ (4.3)

Also, an easy computation now yields

$\displaystyle\bar{\partial}_{+}=-\dfrac{2F(t)}{t}W\partial_{t}-2\sqrt{1-t^{2}}Wt\partial_{r},\leavevmode\nobreak\
\leavevmode\nobreak\
\bar{\partial}_{-}=-\dfrac{2F(t)}{t}Z\partial_{t}+2\sqrt{1-t^{2}}Zt\partial_{r},$
(4.4)

where $F(t)=\dfrac{(1-t^{2})\hat{F}_{1}(t)}{4i^{2}(t)\hat{w}^{2}(t)}>1.$

As a result, system (2.32) can be represented as a semi-linear system in the
$(t,r)$ plane, which takes the form

$\displaystyle\begin{cases}&W_{t}-\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}W_{r}=-\Bigg{[}1+\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}\Bigg{]}\dfrac{W}{2ZF(t)}\left(\dfrac{W+Z}{t}\right)\\\
&\hskip
56.9055pt+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}+2(1-t^{2})\Bigg{]}\dfrac{Wt}{F(t)},\\\
&Z_{t}+\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}Z_{r}=-\Bigg{[}1+\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}\Bigg{]}\dfrac{Z}{2WF(t)}\left(\dfrac{W+Z}{t}\right)\\\
&\hskip
56.9055pt+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}+2(1-t^{2})\Bigg{]}\dfrac{Zt}{F(t)}.\\\
\end{cases}$ (4.5)

We now derive the boundary information of $W$ and $Z$ in the $(t,r)$ plane
using the boundary data (3.4) and (3.6). According to the assumption (3.7), it
is easy to see that $r=\hat{\theta}(x)$ is strictly decreasing, which means
that an inverse function can be defined as $x=\hat{x}(r),r\in[r_{1},r_{2}]$,
where $r_{1}=\hat{\theta}(x_{C})$ and $r_{2}=\hat{\theta}(x_{B})$. Therefore
the sonic curve $\widehat{BC}:y=\varphi(x)$ can be transformed into a
degenerate curve $\widehat{B^{\prime}C^{\prime}}:t=0,\leavevmode\nobreak\
r\in(r_{1},r_{2}]$ in the $(t,r)$ plane such that

$\displaystyle(W,Z)|_{\widehat{B^{\prime}C^{\prime}}}=(-\hat{a}_{0},\hat{a}_{0})(r),\leavevmode\nobreak\
r\in(r_{1},r_{2}],$ (4.6)

where $\hat{a}_{0}(r)=\hat{a}_{0}(\hat{x}(r))$.

Again by differentiating the last equality of (3.2) and noticing the
assumptions on $\widehat{AB}$, it is obvious that
$\tilde{\varpi}^{\prime}(y)>0$ for all $y\in(y_{A},y_{B}]$, which implies that
$\tilde{\theta}^{\prime}(y)<0$ on $\widehat{BA}/\\{B\\}$ by (3.3), which
implies the curve $\widehat{BA}:x=\psi(y)$ is transformed into a curve
$\widehat{B^{\prime}A^{\prime}}:y=\tilde{y}(r),\leavevmode\nobreak\
r\in[r_{2},r_{3}),\leavevmode\nobreak\ r_{3}=\tilde{\theta}(y_{A})$ such that
$W|_{\widehat{B^{\prime}A^{\prime}}}=\tilde{b}_{0}(\tilde{y}(r))$.

We now proceed to prove that the curve $\widehat{A^{\prime}B^{\prime}}$ is
actually a $C_{+}$ characteristic curve in the $(t,r)$ plane:

######  Lemma 4.1.

The map $(x,y)\longrightarrow(t,r)$ transforms the positive characteristic
curve $\widehat{AB}$ into a positive characteristic curve
$\widehat{A^{\prime}B^{\prime}}$ of the system (4.5) defined by

$\displaystyle
r=r_{2}+\displaystyle\int_{0}^{t}\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}dt=:\bar{r}(t).$
(4.7)

###### Proof.

Noting that $x(t,r)=\psi(y(t,r))$ on $\widehat{A^{\prime}B^{\prime}}$, we
differentiate $x(t,r)$ wrt $t$ and use
$\psi^{\prime}=\cot\alpha=\cot(\theta+\omega)$ to yield

$\displaystyle\dfrac{dr}{dt}=\dfrac{y_{t}\cot(\theta+\omega)-x_{t}}{x_{r}-y_{r}\cot(\theta+\omega)},$

which along with the fact that
$x_{t}=\dfrac{\theta_{y}}{J},\leavevmode\nobreak\
x_{r}=\dfrac{-t_{y}}{J},\leavevmode\nobreak\
y_{t}=-\dfrac{\theta_{x}}{J},\leavevmode\nobreak\ y_{r}=\dfrac{t_{x}}{J}$
implies

$\displaystyle\dfrac{dr}{dt}$
$\displaystyle=-\dfrac{\cot(\theta+\omega)\theta_{x}+\theta_{y}}{\sin\omega(\omega_{y}+\cot(\theta+\omega)\omega_{x}}$
$\displaystyle=-\dfrac{\bar{\partial}_{+}\theta}{\bar{\partial}_{+}\omega\sin\omega}=\dfrac{4i^{2}(t)w^{2}(t)t^{2}}{\hat{F}_{1}(t)\sqrt{1-t^{2}}}=\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)},$

which follows that $\widehat{A^{\prime}B^{\prime}}$ is a $C_{+}$
characteristic curve of (4.5). Integrating the above from $0$ to $t$ and
noting that $r(0)=r_{2}$ leads to the expression of (4.7) and hence the Lemma
is proved. ∎

Now noting that

$\displaystyle F(0)$
$\displaystyle=1+\dfrac{\left(i(0)+\frac{H^{2}(0)}{\gamma^{2}(0)}\right)\bigg{[}i^{2}(0)p^{\prime\prime}(\rho(0))+\kappa_{0}^{2}n^{2}(0)(1-p^{\prime}(\rho(0))\bigg{]}\gamma_{w(0)}^{2}f(w(0))}{2i^{2}(0)w^{2}(0)}$
$\displaystyle=\bigg{[}1+\dfrac{\left(i^{2}(t)+\frac{H^{2}}{\gamma^{2}(t)}\right)\bigg{[}i^{2}(t)p^{\prime\prime}(\rho(t))+\kappa_{0}^{2}n^{2}(t)(1-p^{\prime}(\rho(t))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}\bigg{]}\Bigg{|}_{t=0}$

and by $W|_{t=0}=-Z|_{t=0},$ we must have

$\displaystyle W_{t}|_{t=0}=Z_{t}|_{t=0}=\dfrac{W+Z}{2t}\bigg{|}_{t=0}.$ (4.8)

In view of the definition of $W$ and $Z$, we see that
$\dfrac{W+Z}{2t}=\bar{\partial}_{0}I$. Therefore, we must find the boundary
value of $\bar{\partial}_{0}I$ on $\widehat{BC}$. It follows from (2.30) that

$\displaystyle\bar{\partial}_{0}I|_{\widehat{BC}}=\dfrac{2i^{2}(\varpi)w^{2}(\varpi)}{F_{1}(\varpi)}\bar{\partial}_{0}\varpi|_{\widehat{BC}},$
(4.9)

Again by adding the two equations of (2.29), we have

$\displaystyle\bar{\partial}_{0}\theta+\dfrac{4i^{2}(\varpi)w^{2}(\varpi)\varpi}{F_{1}(\varpi)}\bar{\partial}_{\perp}\varpi=0.$

Also,
$\bar{\partial}_{\perp}\varpi|_{\widehat{BC}}=-\varpi_{x}(x,\varphi(x))\sin\hat{\theta}+\varpi_{y}(x,\varphi(x))\cos\hat{\theta}=-F(0)\bar{\partial}_{0}\theta|_{\widehat{BC}}=-2F(0)\hat{a}_{0}(x).$
Since $\varpi(x,y)|_{\widehat{BC}}=\hat{\varpi}(x)=1$, we have
$\varpi^{\prime}(x,y)=\varpi_{x}(x,\varphi(x))+\varphi^{\prime}(x,\varphi(x))\varpi_{y}(x,\varphi(x))=0,$
which implies that

$\displaystyle\varpi_{x}(x,\varphi(x))|_{\widehat{BC}}=\dfrac{2\hat{a}_{0}(x)F(0)\varphi^{\prime}(x)}{\cos\hat{\theta}+\varphi^{\prime}\sin\hat{\theta}},\leavevmode\nobreak\
\leavevmode\nobreak\
\varpi_{y}(x,\varphi(x))|_{\widehat{BC}}=-\dfrac{2\hat{a}_{0}(x)F(0)}{\cos\hat{\theta}+\varphi^{\prime}\sin\hat{\theta}}$

or

$\displaystyle\bar{\partial}_{0}\varpi|_{\widehat{BC}}=\dfrac{\hat{a}_{0}(x)F(0)(\varphi^{\prime}\cos\hat{\theta}-\sin\hat{\theta})}{\cos\hat{\theta}+\varphi^{\prime}\sin\hat{\theta}}.$

Therefore, (4.9) yields

$\displaystyle\bar{\partial}_{0}I|_{\widehat{BC}}=\dfrac{\hat{a}_{0}(x)(\varphi^{\prime}\cos\hat{\theta}-\sin\hat{\theta})}{\cos\hat{\theta}+\varphi^{\prime}\sin\hat{\theta}}=:\hat{a}_{1}(x).$
(4.10)

Let us define $\hat{a}_{1}(r)=\hat{a}_{1}(\hat{x}(r))$ and exploit (4.8) to
obtain the boundary information of $W_{t}$ and $Z_{t}$ as
$W_{t}|_{\widehat{B^{\prime}C^{\prime}}}=Z_{t}|_{\widehat{B^{\prime}C^{\prime}}}=\hat{a}_{1}(r)$
such that the function $\hat{a}_{1}(r)\in C^{2}((r_{1},r_{2}])$. In summary,
the boundary conditions for $(W,Z)$ are as follows:

$\displaystyle\begin{cases}W(0,r)=-\hat{a}_{0}(r),\leavevmode\nobreak\
\leavevmode\nobreak\ Z(0,r)=\hat{a}_{0}(r),\leavevmode\nobreak\
\leavevmode\nobreak\ W_{t}(0,r)=Z_{t}(0,r)=\hat{a}_{1}(r)\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{on}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{\widehat{B^{\prime}C^{\prime}}},\\\
W(t,\bar{r}(t))=\bar{b}_{0}(t),\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{on}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{\widehat{A^{\prime}B^{\prime}}},\end{cases}$ (4.11)

where $\bar{b}_{0}(t)=\tilde{b}_{0}(\tilde{y}(\bar{r}(t)))$. Also, according
to the assumptions (3.7), we know that functions
$\hat{a}_{0},\leavevmode\nobreak\ \hat{a}_{1}$ and $\bar{b}_{0}$ satisfy the
following

$\displaystyle\hat{a}_{0}(r)\in C^{2}((r_{1},r_{2}]),\leavevmode\nobreak\
\leavevmode\nobreak\ \hat{a}_{1}(r)\in
C^{2}((r_{1},r_{2}]),\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{b}_{0}\in
C^{2}([0,t_{0})),\leavevmode\nobreak\ \leavevmode\nobreak\
\hat{a}_{0}\leq-\epsilon_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{b}_{0}\geq\epsilon_{0},$ (4.12)

where $t_{0}=\sqrt{1-\tilde{\varpi}^{2}(A)}$. Moreover, in view of the
definitions of $\hat{a}_{0},\hat{a}_{1}$ and $\bar{b}_{0}$ and the boundary
data (4.11), it is easy to see that the compatibility conditions are satisfied
at the point $B^{\prime}$, i.e.,
$\bar{b}_{0}(0)=W(0,r_{2})=-\hat{a}_{0}(r_{2})$ and
$\bar{b}_{0}^{\prime}(0)=W_{t}(0,r_{2})=\hat{a}_{1}(r_{2})$. Hence the problem
in terms of the $(t,r)$ plane can be reformulated as follows:

Reforumlated problem: Construct a local classical solution for (4.5) with the
boundary data (4.11) in the region $t>0$ in the neighbourhood of the point
$B^{\prime}(0,r_{2})$ under the assumption (4.12).

###  4.2 Solution in the partial hodograph plane

We now construct solution to the reformulated problem in this subsection. Let
us denote

$\displaystyle\overline{W}=\dfrac{1}{W},\leavevmode\nobreak\
\leavevmode\nobreak\ \overline{Z}=-\dfrac{1}{Z}$ (4.13)

to linearize the semi-linear system (4.5). In terms of $\overline{W}$ and
$\overline{Z}$, (4.5) is transformed into the following linear system

$\displaystyle\begin{cases}&\overline{W}_{t}-\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}\overline{W}_{r}=\dfrac{\overline{W}-\overline{Z}}{2t}+tG_{1}(\overline{W},\overline{Z},t),\vspace{0.2
cm}\\\
&\overline{Z}_{t}+\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}\overline{Z}_{r}=\dfrac{\overline{Z}-\overline{W}}{2t}+tG_{2}(\overline{W},\overline{Z},t),\\\
\end{cases}$ (4.14)

where

$\displaystyle G_{1}=$
$\displaystyle\bigg{[}\dfrac{\left(i^{2}(t)+\frac{H^{2}}{\gamma^{2}(t)}\right)\bigg{[}i^{2}(t)p^{\prime\prime}(\rho(t))+\kappa_{0}^{2}n^{2}(t)(1-p^{\prime}(\rho(t))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}+2-t^{2}\bigg{]}\dfrac{(\overline{W}-\overline{Z})}{2F(t)}$
$\displaystyle-\bigg{[}\dfrac{\left(i^{2}(t)+\frac{H^{2}}{\gamma^{2}(t)}\right)\bigg{[}i^{2}(t)p^{\prime\prime}(\rho(t))+\kappa_{0}^{2}n^{2}(t)(1-p^{\prime}(\rho(t))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}+2(1-t^{2})\bigg{]}\dfrac{\overline{W}}{F(t)},$
$\displaystyle G_{2}=$
$\displaystyle\bigg{[}\dfrac{\left(i^{2}(t)+\frac{H^{2}}{\gamma^{2}(t)}\right)\bigg{[}i^{2}(t)p^{\prime\prime}(\rho(t))+\kappa_{0}^{2}n^{2}(t)(1-p^{\prime}(\rho(t))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}+2-t^{2}\bigg{]}\dfrac{(\overline{Z}-\overline{W})}{2F(t)}$
$\displaystyle-\bigg{[}\dfrac{\left(i^{2}(t)+\frac{H^{2}}{\gamma^{2}(t)}\right)\bigg{[}i^{2}(t)p^{\prime\prime}(\rho(t))+\kappa_{0}^{2}n^{2}(t)(1-p^{\prime}(\rho(t))\bigg{]}\gamma_{w(t)}^{2}f(w(t))}{2i^{2}(t)w^{2}(t)}+2(1-t^{2})\bigg{]}\dfrac{\overline{Z}}{F(t)}.$

We now use the transformation $\upsilon=t,\leavevmode\nobreak\
\leavevmode\nobreak\ \chi=\bar{r}(t)-r$ to reduce (4.14) into the following
form

$\displaystyle\begin{cases}&\widetilde{W}_{\upsilon}+\dfrac{2\upsilon^{2}\sqrt{1-\upsilon^{2}}}{F(\upsilon)}\widetilde{W}_{\chi}=\dfrac{\widetilde{W}-\widetilde{Z}}{2\upsilon}+\upsilon
G_{1}(\widetilde{W},\widetilde{Z},\upsilon),\vspace{0.2 cm}\\\
&\widetilde{Z}_{\upsilon}=\dfrac{\widetilde{Z}-\widetilde{W}}{2\upsilon}+\upsilon
G_{2}(\widetilde{W},\widetilde{Z},\upsilon),\end{cases}$ (4.15)

where $\widetilde{W}(\upsilon,\chi)=\overline{W}(\upsilon,\bar{r}(t)-\chi)$
and $\widetilde{Z}(\upsilon,\chi)=\overline{Z}(\upsilon,\bar{r}(t)-\chi)$.
Recalling the boundary data (4.11), we now have the boundary data of
$\widetilde{W}(\upsilon,\chi),\widetilde{Z}(\upsilon,\chi)$ as follows:

$\displaystyle\begin{cases}\widetilde{W}(0,\chi)=\widetilde{Z}(0,\chi)=a_{0}(\chi),\leavevmode\nobreak\
\leavevmode\nobreak\
\widetilde{W}_{\upsilon}(0,\chi)=-\widetilde{Z}_{\upsilon}(0,\chi)=a_{1}(\chi)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{on}\leavevmode\nobreak\ \leavevmode\nobreak\
\upsilon=0,\leavevmode\nobreak\ \leavevmode\nobreak\ 0\leq\chi<r_{2}-r_{1},\\\
\widetilde{W}(\chi,0)=b_{0}(\upsilon)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{on}\leavevmode\nobreak\
\leavevmode\nobreak\ \chi=0,\leavevmode\nobreak\ \leavevmode\nobreak\
0\leq\upsilon<t_{0},\end{cases}$ (4.16)

where

$\displaystyle
a_{0}(\chi)=-\dfrac{1}{\hat{a}_{0}(r_{2}-\chi)},\leavevmode\nobreak\
\leavevmode\nobreak\
a_{1}(\chi)=\dfrac{\hat{a}_{1}}{\hat{a}_{0}^{2}}(r_{2}-\chi),\leavevmode\nobreak\
\leavevmode\nobreak\ b_{0}(\upsilon)=\dfrac{1}{\bar{b}_{0}(\upsilon)}.$ (4.17)

We now denote

$\displaystyle
U=\widetilde{W}-a_{0}(\chi)+a_{1}(\chi)\upsilon,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ V=\widetilde{Z}-a_{0}(\chi)-a_{1}(\chi)\upsilon$ (4.18)

to homogenize the boundary conditions (4.11) such that

$\displaystyle\begin{cases}U(0,\chi)=V(0,\chi)=U_{\upsilon}(0,\chi)=V_{\upsilon}(0,\chi)=0,\forall\chi\leavevmode\nobreak\
\in[0,r_{2}-r_{1}),\\\ U(\upsilon,0)=b_{1}(\upsilon),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \upsilon\in[0,t_{0}),\end{cases}$ (4.19)

where the function $b_{1}(\upsilon)$ satisfies
$b_{1}(\upsilon)=b_{0}(\upsilon)-a_{0}(0)+a_{1}(0)\upsilon$, which implies
that $b_{1}^{\prime}(\upsilon)=b_{0}^{\prime}(\upsilon)+a_{1}(0)$. Therefore,
$b_{1}^{\prime}(0)=b_{1}(0)=0$.

By a direct calculation, one can convert the system (4.15) in terms of $(U,V)$
as follows:

$\displaystyle\begin{cases}&U_{\upsilon}+\dfrac{2\upsilon^{2}\sqrt{1-\upsilon^{2}}}{F(\upsilon)}U_{\chi}=\dfrac{U-V}{2\upsilon}+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}w^{2}}+2-\upsilon^{2}\Bigg{]}\dfrac{(U-V)\upsilon}{2F(\upsilon)}\\\
&\hskip
28.45274pt-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}w^{2}}+2(1-\upsilon^{2})\Bigg{]}\dfrac{U\upsilon}{F(\upsilon)}+E_{1}(\upsilon,\chi)\upsilon,\vspace{0.2
cm}\\\
&V_{\upsilon}=\dfrac{V-U}{2\upsilon}+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}w^{2}}+2-\upsilon^{2}\Bigg{]}\dfrac{(V-U)\upsilon}{2F(\upsilon)}\\\
&\hskip
28.45274pt-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}w^{2}}+2(1-\upsilon^{2})\Bigg{]}\dfrac{V\upsilon}{F(\upsilon)}+E_{2}(\upsilon,\chi)\upsilon,\end{cases}$
(4.20)

where

$\displaystyle E_{1}(\upsilon,\chi)$
$\displaystyle=-\bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\left(i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\right)\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\bigg{]}\dfrac{a_{0}}{F(\upsilon)}$
$\displaystyle-\dfrac{2\upsilon\sqrt{1-\upsilon^{2}}}{F}(a_{0}^{\prime}-a_{1}^{\prime}\upsilon)-\dfrac{a_{1}\upsilon^{3}}{F(\upsilon)}$

and

$\displaystyle E_{2}(\upsilon,\chi)$
$\displaystyle=-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}a_{0}+\dfrac{a_{1}\upsilon^{3}}{F(\upsilon)}.$

In view of the regularity of $a_{0}(\chi),a_{1}(\chi)$, it is easy to see that
$F_{1}$ and $F_{2}$ have continuous derivatives wrt $\chi$. Therefore the
problem is converted to seek a classical solution for a degenerate linear
initial-characteristic problem, which has two eigenvalues given by

$\displaystyle\lambda_{-}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda_{+}=\dfrac{2\upsilon^{2}\sqrt{1-\upsilon^{2}}}{F(\upsilon)}.$ (4.21)

We now denote $\chi=\bar{\chi}(\upsilon)$ as the $C_{+}$ characteristic
passing through the origin, which can be given by

$\displaystyle\bar{\chi}(\upsilon)=\displaystyle\int_{0}^{\upsilon}\dfrac{2s^{2}\sqrt{1-s^{2}}}{F(s)}ds.$

Further, we define a square region
$\Omega:=\\{(\upsilon,\chi)|0\leq\upsilon\leq\delta,\leavevmode\nobreak\
0\leq\chi\leq\delta\\}$ for a positive constant $\delta$. Then we integrate
the system (4.20) with the boundary conditions (4.19) along the
characteristics to yield a system of integral equations given by

$\displaystyle\begin{cases}V(\tau,\zeta)&=\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{V-U}{2\upsilon}+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(V-U)\upsilon}{2F}\\\
&\hskip
56.9055pt-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{V\upsilon}{F}+E_{2}\upsilon\Bigg{\\}}(\upsilon,\chi_{-}(\upsilon))d\upsilon,\vspace{0.3
cm}\\\
U(\tau,\zeta)&=\begin{cases}\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{U-V}{2\upsilon}+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(U-V)\upsilon}{2F}\\\
\hskip
14.22636pt-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{U\upsilon}{F(\upsilon)}+E_{1}\upsilon\Bigg{\\}}(\upsilon,\chi_{+}(\upsilon))d\upsilon,\\\
\hskip 341.43306pt\zeta\geq\bar{\chi}(\tau),\tau\geq 0,\vspace{0.3 cm}\\\
b_{1}(\tau_{1})+\displaystyle\bigintssss_{\tau_{1}}^{\tau}\Bigg{\\{}\dfrac{U-V}{2\upsilon}+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(U-V)\upsilon}{2F}\\\
\hskip
14.22636pt-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{U\upsilon}{F(\upsilon)}+E_{1}\upsilon\Bigg{\\}}(\upsilon,\chi_{+}(\upsilon))d\upsilon\\\
\hskip 341.43306pt0\leq\zeta<\bar{\chi}(\tau),\tau\geq
0,\end{cases}\end{cases}$ (4.22)

where
$\chi_{+}(\upsilon)=\chi_{+}(\upsilon;\tau,\zeta),\chi_{-}(\upsilon)=\chi_{-}(\upsilon;\tau,\zeta)$
are the positive and negative characteristics passing through a point
$(\tau,\zeta)$, respectively and $\tau_{1}\in[0,\tau)$ is a function of
$(\tau,\zeta)$ defined as

$\displaystyle\zeta=\displaystyle\int_{\tau_{1}}^{\tau}\dfrac{2\upsilon^{2}\sqrt{1-\upsilon^{2}}}{F(\upsilon)}d\upsilon.$
(4.23)

Moreover, by using the fact that $\lambda_{-}=0$ we must have
$\chi_{-}(\upsilon;\tau,\zeta)=\zeta$.

We now prove the existence of solution for the system (4.22) by exploiting the
method of iterations. We denote $U^{0}(\upsilon,\chi)=V^{0}(\upsilon,\chi)=0$
and define $U^{(k)}$ and $V^{(k)},(k\geq 1)$ as follows:

$\displaystyle\begin{cases}V^{(k)}(\tau,\zeta)&=\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{V^{(k-1)}-U^{(k-1)}}{2\upsilon}-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{V^{(k-1)}\upsilon}{F}\\\
&+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(V^{(k-1)}-U^{(k-1)})\upsilon}{2F}+E_{2}\upsilon\Bigg{\\}}(\upsilon,\chi_{-}(\upsilon))d\upsilon,\vspace{0.3
cm}\\\
U^{(k)}(\tau,\zeta)&=\begin{cases}\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{U^{(k-1)}-V^{(k-1)}}{2\upsilon}-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{U^{(k-1)}\upsilon}{F(\upsilon)}\\\
+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(U^{(k-1)}-V^{(k-1)})\upsilon}{2F}+E_{1}\upsilon\Bigg{\\}}(\upsilon,\chi_{+}(\upsilon))d\upsilon,\\\
\hskip 341.43306pt\zeta\geq\bar{\chi}(\tau),\tau\geq 0,\vspace{0.3 cm}\\\
\displaystyle\bigintssss_{\tau_{1}}^{\tau}\Bigg{\\{}\dfrac{U^{(k-1)}-V^{(k-1)}}{2\upsilon}-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{U^{(k-1)}\upsilon}{F(\upsilon)}\\\
+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(U^{(k-1)}-V^{(k-1)})\upsilon}{2F}+E_{1}\upsilon\Bigg{\\}}(\upsilon,\chi_{+}(\upsilon))d\upsilon\\\
+b_{1}(\tau_{1}),\hskip 312.9803pt0\leq\zeta<\bar{\chi}(\tau),\tau\geq
0.\end{cases}\end{cases}$ (4.24)

Now we proceed to prove that the sequences $\\{(U^{(k)},V^{(k)})\\}$ are
uniformly convergent in the domain $\Omega$ for a small positive constant
$\delta.$ In view of the expression of $F(t)$ and noting that $0<a^{2}<1$,
$0<q<\hat{q}<1$, it is straightforward to observe that there exists a
sufficiently small $\delta>0$ and $\upsilon\in[0,\delta]$ such that
$0<\dfrac{\sqrt{1-\upsilon^{2}}}{F(\upsilon)}\leq\dfrac{1}{F(\upsilon)}\leq
K_{\delta}<\infty$, where $K_{\delta}$ is a uniform small positive constant.
Moreover, we must have

$\displaystyle
0<\dfrac{1}{F(\upsilon)}\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}w^{2}}+2(1-\upsilon^{2})\right)\leq
K_{\delta},$ (4.25) $\displaystyle
0<\dfrac{1}{F(\upsilon)}\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}w^{2}}+2-\upsilon^{2}\right)\leq
K_{\delta},$ (4.26)

For simplicity of notations, let $M>0$ be a uniform constant depending only on
the $C^{2}$ norms of $a_{0},a_{1}$ and $b_{0}$ which can change between
different expressions in the analysis of this article. From the expressions of
$F_{1}$ and $F_{2}$, it is clear that

$\displaystyle|E_{1}|+|E_{1\chi}|+|E_{2}|+|E_{2\chi}|\leq M.$ (4.27)

For any point $P(\tau,\zeta)$ in
$\Omega=\\{(\upsilon,\chi)|0<\upsilon\leq\delta,\leavevmode\nobreak\
0<\chi\leq\delta\\}$, by the expression of $b_{1}(.)$ it is clear that
$|b_{1}(\tau_{1})|=|b_{0}(\tau_{1})-a_{0}(0)+a_{1}(0)\tau_{1}|\leq\dfrac{M}{2}\tau_{1}^{2}$
if $\zeta<\bar{\chi}(\tau)$. Noting that $\tau_{1}\leq\tau$, we must have

$\displaystyle|b_{1}(\tau_{1})|\leq\dfrac{M\tau^{2}}{2}.$ (4.28)

Further, employing (4.23) for any $\bar{\tau}\in[0,\tau]$, we must have

$\displaystyle|\chi_{+}(\bar{\tau})-\chi_{-}(\bar{\tau})|$
$\displaystyle=|\chi_{+}(\bar{\tau};\tau,\zeta)-\chi_{-}(\bar{\tau};\tau,\zeta)|$
$\displaystyle\leq|\chi_{+}(0;\tau,\zeta)-\zeta|=\bigg{|}\displaystyle\int_{0}^{\tau}\dfrac{2\upsilon^{2}\sqrt{1-\upsilon^{2}}}{F(\upsilon)}d\upsilon\bigg{|}\leq\dfrac{2K_{\delta}\tau^{3}}{3}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{if}\leavevmode\nobreak\ \leavevmode\nobreak\
\zeta\geq\bar{\chi}(\tau)$ (4.29)

and for any $\bar{\tau}\in[\bar{\tau}_{1},\tau]$, we have

$\displaystyle|\chi_{+}(\bar{\tau})-\chi_{-}(\bar{\tau})|$
$\displaystyle=|\chi_{+}(\bar{\tau};\tau,\zeta)-\chi_{-}(\bar{\tau};\tau,\zeta)|$
$\displaystyle\leq|\chi_{+}(\bar{\tau_{1}};\tau,\zeta)-\zeta|=\bigg{|}\displaystyle\int_{\tau_{1}}^{\tau}\dfrac{2\upsilon^{2}\sqrt{1-\upsilon^{2}}}{F(\upsilon)}d\upsilon\bigg{|}\leq\dfrac{2K_{\delta}\tau^{3}}{3}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{if}\leavevmode\nobreak\ \leavevmode\nobreak\
\zeta<\bar{\chi}(\tau).$ (4.30)

Based on the above discussions, one can derive the following Lemmas:

######  Lemma 4.2.

For any $k\geq 1$ and $(\tau,\zeta)\in\Omega$, the following inequalities hold
true

$\displaystyle\begin{cases}|U^{(k)}(\tau,\zeta)|;|V^{(k)}(\tau,\zeta)|\leq
M\tau^{2}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j},\\\
|U^{(k)}(\tau,\zeta)-V^{(k)}(\tau,\zeta)|\leq
M\tau^{2}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}.\end{cases}$ (4.31)

###### Proof.

We use the induction argument to prove this Lemma. The proof can be discussed
into two different cases depending on whether $\zeta<\bar{\chi}(\tau)$ or
$\zeta\geq\bar{\chi}(\tau)$. In this proof, we analyze the first case
$\zeta<\bar{\chi}(\tau)$ only as the second case can be analyzed similarly.

Now for $n=1$, we have

$\displaystyle|V^{(1)}(\tau,\zeta)|\leq\displaystyle\int_{0}^{\tau}|\upsilon
E_{2}(\upsilon)|d\upsilon\leq\displaystyle\int_{0}^{\tau}M\upsilon
d\upsilon\leq\dfrac{M\tau^{2}}{2}\leq
M\tau^{2}\sum_{j=0}^{1}\left(\dfrac{2}{3}\right)^{j}.$ (4.32)

Similarly we estimate $U^{(1)}$ for $\zeta<\bar{\chi}(\tau)$ as

$\displaystyle|U^{(1)}(\tau,\zeta)|$
$\displaystyle\leq\displaystyle\int_{\tau_{1}}^{\tau}|\upsilon
E_{1}(\upsilon)|d\upsilon+|b_{1}(\tau_{1})|\leq\displaystyle\int_{\tau_{1}}^{\tau}M\upsilon
d\upsilon+\dfrac{M\tau^{2}}{2}$
$\displaystyle\leq\dfrac{M(\tau^{2}-\tau_{1}^{2})}{2}+\dfrac{M\tau^{2}}{2}\leq
M\tau^{2}\sum_{j=0}^{1}\left(\dfrac{2}{3}\right)^{j}.$ (4.33)

Now we proceed to estimate $|U^{(1)}(\tau,\zeta)-V^{(1)}(\tau,\zeta)|$. Now by
direct calculation, one can compute that

$\displaystyle|U^{(1)}(\tau,\zeta)$ $\displaystyle-V^{(1)}(\tau,\zeta)|$
$\displaystyle\leq\displaystyle\int_{\tau_{1}}^{\tau}|E_{1}(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))-E_{2}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|\upsilon
d\upsilon+\displaystyle\int_{0}^{\tau_{1}}|E_{2}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|\upsilon
d\upsilon+|b_{1}(\tau_{1})|.$ (4.34)

Hence, it is clear that we need to estimate
$|E_{1}(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))-E_{2}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|$
in order to estimate $|U^{(1)}(\tau,\zeta)-V^{(1)}(\tau,\zeta)|$. Now

$\displaystyle|E_{1}(\upsilon,$
$\displaystyle\chi_{+}(\upsilon;\tau,\zeta))-E_{2}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|$
$\displaystyle\leq\dfrac{2\upsilon\sqrt{1-\upsilon^{2}}}{F}(|a_{0}^{\prime}|+|a_{1}^{\prime}|\upsilon)+\dfrac{|a_{1}(\chi_{+}(\upsilon;\tau,\zeta))+a_{1}(\chi_{-}(\upsilon;\tau,\zeta))|}{F}\upsilon^{3}$
$\displaystyle+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{|a_{0}(\chi_{+}(\upsilon;\tau,\zeta))-a_{0}(\chi_{-}(\upsilon;\tau,\zeta))|}{F}$
$\displaystyle\leq
2K_{\delta}(M+M\upsilon)\upsilon+2MK_{\delta}\upsilon^{3}+K_{\delta}M|\chi_{+}-\chi_{-}|$
$\displaystyle\leq
2K_{\delta}(M+M\tau)\tau+2MK_{\delta}\tau^{3}+\dfrac{2}{3}MK_{\delta}\tau^{3}$
$\displaystyle\leq K_{\delta}M\tau,$

which is exploited in (4.2) along with (4.28) to obtain

$\displaystyle|U^{(1)}(\tau,\zeta)-V^{(1)}(\tau,\zeta)|$
$\displaystyle\leq\displaystyle\int_{\tau_{1}}^{\tau}MK_{\delta}\tau\upsilon
d\upsilon+\displaystyle\int_{0}^{\tau_{1}}M\upsilon
d\upsilon+\dfrac{1}{2}M\tau^{2}$
$\displaystyle\leq\dfrac{MK_{\delta}}{2}\tau(\tau^{2}-\tau_{1}^{2})+\dfrac{1}{2}M\tau_{1}^{2}+\dfrac{1}{2}M\tau^{2}$
$\displaystyle\leq\dfrac{MK_{\delta}}{2}\tau^{3}+\dfrac{1}{2}M\tau^{2}+\dfrac{1}{2}M\tau^{2}\leq
M\tau^{2}\left(\dfrac{\delta K_{\delta}}{2}+1\right)$ $\displaystyle\leq
M\tau^{2}\sum_{j=0}^{1}\left(\dfrac{2}{3}\right)^{j}.$ (4.35)

Now we assume that the induction argument is true for $n=k$, and then we need
to prove that the statement is true for $n=k+1$. Now we use the induction
assumption to compute

$\displaystyle|V^{(k+1)}(\tau,\zeta)|$
$\displaystyle\leq\displaystyle\bigintssss_{0}^{\tau}\Bigg{|}\dfrac{V^{(k)}(\upsilon,\chi_{-}(\upsilon))-U^{(k)}(\upsilon,\chi_{-}(\upsilon))}{2\upsilon}+E_{2}(\upsilon,\chi_{-}(\upsilon))\upsilon\Bigg{|}d\upsilon$
$\displaystyle+\displaystyle\bigintssss_{0}^{\tau}\Bigg{|}\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{V^{(k)}(\upsilon,\chi_{-}(\upsilon))\upsilon}{F}\Bigg{|}d\upsilon$
$\displaystyle+\displaystyle\bigintssss_{0}^{\tau}\Bigg{|}\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{\left(U^{(k)}(\upsilon,\chi_{-}(\upsilon))-V^{(k)}(\upsilon,\chi_{-}(\upsilon))\right)\upsilon}{2F}\bigg{|}d\upsilon$
$\displaystyle\leq\displaystyle\int_{0}^{\tau}\bigg{\\{}\bigg{[}\dfrac{M\upsilon}{2}+2K_{\delta}M\upsilon^{3}\bigg{]}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+M\upsilon\bigg{\\}}d\upsilon=M\tau^{2}\bigg{\\{}\dfrac{1}{2}+\bigg{[}\dfrac{1}{4}+\dfrac{\delta^{2}K_{\delta}}{2}\bigg{]}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}\bigg{\\}}$
$\displaystyle\leq M\tau^{2}\sum_{j=0}^{k+1}\left(\dfrac{2}{3}\right)^{j}.$

One can find the following estimate of $U^{(k+1)}$ in a similar manner

$\displaystyle|U^{(k+1)}(\tau,\zeta)|\leq
M\tau^{2}\sum_{j=0}^{k+1}\left(\dfrac{2}{3}\right)^{j}.$ (4.36)

Now we compute the estimates of
$|U^{(k+1)}(\tau,\zeta)-V^{(k+1)}(\tau,\zeta)|$. A straightforward calculation
yields

$\displaystyle|U^{(k+1)}(\tau,\zeta)-V^{(k+1)}(\tau,\zeta)|\leq
I_{1}+I_{2}+|b_{1}(\tau_{1})|,$ (4.37)

where

$\displaystyle I_{1}$
$\displaystyle=\displaystyle\bigintssss_{\tau_{1}}^{\tau}\Bigg{\\{}\dfrac{|(U^{(k)}-V^{(k)})|(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))-|U^{(k)}-V^{(k)}|(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))}{2\upsilon}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)$
$\displaystyle\hskip
113.81102pt\dfrac{\bigg{[}|U^{(k)}-V^{(k)}|(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))+|U^{(k)}-V^{(k)}|(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))\bigg{]}\upsilon}{2F}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{\bigg{[}|U^{(k)}(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))|+|V^{(k)}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|\bigg{]}\upsilon}{F}$
$\displaystyle\hskip
28.45274pt+|E_{1}(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))-E_{2}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|\upsilon\Bigg{\\}}d\upsilon$

and

$\displaystyle I_{2}$
$\displaystyle=\displaystyle\bigintssss_{0}^{\tau_{1}}\Bigg{\\{}\dfrac{|(U^{(k)}-V^{(k)})(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|}{2\upsilon}+|E_{2}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|\upsilon$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{|(U^{(k)}-V^{(k)})(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|\upsilon}{2F}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{|V^{(k)}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))|\upsilon}{F}\Bigg{\\}}d\upsilon.$

Clearly, we need to find the estimates of $I_{1}$ and $I_{2}$ in order to find
the estimate of $|U^{(k+1)}(\tau,\zeta)-V^{(k+1)}(\tau,\zeta)|$.

Now using the induction assumption, we have

$\displaystyle I_{1}$
$\displaystyle\leq\displaystyle\int_{\tau_{1}}^{\tau}\bigg{\\{}M\upsilon\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+2K_{\delta}M\upsilon^{3}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+K_{\delta}M\upsilon^{3}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+K_{\delta}M\tau\upsilon\bigg{\\}}d\upsilon$
$\displaystyle\leq\dfrac{M}{2}(\tau^{2}-\tau_{1}^{2})\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+\dfrac{3K_{\delta}M}{4}(\tau^{4}-\tau_{1}^{4})\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+\dfrac{K_{\delta}M\tau^{3}}{2},$

and

$\displaystyle I_{2}$
$\displaystyle\leq\displaystyle\int_{0}^{\tau_{1}}\bigg{\\{}\dfrac{M\upsilon}{2}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+2K_{\delta}M\upsilon^{3}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+M\upsilon\bigg{\\}}d\upsilon$
$\displaystyle\leq\dfrac{M\tau_{1}^{2}}{4}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+\dfrac{K_{\delta}M\tau_{1}^{4}}{2}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+\dfrac{M\tau_{1}^{2}}{2},$

which implies

$\displaystyle|U^{(k+1)}(\tau,\zeta)-V^{(k+1)}(\tau,\zeta)|$
$\displaystyle\leq\bigg{\\{}\dfrac{M}{2}(\tau^{2}-\tau_{1}^{2})\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+\dfrac{3K_{\delta}M}{4}(\tau^{4}-\tau_{1}^{4})\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+\dfrac{K_{\delta}M\tau^{3}}{2}\bigg{\\}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\bigg{\\{}\dfrac{M\tau_{1}^{2}}{4}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+\dfrac{K_{\delta}M\tau_{1}^{4}}{2}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}+\dfrac{M\tau_{1}^{2}}{2}\bigg{\\}}+\dfrac{M}{2}\tau^{2}$
$\displaystyle\leq M\tau^{2}\bigg{\\{}\dfrac{1}{2}+\dfrac{\delta
K_{\delta}}{2}+\bigg{[}\dfrac{3\delta^{2}K_{\delta}}{4}+\dfrac{1}{2}\bigg{]}\sum_{j=0}^{k}\left(\dfrac{2}{3}\right)^{j}\bigg{\\}}\leq
M\tau^{2}\sum_{j=0}^{k+1}\left(\dfrac{2}{3}\right)^{j},$

which shows that the proof of the Lemma is complete. ∎

######  Lemma 4.3.

For all $k\geq 1$ and any $(\tau,\zeta)\in\Omega$, the following inequalities
hold true

$\displaystyle\begin{cases}|U^{(k+1)}(\tau,\zeta)-U^{(k)}(\tau,\zeta)|;|V^{(k+1)}(\tau,\zeta)-V^{(k)}(\tau,\zeta)|\leq
M\tau^{2}\left(\dfrac{2}{3}\right)^{j},\\\
|U^{(k+1)}(\tau,\zeta)-V^{(k+1)}(\tau,\zeta)-U^{(k)}(\tau,\zeta)+V^{(k)}(\tau,\zeta)|\leq
M\tau^{2}\left(\dfrac{2}{3}\right)^{j}.\end{cases}$ (4.38)

###### Proof.

We again employ the induction technique to prove this Lemma as well.

For $n=1$, one has

$\displaystyle|U^{(2)}$
$\displaystyle-U^{(1)}|\leq\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{|U^{(1)}-V^{(1)}|}{2\upsilon}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{|U^{(1)}-V^{(1)}|\upsilon}{2F}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{|U^{(1)}|\upsilon}{F}\Bigg{\\}}(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))d\upsilon$
$\displaystyle\leq\sum_{j=0}^{1}\left(\dfrac{2}{3}\right)^{j}\displaystyle\int_{0}^{\tau}\left(\dfrac{M\upsilon}{2}+2K_{\delta}M\upsilon^{3}\right)(\upsilon,\chi_{+}(\upsilon))d\upsilon$
$\displaystyle\leq
M\tau^{2}\bigg{[}\dfrac{1}{4}+\dfrac{\delta^{2}K_{\delta}}{2}\bigg{]}\sum_{j=0}^{1}\left(\dfrac{2}{3}\right)^{j}\leq
M\tau^{2}\left(\dfrac{2}{3}\right).$

Similarly one can easily prove that $|V^{(2)}-V^{(1)}|\leq
M\tau^{2}\left(\dfrac{2}{3}\right)$ and $|U^{(2)}-V^{(2)}-U^{(1)}+V^{(1)}|\leq
M\tau^{2}\left(\dfrac{2}{3}\right)$.

We now assume that the Lemma is true for $n=k$, and then we proceed to prove
that the Lemma is true for $n=k+1$.

We first use (4.24) to obtain

$\displaystyle|V^{(k+1)}(\tau,\zeta)-V^{(k)}(\tau,\zeta)|$
$\displaystyle\leq\displaystyle\bigintssss_{0}^{\tau}\bigg{\\{}\dfrac{|V^{(k)}-U^{(k)}-V^{(k-1)}+U^{(k-1)}|}{2\upsilon}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{|U^{(k)}-V^{(k)}-U^{(k-1)}+V^{(k-1)}|\upsilon}{2F}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{|V^{(k)}-V^{(k-1)}|\upsilon}{F}\bigg{\\}}(\upsilon,\chi_{-}(\upsilon;\tau,\zeta))d\upsilon$
$\displaystyle\leq\displaystyle\bigintssss_{0}^{\tau}\bigg{\\{}M\upsilon\left(\dfrac{2}{3}\right)^{k-1}+3K_{\delta}M\upsilon^{3}\left(\dfrac{2}{3}\right)^{k-1}\bigg{\\}}d\upsilon=\left(\dfrac{2}{3}\right)^{k-1}\bigg{[}\dfrac{M\tau^{2}}{2}+\dfrac{3MK_{\delta}\tau^{4}}{4}\bigg{]}$
$\displaystyle\leq
M\tau^{2}\bigg{[}\dfrac{1}{2}+\dfrac{3K_{\delta}\delta^{2}}{4}\bigg{]}\left(\dfrac{2}{3}\right)^{k-1}\leq
M\tau^{2}\left(\dfrac{2}{3}\right)^{k}.$

The estimate of $|U^{(k+1)}-U^{(k)}|$ can be obtained in a similar manner. For
the sake of brevity, we omit the details here.

Now we proceed to estimate the quantity
$|U^{(k+1)}-V^{(k+1)}-U^{(k)}+V^{(k)}|$. A straightforward calculation leads
us to

$\displaystyle|U^{(k+1)}-V^{(k+1)}-U^{(k)}+V^{(k)}|\leq I_{3}+I_{4},$ (4.39)

where

$\displaystyle I_{3}$
$\displaystyle=\displaystyle\bigintssss_{\tau_{1}}^{\tau}\bigg{\\{}\dfrac{|U^{(k)}-V^{(k)}-U^{(k-1)}+V^{(k-1)}|}{2\upsilon}(\upsilon,\chi_{+}(\upsilon))+\dfrac{|U^{(k)}-V^{(k)}-U^{(k-1)}+V^{(k-1)}|}{2\upsilon}(\upsilon,\chi_{-}(\upsilon))$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{|U^{(k)}-V^{(k)}-U^{(k-1)}+V^{(k-1)}|\upsilon}{2F}(\upsilon,\chi_{+}(\upsilon))$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{|U^{(k)}-V^{(k)}-U^{(k-1)}+V^{(k-1)}|\upsilon}{2F}(\upsilon,\chi_{-}(\upsilon))$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{|U^{(k)}-U^{(k-1)}|\upsilon}{F}(\upsilon,\chi_{+}(\upsilon)$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{|V^{(k)}-V^{(k-1)}|\upsilon}{F}(\upsilon,\chi_{-}(\upsilon)\bigg{\\}}d\upsilon$

and

$\displaystyle I_{4}$
$\displaystyle=\displaystyle\bigintssss_{0}^{\tau_{1}}\bigg{\\{}\dfrac{|V^{(k)}-U^{(k)}-V^{(k-1)}+U^{(k-1)}|}{2\upsilon}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{|U^{(k)}-V^{(k)}-U^{(k-1)}+V^{(k-1)}|\upsilon}{2F}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{|V^{(k)}-V^{(k-1)}|\upsilon}{F}\bigg{\\}}(\upsilon,\chi_{+}(\upsilon))d\upsilon.$

Hence by the induction assumption, we have

$\displaystyle I_{3}$
$\displaystyle\leq\displaystyle\bigintssss_{\tau_{1}}^{\tau}\bigg{\\{}M\upsilon\left(\dfrac{2}{3}\right)^{k-1}+6K_{\delta}M\upsilon^{3}\left(\dfrac{2}{3}\right)^{k-1}\bigg{\\}}d\upsilon=\left(\dfrac{2}{3}\right)^{k-1}\bigg{\\{}\dfrac{M}{2}(\tau^{2}-\tau_{1}^{2})+\dfrac{3}{2}K_{\delta}M(\tau^{4}-\tau_{1}^{4})\bigg{\\}},$
$\displaystyle I_{4}$
$\displaystyle\leq\displaystyle\int_{0}^{\tau}\bigg{\\{}\dfrac{M\upsilon}{2}\left(\dfrac{2}{3}\right)^{k-1}+3K_{\delta}M\upsilon^{3}\left(\dfrac{2}{3}\right)^{k-1}\bigg{\\}}d\upsilon=\left(\dfrac{2}{3}\right)^{k-1}\bigg{\\{}\dfrac{M}{4}\tau_{1}^{2}+\dfrac{3K_{\delta}M\tau_{1}^{4}}{4}\bigg{\\}}.$

which is used in (4.39) to obtain

$\displaystyle|U^{(k+1)}-V^{(k+1)}-U^{(k)}+V^{(k)}|$
$\displaystyle\leq\left(\dfrac{2}{3}\right)^{k-1}\bigg{\\{}\dfrac{M}{2}(\tau^{2}-\tau_{1}^{2})+\dfrac{3}{2}K_{\delta}M(\tau^{4}-\tau_{1}^{4})+\dfrac{M}{4}\tau_{1}^{2}+\dfrac{3K_{\delta}M}{4}\tau_{1}^{4}\bigg{\\}}$
$\displaystyle=\left(\dfrac{2}{3}\right)^{k-1}\bigg{\\{}\dfrac{M\tau^{2}}{2}+\dfrac{3}{2}K_{\delta}M\tau^{4}-\dfrac{M}{4}\tau_{1}^{2}-\dfrac{3K_{\delta}M}{4}\tau_{1}^{4}\bigg{\\}}$
$\displaystyle\leq
M\tau^{2}\left(\dfrac{1}{2}+\dfrac{3}{2}\delta^{2}K_{\delta}\right)\left(\dfrac{2}{3}\right)^{k-1}\leq
M\tau^{2}\left(\dfrac{2}{3}\right)^{k}.$

Combining all these estimates proves the lemma. ∎

###  4.3 Existence and uniqueness of the solution in the $(t-r)$ plane

We now discuss the properties of $(U^{(k)},V^{(k)})(\tau,\zeta)$ defined by
(4.24). It is easy to see that from Lemma 4.3 that
$(U^{(k)},V^{(k)})(\tau,\zeta)$ are uniformly convergent sequences of
continuous functions which means that the limit function $(U,V)$ must be
continuous too and it must satisfy

$\displaystyle|U(\tau,\zeta)|\leq 3M\tau^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ |V(\tau,\zeta)|\leq 3M\tau^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ |U(\tau,\zeta)-V(\tau,\zeta)|\leq 3M\tau^{2}$ (4.40)

for all $(\tau,\zeta)\in\Omega$. Further, the functions $(U,V)$ satisfy (4.24)
and the boundary conditions $(U(0,\zeta))=V(0,\zeta)=0$. Also, in view of
(4.23) we find that $\zeta=0$ if and only if $\tau=\tau_{1}$. Applying this in
(4.24) implies that $U(\tau,0)=b_{1}(\tau)$.

We now verify that the boundary conditions
$U_{\tau}(0,\zeta)=V_{\tau}(0,\zeta)=0$ are satisfied by $(U,V)$. In view of
(4.40) and the fundamental theorem of calculus, it is easy to observe that the
derivatives of $U(\tau,\zeta)$ and $V(\tau,\zeta)$ with respect to $\tau$ are
continuous. We now prove the existence of $(U_{\zeta},V_{\zeta})$ near
$\tau=0$. For the sake of brevity, we only discuss the case
$\zeta\geq\bar{\chi}(\tau)$ as the other case can be discussed in a similar
manner. Now we differentiate (4.24) wrt $\zeta$ to yield the system of
integral equations of the form

$\displaystyle\begin{cases}V_{\zeta}^{(k)}(\tau,\zeta)&=\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{V_{\chi}^{(k-1)}-U_{\chi}^{(k-1)}}{2\upsilon}+E_{2\chi}\upsilon\\\
&+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{(V_{\chi}^{(k-1)}-U_{\chi}^{(k-1)})\upsilon}{2F}\\\
&-\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{V_{\chi}^{(k-1)}\upsilon}{F}\Bigg{\\}}\dfrac{\partial\chi_{-}}{\partial\zeta}(\upsilon,\chi_{-}(\upsilon))d\upsilon,\vspace{0.3
cm}\\\
U_{\zeta}^{(k)}(\tau,\zeta)&=\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{U_{\chi}^{(k-1)}-V_{\chi}^{(k-1)}}{2\upsilon}+E_{1\chi}\upsilon\\\
&+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\right)\dfrac{(U_{\chi}^{(k-1)}-V_{\chi}^{(k-1)})\upsilon}{2F}\\\
&-\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{U_{\chi}^{(k-1)}\upsilon}{F(\upsilon)}\Bigg{\\}}\dfrac{\partial\chi_{+}}{\partial\zeta}(\upsilon,\chi_{+}(\upsilon))d\upsilon,\end{cases}$
(4.41)

where

$\displaystyle\dfrac{\partial\chi_{\pm}}{\partial\zeta}=\exp\left(\displaystyle\int_{\tau}^{\upsilon}\dfrac{\partial\chi_{\pm}}{\partial\chi}(s,\chi_{\pm}(s;\tau,\zeta))ds\right).$
(4.42)

Recalling the expressions of $\lambda_{\pm}$, we find
$\dfrac{\partial\chi_{\pm}}{\partial\chi}=0$ and hence
$\dfrac{\partial\chi_{\pm}}{\partial\zeta}=1$. Also by recalling the
expressions of $E_{1},E_{2}$ and the regularities of $a_{0},a_{1}$, we also
have

$\displaystyle|E_{1\chi}(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))-E_{2\chi}(\upsilon,\chi_{+}(\upsilon;\tau,\zeta))|$
$\displaystyle\leq\dfrac{2\upsilon\sqrt{1-\upsilon^{2}}}{F}(|a_{0}^{\prime\prime}|+|a_{1}^{\prime\prime}|\upsilon)+\dfrac{|a_{1}^{\prime}(\chi_{+}(\upsilon;\tau,\zeta))|+|a_{1}^{\prime}(\chi_{-}(\upsilon;\tau,\zeta))|}{F}\upsilon^{3}$
$\displaystyle+\left(\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w}^{2}f(w)}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\right)\dfrac{|a_{0}^{\prime}(\chi_{+}(\upsilon;\tau,\zeta))-a_{0}^{\prime}(\chi_{-}(\upsilon;\tau,\zeta))|}{F}$
$\displaystyle\leq
K_{\delta}M|\chi_{+}(\upsilon;\tau,\zeta)-\chi_{-}(\upsilon;\tau,\zeta)|+2K_{\delta}\upsilon(M+M\upsilon)+2K_{\delta}M\upsilon^{3}$
$\displaystyle\leq\dfrac{2K_{\delta}\tau^{3}}{3}K_{\delta}M+2K_{\delta}\tau(M+M\tau)+2K_{\delta}M\tau^{3}$
$\displaystyle\leq
2K_{\delta}M\tau\left(\dfrac{\delta^{2}K_{\delta}}{3}+1+\delta+\delta^{2}\right)\leq
K_{\delta}M\tau.$

One can perform the same iterative process used in Lemma 4.2 and Lemma 4.3 to
conclude the uniform convergence of $(U_{\zeta}^{(k)},V_{\zeta}^{(k)})$ which
is enough to prove that the limit functions $(U_{\zeta}^{(k)},V_{\zeta})$ are
continuous and satisfy $U_{\zeta}(0,\zeta)=V_{\zeta}(0,\zeta)=0$. Since all
partial derivatives of $(U,V)$ are continuous, $(U,V)$ must be differentiable.
Also since $(U,V)$ satisfies the boundary conditions (4.19), $(U,V)$ must be a
solution of the system (4.22) with the boundary data (4.19).

Now we proceed to prove the uniqueness of the solution. On the contrary let us
assume that there exist two solutions of the system given by $(U_{1},V_{1})$
and $(U_{2},V_{2})$. Then we denote $\tilde{U}=U_{2}-U_{1}$ and
$\tilde{V}=V_{2}-V_{1}$ to observe that the functions $(\tilde{U},\tilde{V})$
satisfy the following integral equations

$\displaystyle\begin{cases}\tilde{V}(\tau,\zeta)&=\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{\tilde{V}-\tilde{R}}{2\upsilon}+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(\tilde{V}-\tilde{U})\upsilon}{2F}\\\
&\hskip
56.9055pt-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{\tilde{V}\upsilon}{F}\Bigg{\\}}(\upsilon,\chi_{-}(\upsilon))d\upsilon,\vspace{0.3
cm}\\\
\tilde{U}(\tau,\zeta)&=\begin{cases}\displaystyle\bigintssss_{0}^{\tau}\Bigg{\\{}\dfrac{\tilde{U}-\tilde{V}}{2\upsilon}+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(\tilde{U}-\tilde{V})\upsilon}{2F}\\\
\hskip
14.22636pt-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{\tilde{U}\upsilon}{F(\upsilon)}\Bigg{\\}}(\upsilon,\chi_{+}(\upsilon))d\upsilon,\\\
\hskip 341.43306pt\zeta\geq\bar{\chi}(\tau),\vspace{0.3 cm}\\\
b_{1}(\tau_{1})+\displaystyle\bigintssss_{\tau_{1}}^{\tau}\Bigg{\\{}\dfrac{\tilde{U}-\tilde{V}}{2\upsilon}+\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2-\upsilon^{2}\Bigg{]}\dfrac{(\tilde{U}-\tilde{V})\upsilon}{2F}\\\
\hskip
14.22636pt-\Bigg{[}\dfrac{\left(i+\frac{H^{2}}{\gamma^{2}}\right)\bigg{[}i^{2}p^{\prime\prime}(\rho)+\kappa_{0}^{2}n^{2}(1-p^{\prime}(\rho))\bigg{]}\gamma_{w(\upsilon)}^{2}f(w(\upsilon))}{2i^{2}(\upsilon)w^{2}(\upsilon)}+2(1-\upsilon^{2})\Bigg{]}\dfrac{\tilde{U}\upsilon}{F(\upsilon)}\Bigg{\\}}(\upsilon,\chi_{+}(\upsilon))d\upsilon,\\\
\hskip 341.43306pt0\leq\zeta<\bar{\chi}(\tau).\end{cases}\end{cases}$ (4.43)

It can be observed that $(\tilde{U},\tilde{V})$ satisfies the inequalities
(4.38) for $k\geq 1$, therefore a repeated insertion of these inequalities in
the right-hand side of (4.43) asserts that
$|\tilde{U}|\leq\widetilde{M}\left(\dfrac{2}{3}\right)^{k}$ and
$|\tilde{V}|\leq\widetilde{M}\left(\dfrac{2}{3}\right)^{k}$ for some positive
constant $\widetilde{M}>0$ and for any $k$. Therefore, we must have
$\tilde{U}=\tilde{V}=0$, which proves the uniqueness of the solution.

Finally, we note by (4.13), (4.15) and (4.16) that the initial boundary value
problem (4.5), (4.11) is equivalent to the initial boundary value problem
(4.19), (4.20). Therefore, we have the following result in the partial
hodograph plane:

######  Theorem 4.1.

Under the assumption (4.12), the boundary value problem (4.5) and (4.11) have
a unique classical solution in the neighbourhood of the point
$B^{\prime}(0,r_{2})$ in the region $A^{\prime}B^{\prime}C^{\prime}$.

##  5 Solutions in the physical plane

We now recover a solution to our main problem in the physical plane by
utilizing the solution obtained in Theorem 4.1 with the help of an inverse
transformation.

###  5.1 Inversion

In view of (4.1) one has

$\displaystyle\dfrac{\partial x}{\partial
t}=\dfrac{\theta_{y}}{J},\leavevmode\nobreak\ \leavevmode\nobreak\
\dfrac{\partial y}{\partial t}=-\dfrac{\theta_{x}}{J},\leavevmode\nobreak\
\leavevmode\nobreak\ \dfrac{\partial x}{\partial
r}=\dfrac{\sin\omega\omega_{y}}{J},\leavevmode\nobreak\ \leavevmode\nobreak\
\dfrac{\partial y}{\partial r}=-\dfrac{\sin\omega\omega_{x}}{J},$ (5.1)

where $J$ is the Jacobian defined in (4.1). Therefore, a straightforward
calculation leads us to

$\displaystyle\begin{cases}\theta_{x}=t\sin r(W+Z)-\sqrt{1-t^{2}}\cos
r(W-Z),\vspace{0.2 cm}\\\ \theta_{y}=-t\cos r(W+Z)-\sqrt{1-t^{2}}\sin
r(W-Z),\vspace{0.2 cm}\\\
\varpi_{x}=-\dfrac{\hat{F}_{1}(t)}{4i^{2}(t)w^{2}(t)t}\left(t\sin
r(W-Z)-\sqrt{1-t^{2}}\cos r(W+Z)\right),\vspace{0.2 cm}\\\
\varpi_{y}=\dfrac{\hat{F}_{1}(t)}{4i^{2}(t)w^{2}(t)t}\left(t\cos
r(W-Z)+\sqrt{1-t^{2}}\sin r(W+Z)\right).\end{cases}$ (5.2)

Therefore, (5.1) implies

$\displaystyle\begin{cases}x_{t}=-\dfrac{t\cos r(W+Z)+\sqrt{1-t^{2}}\sin
r(W-Z)}{4FWZ},\vspace{0.2 cm}\\\ y_{t}=-\dfrac{t\sin r(W+Z)-\sqrt{1-t^{2}}\cos
r(W-Z)}{4FWZ},\vspace{0.2 cm}\\\
x_{r}=\dfrac{1}{4t\sqrt{1-t^{2}}WZ}\left(t\cos r(W-Z)+\sqrt{1-t^{2}}\sin
r(W-Z)\right),\vspace{0.2 cm}\\\
y_{r}=\dfrac{1}{4t\sqrt{1-t^{2}}WZ}\left(t\sin r(W-Z)-\sqrt{1-t^{2}}\cos
r(W+Z)\right).\end{cases}$ (5.3)

Since

$\displaystyle\dfrac{dx(t,r_{-}(t))}{dt}=x_{t}+x_{r}\dfrac{dr_{-}(t)}{dt},\leavevmode\nobreak\
\leavevmode\nobreak\
\dfrac{dy(t,r_{-}(t))}{dt}=y_{t}+y_{r}\dfrac{dr_{-}(t)}{dt}$

where
$\dfrac{dr_{-}(t)}{dt}=\lambda_{-}(t)=\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}$.

Clearly

$\displaystyle\begin{cases}\dfrac{dx(t,r_{-}(t))}{dt}=-\dfrac{\sqrt{1-t^{2}}\sin
r+t\cos r}{2F(t)Z(t,r)},\\\ \dfrac{dy(t,r_{-}(t))}{dt}=-\dfrac{t\sin
r-\sqrt{1-t^{2}}\cos r}{2F(t)Z(t,r)}.\end{cases}$ (5.4)

Let $\Omega^{\prime}:=\\{(t,r)|0\leq t\leq\delta,\bar{r}(t)-\delta\leq
r\leq\bar{r}(t)\\}$ be the image of the region $\Omega$ in the $(t-r)$ plane.
Then we can use (5.4) to define $\hat{x}=x(\hat{t},\hat{r})$ for any point
$(\hat{t},\hat{r})\in\Omega^{\prime}$ given by:

$\displaystyle
x(\hat{t},\hat{r})=\begin{cases}\hat{\theta}^{-1}(\tilde{r})-\displaystyle\int_{0}^{\hat{t}}\dfrac{t(\sqrt{1-t^{2}}\sin
r_{-}(t;\hat{t},\hat{r}))+t\cos
r_{-}(t;\hat{t},\hat{r})}{2F(t)Z(t;r_{-}(t;\hat{t},\hat{r}))}dt,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \hat{r}\leq
r(\hat{t}),\\\
\psi^{-1}(\tilde{\theta}^{-1}(\tilde{r}))-\displaystyle\int_{\tilde{t}}^{\hat{t}}\dfrac{t(\sqrt{1-t^{2}}\sin
r_{-}(t;\hat{t},\hat{r}))+t\cos
r_{-}(t;\hat{t},\hat{r})}{2F(t)Z(t;r_{-}(t;\hat{t},\hat{r}))}dt,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\hat{r}>r(\hat{t}),\end{cases}$ (5.5)

where $\hat{\theta}^{-1},\tilde{\theta}^{-1}$ and $\psi^{-1}$ denote the
inverses of $\hat{\theta},\tilde{\theta}$ and $\psi$, respectively with

$\displaystyle
r_{-}(t;\hat{t},\hat{r})=\hat{r}+\displaystyle\int_{t}^{\hat{t}}\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}dt,\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{r}=\hat{r}+\displaystyle\int_{0}^{\hat{t}}\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}dt,\leavevmode\nobreak\
\leavevmode\nobreak\
r(\hat{t})=r_{2}-\displaystyle\int_{0}^{\hat{t}}\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}dt,$
(5.6)

where the numbers $\tilde{t}$ and $\tilde{r}$ can be determined by

$\displaystyle\tilde{r}=r_{2}+\displaystyle\int_{0}^{\tilde{t}}\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}dt,\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{r}=r_{2}+\displaystyle\int_{\tilde{t}}^{\hat{t}}\dfrac{t^{2}\sqrt{1-t^{2}}}{F(t)}dt.$
(5.7)

Note that the intersection point of the negative characteristic
$r=r_{-}(t;\hat{t},\hat{r})$ and positive characteristic $r=r_{+}(t;0,r_{2})$
is denoted by the point $(\tilde{t},\tilde{r})$.

One can define the value $\hat{y}=y(\hat{t},\hat{r})$ in a similar fashion,
which is given by

$\displaystyle
y(\hat{t},\hat{r})=\begin{cases}\varphi(\hat{\theta}^{-1}(\tilde{r}))-\displaystyle\int_{0}^{\hat{t}}\dfrac{t(t\sin
r_{-}(t;\hat{t},\hat{r})-\sqrt{1-t^{2}}\cos
r_{-}(t;\hat{t},\hat{r}))}{2F(t)Z(t;r_{-}(t;\hat{t},\hat{r}))}dt,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \hat{r}\leq
r(\hat{t}),\\\
\tilde{\theta}^{-1}(\tilde{r})-\displaystyle\int_{\tilde{t}}^{\hat{t}}\dfrac{t(t\sin
r_{-}(t;\hat{t},\hat{r})-\sqrt{1-t^{2}}\cos
r_{-}(t;\hat{t},\hat{r}))}{2F(t)Z(t;r_{-}(t;\hat{t},\hat{r}))}dt,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\hat{r}>r(\hat{t}).\end{cases}$ (5.8)

Noting the expressions of $y_{t},y_{r},x_{t}$ and $x_{r}$, it is clear that

$j:=\dfrac{\partial(x,y)}{\partial(t,r)}=\dfrac{t}{4F(t)W(t,r)Z(t,r))}\neq
0,\leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\ t\in(0,\delta],$

which shows that the transfromation $(t,r)\longrightarrow(x,y)$ is injective
for $t\in(0,\delta]$ and therefore one should have $t=t(x,y)$ and $r=r(x,y)$
so that we can define the functions

$\displaystyle\theta=\theta(t(x,y),r(x,y)),\leavevmode\nobreak\
\leavevmode\nobreak\ \varpi=\sqrt{1-t^{2}(x,y)},$ (5.9)

We claim that the constructed functions $(\theta,\varpi)(x,y)$ is actually a
solution to the boundary value problem for the system (2.28).

###  5.2 Verification of solutions in the physical plane

In view of expression of $\bar{\partial}_{+}$ and (5.2), we have

$\displaystyle\bar{\partial}_{+}\theta$
$\displaystyle=\cos\alpha\theta_{x}+\sin\alpha\theta_{y}$
$\displaystyle=\cos\alpha[t\sin r(W+Z)-\sqrt{1-t^{2}}\cos
r(W-Z)]+\sin\alpha[-t\cos r(W+Z)-\sqrt{1-t^{2}}\sin r(W-Z)]$

and

$\displaystyle\dfrac{4i^{2}(\varpi)w^{2}(\varpi)\cos\omega}{F_{1}(\varpi)}\bar{\partial}_{+}\varpi$
$\displaystyle=\cos\alpha\varpi_{x}+\sin\alpha\varpi_{y}$
$\displaystyle=\bigg{\\{}\sin\alpha\left(t\cos r(W-Z)+\sqrt{1-t^{2}}\sin
r(W+Z)\right)$ $\displaystyle\hskip 56.9055pt-\cos\alpha\left(t\sin
r(W-Z)-\sqrt{1-t^{2}}\cos r(W+Z)\right)\bigg{\\}}.$

Therefore, we have

$\displaystyle\bar{\partial}_{+}\theta+\dfrac{4i^{2}(\varpi)w^{2}(\varpi)\cos\omega}{F_{1}(\omega)}\bar{\partial}_{+}\varpi$
$\displaystyle=2Z\left(\sqrt{1-t^{2}}\cos(\alpha-\theta)-t\sin(\alpha-\theta)\right)=0.$

Clearly, $\theta(x,y)$ and $\varpi(x,y)$ satisfy first equation of (2.28).
Similarly, we can show that the functions $\theta(x,y)$ and $\varpi(x,y)$
satisfy the second equation of (2.28), which shows that $\theta(x,y)$ and
$\varpi(x,y)$ is the solution of (2.28) with boundary data (3.2).

In view of $\dfrac{dw}{dq}<0$ and $\dfrac{d\gamma}{dq}>0$, we must have
$\dfrac{dw}{d\gamma}<0$ which implies $\gamma:=\gamma(w)$. Thus we combine
$w=w(\varpi)$ and (5.9) together with the relations
$u=\dfrac{w\gamma_{w}\cos\theta}{\gamma\sin\omega},v=\dfrac{w\gamma_{w}\sin\theta}{\gamma\sin\omega}$
to define the functions $(w,u,v)(x,y)$ satisfying

$\displaystyle w(x,y)=w(\varpi(x,y)),\leavevmode\nobreak\
u=\dfrac{w(\varpi(x,y))\gamma_{w(\varpi(x,y))}\cos\theta(x,y)}{\gamma(w(\varpi(x,y)))\varpi(x,y)},\leavevmode\nobreak\
v=\dfrac{w(\varpi(x,y))\gamma_{w(\varpi(x,y))}\sin\theta(x,y)}{\gamma(w(\varpi(x,y)))\varpi(x,y)}$
(5.10)

is the solution to our main problem. We summarise the main result of our
article in the following theorem:

######  Theorem 5.1.

Let $\widehat{AB}:x=\psi(y)$ be a locally concave characteristic curve of
relativistic magnetohydrodynamics flow and $\widehat{BC}:y=\varphi(x)$ be a
given sonic curve satisfying $\psi^{\prime\prime}(y_{B})<0$ and
$\varphi(x_{B})=1$. Further,
$(\theta,\varpi)|_{\widehat{BC}}=(\hat{\theta}(x),\hat{\varpi}(x))$ and
$(\theta,\varpi)|_{\widehat{AB}}=(\tilde{\theta}(x),\tilde{\varpi}(x))$ are
given boundary data on these curves such that $\theta^{\prime}(x_{B})<0$ and
$(\varphi^{\prime}\sin\hat{\theta}+\cos\hat{\theta})(x_{B})>0$, then the
degenerate boundary value problem for the system (2.28) admits a unique
classical solution $(\theta,\varpi)$ in the angular domain $ABC$ in the
neighbourhood of the point $B$.

######  Remark 1.

Note that, in the Newtonian limit $(c\longrightarrow\infty)$ the system (1.7)
actually reduces to the 2-D steady magnetogasdynamics system which is given by

$\displaystyle\begin{cases}(\rho u)_{x}+(\rho v)_{y}=0,\\\ \left(\rho
u^{2}+p+\dfrac{1}{2}\kappa_{0}^{2}\rho^{2}\right)_{x}+\left(\rho
uv\right)_{y}=0,\\\ \left(\rho uv\right)_{x}+\left(\rho
v^{2}+p+\dfrac{1}{2}\kappa_{0}^{2}\rho^{2}\right)_{y}=0.\end{cases}$

Therefore, our results extend the results available in the work of Li and Hu
[1] to the magnetogasdynamics system as well especially when the pressure law
satisfies $p=A\rho^{\gamma}$. In particular, if we choose $\kappa_{0}=0$, one
can recover the main result from [1].

## Acknowledgments

The second author (TRS) expresses his gratitude towards SERB, DST, Government
of India (Ref. No. CRG/2022/006297) for its financial support through the core
research grant.

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