# Reflectionless wave propagation on shallow water with variable bathymetry
and current

Semyon M. Churilov1  Yury A. Stepanyants2,3<EMAIL_ADDRESS>1
Institute of Solar-Terrestrial Physics of the Siberian Branch of the Russian
Academy of Sciences, PO Box 291, Irkutsk, 664033, Russia 2 Scool of Sciences,
University of Southern Queensland, West St., Toowoomba, QLD, 4350, Australia
3 Department of Applied Mathematics, Nizhny Novgorod State Technical
University
n.a. R. E. Alekseev, 24 Minin St., Nizhny Novgorod, 603950, Russia

###### Abstract

In the linear approximation, we study a one-dimensional problem of the
reflectionless wave propagation on a surface of a shallow duct with the
spatially varying water depth, duct width, and current. We show that both
global and bounded exact solutions describing reflectionless propagation in
opposite directions of long waves of arbitrary shape exist for the particular
variations of these parameters. A general analysis of the problem is
illustrated by a few solutions constructed for the specific cases of spatial
profiles of the flow velocity. The results obtained can be of interest to
mitigate the possible impact of waves on ships, marine engineering
constructions, and human activity in the coastal zones.

###### keywords:

##  1 Introduction

Wave propagation in an inhomogeneous ocean is one of the important and topical
problems of physical oceanography and fluid mechanics, in general. However, it
is a difficult problem from the mathematical point of view in the general
statement due to the nonlinearity, dispersion, and strong inhomogeneity.
Analytical solutions either in the exact or approximate forms are possible in
exceptional cases; see, for example, (Stoker, 1957; Sretensky, 1977) in
application to water waves. In these books, exact solutions for small-
amplitude surface waves on water of arbitrary depth with the sloping bottom
are presented. When the characteristic length of a small-amplitude wave is
short in comparison with the typical scale of inhomogeneity, the well-known
WKB approximation can be used for the description of quasi-monochromatic
waves. However, in some special cases of media parameters variation, exact
solutions can be derived for waves of arbitrary length both in the linear and
even in the nonlinear cases. There is a vast volume of publications on this
theme in the past two decades; here we refer only to the most recent and
relevant, others can be found in the cited publications (Didenkulova et al.,
2009; Didenkulova & Pelinovsky, 2009; Didenkulova, 2009; Dobrokhotov &
Tirozzi, 2010; Dobrokhotov et al., 2011; Grimshaw et al., 2010b; Didenkulova &
Pelinovsky, 2011; Dobrokhotov et al., 2013; Pelinovsky et al., 2017a, b,
2019). The outcomes of theoretical predictions for shallow-water waves were
validated in the numerical modelling (see, for example, (Choi et al., 2008;
Vlasenko, 1987; Pudjaprasetya et al., 2021)). The important feature of such
solutions is they describe the reflectionless wave propagation when the wave
energy is transmitted through the inhomogeneous zone the most effectively
without the energy losses on the wave reflection. Such situations can be the
most dangerous from the point of view of their possible impact on ships and
engineering constructions in the coastal zones.

The problem of reflectionless wave propagation in inhomogeneous media has a
long history; it was studied for monochromatic waves in plasma physics
(Ginzburg, 1970; Petrukhin et al., 2020), acoustics (Brekhovskih, 1980),
solids (Clements & Rogers, 1974), fluids (Magaard, 1962; Vlasenko, 1987), etc.
A more general analysis of reflectionless propagation of water waves of
different shapes was presented in the cited above papers by Didenkulova,
Pelinovsky, Dobrokhotov et al. for the particular configurations of a bottom
profile. In the cited publications, wave propagation was studied basically for
the fluid without mean currents (in the paper by Dobrokhotov et al., 2013, a
particular case of a variable mean current was taken into account). However,
in many cases, influence of currents on wave propagation can be significant
and therefore, should be studied from the general point of view.

In this paper, we fill the gap and study in the long-wave approximation the
reflectionless linear wave propagation in canals with the variable depth,
width, and currents. The problem is formulated in terms of the velocity
potential which allows us to find spatially varying mean flow profile, as well
as variation of the duct width and depth, which admit the reflectionless wave
propagation in ducts. We show that both the global solutions defined on the
entire $x$-axis and bounded solutions defined only on the semi-infinite
$x$-interval do exist and present particular examples. Then, we find the
conditions which secure the existence of global solutions. The results
obtained are illustrated graphically.

To solve this problem, we employ the transformation technique which has been
used in many publications (see, for example, Grimshaw et al., 2010a;
Pelinovsky et al., 2017b, and references therein). Such a technique is one of
the particular cases of the general methods of reduction of linear
differential equations to some reference equations which can be solved
analytically. These methods develop starting back from the works of Euler,
Laplace, Poisson, Darboux, et al., until now (see, for example, Bluman, 1983;
Varley & Semyr, 1988; Chirkunov et al., 2014; Kaptsov & Mirzaokhmedov, 2021,
and references therein).

The paper is organized as follows. In Section 2, we formulate the basic
equations and derive conditions under which the wave propagation is
reflectionless. The Section ends with a simple condition that allows us to
distinguish a particular class of global solutions. Section 3 is devoted to
the general analysis of the properties of solutions belonging to another class
among which there are both global and bounded solutions. Particular solutions
of this class are obtained and analyzed in Section 4. In Section 5, we explore
the existence of smooth global solutions belonging to the second class and
derive sufficient conditions of their existence. The possibility to construct
global solutions by matching bounded solutions is considered in Section 6.
Finally, Section 7 contains a discussion of results and concluding remarks.
The technical aspects of matching solutions are considered in Appendix.

##  2 Problem statement and general analysis

Let us consider the propagation of surface waves on a shallow water flow in a
duct with a width $W(x)$ gradually varying along the direction of the flow
with the spatially varying depth $H(x)$ and bottom profile $z_{B}=B(x)$ as
shown in figure 1 (note that the water surface is not horizontal in the
presence of spatially inhomogeneous flow). In a stationary flow, the current
velocity $U(x)$ is related to the duct parameters by the flux conservation
law:

$\Phi\equiv U(x)H(x)W(x)=\mbox{const}$ (1)

and the Bernoulli equation:

$\frac{1}{2}\,U^{2}(x)+g\Bigl{[}H(x)+B(x)\Bigr{]}={\rm const},$ (2)

where $g$ is the acceleration due to gravity. It is easy to see that by
appropriately choosing profiles $B(x)$ and $W(x)$ one can provide the desired
(and independent) variation along the canal of the flow velocity $U(x)$ and
the speed of long waves $c(x)=\sqrt{gH(x)}$; this velocities are assumed to be
positive everywhere.

Figure 1: (Color online.) Sketch of the flow configuration in the vertical
plane.

In the shallow-water theory, the linearised Euler equation is:

$\displaystyle\frac{\partial\tilde{u}}{\partial
t}+\displaystyle\frac{\partial(U\tilde{u})}{\partial
x}=-g\displaystyle\frac{\partial\eta}{\partial x},$ (3)

where $\tilde{u}(x,t)$ is the perturbation of the longitudinal component of
fluid velocity, and $\eta(x,t)$ is the deviation of a free surface from the
equilibrium state. Then the mass balance equation is:

$\displaystyle\frac{\partial S}{\partial
t}+\displaystyle\frac{\partial}{\partial x}\Bigl{[}S(U+\tilde{u})\Bigr{]}=0,$
(4)

where $S(x,t)=[H(x)+\eta(x,t)]W(x)$ is a part of the duct cross-section
occupied by water. Linearising this equation with respect to small
perturbations $\eta$ and $\tilde{u}$ and taking into account equation (1) with
$\Phi\neq 0$, we obtain:

$\displaystyle\frac{\partial\eta}{\partial
t}+\displaystyle\frac{1}{W}\,\displaystyle\frac{\partial}{\partial
x}\Bigl{[}W(U\,\eta+H\,\tilde{u})\Bigr{]}\equiv\,\displaystyle\frac{\partial\eta}{\partial
t}+HU\displaystyle\frac{\partial}{\partial
x}\left(\displaystyle\frac{\eta}{H}+\displaystyle\frac{\tilde{u}}{U}\right)=0.$
(5)

Introducing the velocity potential $\varphi$ such that
$\tilde{u}=\partial\varphi/\partial x$, we integrate the linearised Euler
equation (3) and present $\eta$ in terms of $\varphi$; then, a combination of
equations (3) and (5) gives the equation describing propagation of long wave:

$\left(\displaystyle\frac{\partial}{\partial
t}+U\displaystyle\frac{\partial}{\partial
x}-2U\displaystyle\frac{c^{\prime}}{c}\right)\left(\displaystyle\frac{\partial\varphi}{\partial
t}+U\displaystyle\frac{\partial\varphi}{\partial
x}\right)=\displaystyle\frac{g}{W}\,\displaystyle\frac{\partial}{\partial
x}\left(WH\,\displaystyle\frac{\partial\varphi}{\partial x}\right),$

where prime stands for the derivative with respect to $x$. This equation
remains valid in the limiting case when $U(x)\equiv 0$. Using equation (1)
with $\Phi\neq 0$, we rewrite it in the form:

$\left(\displaystyle\frac{\partial}{\partial
t}+U\displaystyle\frac{\partial}{\partial
x}-2U\displaystyle\frac{c^{\prime}}{c}\right)\left(\displaystyle\frac{\partial\varphi}{\partial
t}+U\displaystyle\frac{\partial\varphi}{\partial
x}\right)=c^{2}U\displaystyle\frac{\partial}{\partial
x}\left(\displaystyle\frac{1}{U}\displaystyle\frac{\partial\varphi}{\partial
x}\right).$ (6)

This equation represents a basis for our analysis presented below but it does
not admit the transition to the limiting case $U(x)\equiv 0$.

For what follows, it is convenient to present the velocity potential in the
form: $\varphi(x,t)=a(x)\psi(x,t)$, where the ‘‘amplitude factor’’ $a(x)$ will
be defined later. Such a well-known trick allows one to reduce a linear
equation with variable coefficients to one of the reference equations (see,
e.g., (Grimshaw et al., 2010a; Pelinovsky et al., 2017b) and references
therein). After substitution of expression for $\varphi$ in equation (6), we
arrive at the following equation for $\psi(x,t)$:

$\begin{array}[]{l}\displaystyle\frac{\partial^{2}\psi}{\partial
t^{2}}+(U^{2}-c^{2})\,\displaystyle\frac{\partial^{2}\psi}{\partial
x^{2}}+2U\,\displaystyle\frac{\partial^{2}\psi}{\partial t\partial
x}+2U\,\left(\displaystyle\frac{a^{\prime}}{a}-\displaystyle\frac{c^{\prime}}{c}\right)\displaystyle\frac{\partial\psi}{\partial
t}\\\ \\\
\phantom{wwa}+\,\left[2(U^{2}-c^{2})\,\displaystyle\frac{a^{\prime}}{a}+(U^{2}+c^{2})\,\displaystyle\frac{U^{\prime}}{U}-2U^{2}\displaystyle\frac{c^{\prime}}{c}\right]\displaystyle\frac{\partial\psi}{\partial
x}+Z(x)\psi=0,\end{array}$ (7)

where $Z(x)$ is defined by the equation:

$a(x)\,Z(x)=(U^{2}-c^{2})a^{\prime\prime}+a^{\prime}\Bigl{[}(U^{2}+c^{2})(\ln
U)^{\prime}-2U^{2}(\ln c)^{\prime}\Bigr{]}.$ (8)

Let us consider such a class of fluid flows for which $Z(x)\equiv 0$. Then,
from (8) we obtain the exact differential equation for $a(x)$:

${\rm d}\Bigl{(}\ln a^{\prime}\Bigr{)}={\rm
d}\left(\ln\,\displaystyle\frac{c^{2}U}{c^{2}-U^{2}}\right).$

The first integral of this equation is:

$\displaystyle\frac{{\rm d}a}{{\rm
d}x}=D\displaystyle\frac{\,c^{2}U}{c^{2}-U^{2}},$ (9)

where $D$ is a constant of integration.

Let us consider now a model equation:

$\left(\displaystyle\frac{\partial}{\partial
t}+s_{1}(x)\,\displaystyle\frac{\partial}{\partial
x}+F(x)\right)\left(\displaystyle\frac{\partial}{\partial
t}+s_{2}(x)\,\displaystyle\frac{\partial}{\partial x}\right)f(x,t)=0,$ (10)

where $s_{1}(x)$, $s_{2}(x)$, and $F(x)$ are yet undefined functions. One of
the solutions to this equation describes a travelling wave:

$f(x,t)=f_{1}\left(t-\int\displaystyle\frac{{\rm d}x}{s_{2}(x)}\right),$

where $f_{1}(z)$ is an arbitrary function. Expansion of equation (10) leads
to:

$\displaystyle\frac{\partial^{2}f}{\partial
t^{2}}+s_{1}(x)s_{2}(x)\,\displaystyle\frac{\partial^{2}f}{\partial
x^{2}}+\Bigl{[}s_{1}(x)+s_{2}(x)\Bigr{]}\displaystyle\frac{\partial^{2}f}{\partial
t\partial x}$ ${}+F(x)\,\displaystyle\frac{\partial f}{\partial
t}+\Bigl{[}s_{1}(x)s^{\prime}_{2}(x)+F(x)s_{2}(x)\Bigr{]}\displaystyle\frac{\partial
f}{\partial x}=0.$ (11)

Let us compare this equation with (7) provided that $Z(x)\equiv 0$ and
equation (9) is hold. These equations are identical if we define functions
$s_{1}(x)$, $s_{2}(x)$, and $F(x)$ such that:

$s_{1}(x)s_{2}(x)=U^{2}(x)-c^{2}(x),\quad s_{1}(x)+s_{2}(x)=2U(x),$
$F(x)=2U(x)\,\left(\displaystyle\frac{a^{\prime}(x)}{a(x)}-\displaystyle\frac{c^{\prime}(x)}{c(x)}\right)$
(12)

and

$s_{1}(x)s^{\prime}_{2}(x)+F(x)s_{2}(x)$
${}=2\Bigl{[}U^{2}(x)-c^{2}(x)\Bigr{]}\displaystyle\frac{a^{\prime}(x)}{a(x)}+\Bigl{[}U^{2}(x)+c^{2}(x)\Bigr{]}\displaystyle\frac{U^{\prime}(x)}{U(x)}-2U^{2}(x)\,\displaystyle\frac{c^{\prime}(x)}{c(x)}\,.$
(13)

The first two equations (12) are hold if either $s_{1}=U-c$, $s_{2}=U+c$, or
$s_{1}=U+c$, $s_{2}=U-c$. It is easy to see that in both these cases equation
(13) yields (up to an unimportant numerical factor)

$a(x)=\Bigl{[}c(x)U(x)\Bigr{]}^{1/2}.$ (14)

Using this relation, we can rewrite equation (9) in terms of functions $c(x)$
and $U(x)$ (positive values of the roots $c^{1/2}(x)$ and $U^{1/2}(x)$ are
presumed hereafter):

$\displaystyle\frac{{\rm d}(c\,U)}{{\rm d}x}\equiv
c(x)\,\displaystyle\frac{{\rm d}U}{{\rm d}x}+U(x)\,\displaystyle\frac{{\rm
d}c}{{\rm
d}x}=2D\,\displaystyle\frac{c^{5/2}(x)\,U^{3/2}(x)}{c^{2}(x)-U^{2}(x)}.$ (15)

Thus, when the relationships (12), (14) and (9) are hold, equation (6) can be
presented in one of the following forms:

$\displaystyle\left[\displaystyle\frac{\partial}{\partial
t}+\Bigl{(}U-c\Bigr{)}\,\displaystyle\frac{\partial}{\partial
x}+U\left(\displaystyle\frac{U^{\prime}}{U}-\displaystyle\frac{c^{\prime}}{c}\right)\right]\left[\displaystyle\frac{\partial}{\partial
t}+\Bigl{(}U+c\Bigr{)}\displaystyle\frac{\partial}{\partial x}\right]\psi$
(16) $\displaystyle\equiv$
$\displaystyle\left[\displaystyle\frac{\partial}{\partial
t}+\Bigl{(}U+c\Bigr{)}\,\displaystyle\frac{\partial}{\partial
x}+U\left(\displaystyle\frac{U^{\prime}}{U}-\displaystyle\frac{c^{\prime}}{c}\right)\right]\left[\displaystyle\frac{\partial}{\partial
t}+\Bigl{(}U-c\Bigr{)}\displaystyle\frac{\partial}{\partial x}\right]\psi=0.$

The general solution of this equation can be presented as a superposition of
two waves of an arbitrary form travelling with the different velocities:

$\psi(x,t)=\psi_{1}\left(t-\int\displaystyle\frac{{\rm
d}x}{U(x)+c(x)}\right)+\psi_{2}\left(t-\int\displaystyle\frac{{\rm
d}x}{U(x)-c(x)}\right).$ (17)

The independent propagation of each of these waves in the inhomogeneous fluid
is provided by the single condition (9) which relates the fluid and wave
speeds, $U(x)$ and $c(x)$. Therefore, equation (9) when it holds provides the
reflectionless (RL) wave propagation.

When the functions $a(x)$ and $\psi(x,t)$ are found, then we get the velocity
potential $\varphi$ which allows as to find the velocity $u$ and water surface
elevation $\eta$:

$\varphi(x,t)=a(x)\psi(x,t),\qquad u=\frac{\partial\varphi}{\partial
x},\qquad\eta=-\frac{1}{g}\left(\frac{\partial\varphi}{\partial
t}+U\frac{\partial\varphi}{\partial x}\right).$

A wide class of various RL flows can be obtained even in the simplest case
when $D=0$. In such a case, (9) reduces to the condition $a(x)=\mbox{const}$
or, equivalently (see, for example, (Churilov & Stepanyants, 2021)):

$c(x)U(x)=\mbox{const}.$ (18)

The profile of one of these speeds can be specified as an arbitrary continuous
positively-defined function, and the profile of another speed is uniquely
calculated then. It is important to note that, firstly, such a flow is global,
i.e. is defined on the entire $x$-axis, and, secondly, neither the presence
nor absence of critical points in the flow, at which the velocities $U(x)$ and
$c(x)$ become equal, does not affect in any way such properties as smoothness
and boundedness of the velocities.

##  3 Reflectionless flows in the case $D\neq 0$. A general analysis

If $D\neq 0$ being positive or negative constant, singularities appear in the
right-hand side of equation (15) when either $c=U$, or $U=0$, or $c=0$, and
also when $U(x)$ or $c(x)$ grow with no limit when $x$ approaches some point
$x_{*}$. If at least one of these singularities occurs at a finite point
$x=x_{*}$, then several questions arise: (1) about the continuation of the
flow model through this point, (2) about the contribution of the singular
point to the wave reflection and/or absorption processes, and, more broadly,
(3) about the existence of global RL flows. Below we investigate these issues.

We note first that equations (9) and (15) are invariant with respect to the
simultaneous replacement $x\to-x$ and $D\to-D$. For this reason, we introduce
a variable $\xi=Dx$ which remains the same under such a transformation and
rewrite equations (9) and (15) as:

$\displaystyle\frac{{\rm d}a(\xi)}{{\rm
d}\xi}=\displaystyle\frac{c^{2}(\xi)U(\xi)}{c^{2}(\xi)-U^{2}(\xi)}\quad{\rm
and}\quad c(\xi)\displaystyle\frac{{\rm d}U(\xi)}{{\rm
d}\xi}+U(\xi)\displaystyle\frac{{\rm d}c(\xi)}{{\rm
d}\xi}=\displaystyle\frac{2\,c^{5/2}(\xi)U^{3/2}(\xi)}{c^{2}(\xi)-U^{2}(\xi)}\,.$
(19)

In addition, equations (9), (15), and (19) are invariant with respect to the
scaling transformation $U(x)\ \to\ U(x)/c_{s}$, $c(x)\ \to\ c(x)/c_{s}$, where
$c_{s}=\mbox{const}$. In what follows, we will choose the appropriate scale
$c_{s}$. Note also that these equations possess the translational symmetry
which means that if $c(\xi)$ and $U(\xi)$ satisfy equation (19), then
$c(\xi+b)$ and $U(\xi+b)$, where $b$ is an arbitrary constant, satisfy this
equation too.

Let us introduce the dimensionless functions:
$u(\xi)=w^{-1}(\xi)=\Bigl{[}U(\xi)/c(\xi)\Bigr{]}^{1/2}$, noting that
$c(\xi)u(\xi)=\Bigl{[}c(\xi)U(\xi)\Bigr{]}^{1/2}\equiv a(\xi)$. Then, let us
re-write equation (9) in the forms:

$\displaystyle\frac{{\rm d}u}{{\rm
d}\xi}=\displaystyle\frac{u^{2}}{1-u^{4}}-\displaystyle\frac{u}{c(\xi)}\,\displaystyle\frac{{\rm
d}c(\xi)}{{\rm d}\xi}$ (20)

and

$\displaystyle\frac{{\rm d}}{{\rm d}\xi}\,\ln
a(\xi)=\displaystyle\frac{u(\xi)}{1-u^{4}(\xi)}=\displaystyle\frac{w^{3}(\xi)}{w^{4}(\xi)-1}\,.$
(21)

The right-hand side of equation (21) is positive when $u<1$ ($w>1$) and is
negative when $u>1$ ($w<1$), therefore, $a(\xi)$ increases monotonically in
the subcritical flow and decreases monotonically in the supercritical flow. If
we assume that $c(\xi)$ is a bounded function at any finite $\xi$, i.e.
$0<c(\xi)<\infty$, then the singularities at $U=0$ and $U=\infty$ can be
reached only asymptotically when $\xi\to-\infty$. Similarly, under the
assumption of the boundness of $U(\xi)$, singularities at $c=0$ and $c=\infty$
can be reached only asymptotically when $\xi\to+\infty$. In all these cases,
the right-hand side of equation (21) vanishes, hence the derivative in the
left-hand side vanishes too. As the result of this, when $|\xi|\to\infty$,
function $a(\xi)$ either asymptotically approaches some constant (which may be
zero, in particular) or grows slower than an exponential function, e.g., in a
power-type manner, $a(\xi)\sim|\xi|^{\alpha}$, where $\alpha>0$. In contrast
to this, a singularity at $U=c$ ($u=w=1$) can be achieved (as will be shown
below) at a finite point $\xi$.

To conclude this section, we consider the behaviour of the functions $U(\xi)$
and $c(\xi)$ in the neighborhood of a critical point where $U=c$. Assume that
$\xi=0$ is a critical point, and the normalisation constant $c_{s}$ is chosen
such that $U(0)=c(0)=1$. Let us set

$c(\xi)=1+s(\xi),\qquad U(\xi)=1+v(\xi),$

where $s(\xi)\to 0$ and $v(\xi)\to 0$ when $\xi\to 0$. After substitution in
the second equation of (19), we obtain:

$\Bigl{(}s-v\Bigr{)}\Bigl{(}s^{\prime}+v^{\prime}\Bigr{)}=1+2s+v+O(s^{2}+v^{2}),$

where prime stands for the derivative with respect to $\xi$. From here, we see
that $(s-v)$ and $(s+v)$ are power-type functions of $\xi$:

$(s-v)\sim\xi^{\alpha},\qquad(s+v)\sim\xi^{\beta},$ (22)

where $\alpha+\beta=1$, $\alpha>0$, $\beta>0$.

Since $\xi=0$ is a branching point and $s(\xi)$ and $v(\xi)$ are real
functions, the cases $\xi>0$ and $\xi<0$ should be considered separately.

Case A. If $\alpha=\beta=1/2$, then we readily find for $\xi\geq 0$:

$c(\xi)=1+s_{+}\xi^{1/2}+O(\xi),\qquad U(\xi)=1+v_{+}\xi^{1/2}+O(\xi),$ (23)

where $s_{+}^{2}=v_{+}^{2}+2$.

For $\xi\leq 0$, we find:

$c(\xi)=1+s_{-}(-\xi)^{1/2}+O(\xi),\qquad U(\xi)=1+v_{-}(-\xi)^{1/2}+O(\xi),$
(24)

where $s_{-}^{2}=v_{-}^{2}-2$. Here $s_{\pm}$ and $v_{\pm}$ are constants
which can have any sign, depending on the branch of the solution (subcritical
or supercritical). Moreover, it is not necessary that the both velocities have
singularities at $\xi=0$. In particular, in the domain where $\xi\geq 0$, the
flow velocity can be regular ($v_{+}=0$), while the wave velocity can be
regular ($s_{-}=0$) in the domain where $\xi\leq 0$ .

If $\alpha\neq\beta$ then, we can present $s(x)$ and $u(x)$ as:

$s(\xi)=s_{a\pm}|\xi|^{\alpha}+s_{b\pm}|\xi|^{\beta}+\dots,\qquad
v(\xi)=v_{a\pm}|\xi|^{\alpha}+v_{b\pm}|\xi|^{\beta}+\dots,$

where the indices plus (minus) pertain to the regions $\xi\geq 0$ ($\xi\leq
0$).

Case B. Consider the case when $0<\alpha<\beta$, then from the condition
$(s+v)\sim|\xi|^{\beta}$, we obtain:

$s_{a\pm}+v_{a\pm}=0,\qquad 2\beta s_{a\pm}(s_{b\pm}+v_{b\pm})=\pm 1.$ (25)

Case C. In another case, $\alpha>\beta>0$, from the condition
$(s-v)\sim|\xi|^{\alpha}$, we obtain:

$s_{b\pm}=v_{b\pm},\qquad 2\beta s_{b\pm}(s_{a\pm}-v_{a\pm})=\pm 1.$ (26)

Note that if $\alpha\neq\beta$, then both velocities inevitably have a
singularity in the point $\xi=0$.

##  4 Examples of reflectionless flows

###  4.1 Currents with a constant wave speed

As a first example, consider a current with a constant wave speed
$c(\xi)\equiv 1$. From equation (19) we find:

$\displaystyle\frac{U^{\prime}(\xi)}{U^{3/2}(\xi)}\Bigl{[}1-U^{2}(\xi)\Bigr{]}=2\
\ \Longrightarrow\ \ U^{2}(\xi)+3(\xi-\xi_{0})U^{1/2}(\xi)+3=0.$

Using the substitution $v(\xi)=U^{1/2}(\xi)$, we arrive at the algebraic
equation for $v$:

$F(v)\equiv v^{4}+3(\xi-\xi_{0})v+3=0.$ (27)

This equation has two positive real roots for $\xi<\xi_{*}=\xi_{0}-4/3$, one
of them $v_{1}(\xi)<1$ and another one $v_{2}(\xi)>1$ (see figure 2(a)). The
roots merge into one double root $v=1$ (corresponding to the critical point
where $U=c=1$) when $\xi=\xi_{*}$. If $\xi>\xi_{*}$, then the polynomial (27)
does not have real roots. Without loss of generality, we can set $\xi_{0}=4/3$
($\xi_{*}=0$) bearing in mind that all the other solutions can be obtained by
a shift in $\xi_{0}$. Then for $0\leq-\xi\ll 1$ we have:

$v_{1,2}(\xi)=1\mp\left(-\frac{\xi}{2}\right)^{1/2}-\frac{\xi}{12}+\dots,\quad
U_{1,2}(\xi)=1\mp\Bigl{(}-2\xi\Bigr{)}^{1/2}-\frac{2}{3}\,\xi\dots,$ (28)

and for $\xi\to-\infty$ the asymptotic solutions originated for each root are
(we are recalling that $a(\xi)=c(\xi)\,v(\xi)$):

$\begin{array}[]{ll}a_{1}(\xi)=v_{1}(\xi)\approx\Bigl{(}4/3-\xi\Bigr{)}^{-1},&\
\ a_{2}(\xi)=v_{2}(\xi)\approx\Bigl{(}4-3\xi\Bigr{)}^{1/3},\\\
U_{1}(\xi)\approx\Bigl{(}4/3-\xi\Bigr{)}^{-2},&\ \
U_{2}(\xi)\approx\Bigl{(}4-3\xi\Bigr{)}^{2/3}.\end{array}$ (29)

Figure 2: (Color online.) (a) – Roots of equation (27) for $\xi_{0}-\xi=2/3$
(line 1); $\xi_{0}-\xi=4/3$ (line 2); $\xi_{0}-\xi=7/3$ (line 3). (b) Roots of
equation (31) for $\xi-\xi_{0}=1$ (line 1); $\xi-\xi_{0}=4/3$ (line 2);
$\xi-\xi_{0}=1.75$ (line 3).

The subcritical solution ($U_{1}$) and supercritical solution ($U_{2}$) are
defined only for the negative $\xi\leq 0$. Profiles $v(\xi)$ are shown in
figure 3(a) for different values of $\xi_{0}$. With the known dependences of
$U_{1,2}(\xi)$ and fixed water depth $H_{0}$, we can calculate the dependences
of the duct width $W(\xi)$ and bottom profile $z_{B}=B(\xi)$. Using equations
(1) and (2), we obtain:

$W(\xi)\sim U^{-1}(\xi),\qquad B(\xi)=B(0)-\frac{1}{2g}U^{2}(\xi).$ (30)

Figure 3: (a) Solutions of equation (27) for the different values of $\xi_{0}$
in the subcritical (lover part of the frame) and supercritical (upper part of
the frame) regions. (b) The same for the solutions of equation (31), but
graphs for subcritical flows ($s>1$) are in the upper part, whereas graphs for
supercritical flows ($s<1$) are in the lower part of the frame.

###  4.2 Currents with a constant flow speed

A very similar features has a flow with a constant current speed, $U(\xi)=1$.
In this case equation (19) reduces to:

$\displaystyle\frac{c^{\prime}(\xi)}{c^{5/2}(\xi)}\Bigl{[}c^{2}(\xi)-1\Bigr{]}=2\
\ \Longrightarrow\ \ c^{2}(\xi)-(\xi-\xi_{0})c^{3/2}(\xi)+\frac{1}{3}=0,$

or equivalently to:

$S(s)\equiv s^{4}-(\xi-\xi_{0})s^{3}+\frac{1}{3}=0,$ (31)

where $s(\xi)=c^{1/2}(\xi)$. There are two positive real roots for
$\xi>\xi_{*}=\xi_{0}+4/3$, $s_{1}(\xi)>1$ and $s_{2}(\xi)<1$ corresponding to
the subcritical and supercritical flows, respectively. These roots merge at
$s=1$ when $\xi=\xi_{*}$. The polynomial (31) does not have real roots if
$\xi<\xi_{*}$ (see figure 2(b). Setting $\xi_{0}=-4/3$ ($\xi_{*}=0$), we find
for $0\leq\xi\ll 1$:

$s_{1,2}(\xi)=1\pm\left(\frac{\xi}{2}\right)^{1/2}+\frac{5}{12}\,\xi+\dots,\quad
c_{1,2}(\xi)=1\pm\Bigl{(}2\xi\Bigr{)}^{1/2}+\frac{4}{3}\,\xi+\dots$ (32)

Then, for $\xi\to+\infty$, we have (here $a(\xi)=s(\xi)\,U(\xi)$):

$\begin{array}[]{ll}a_{1}(\xi)=s_{1}(\xi)\approx\xi+4/3,&\
a_{2}(\xi)=s_{2}(\xi)\approx\Bigl{(}3\xi+4\Bigr{)}^{-1/3},\\\
c_{1}(\xi)\approx\Bigl{(}\xi+4/3\Bigr{)}^{2},&\
c_{2}(\xi)\approx\Bigl{(}3\xi+4\Bigr{)}^{-2/3}.\end{array}$ (33)

Note that, in contrast to the previous example, the solutions are defined only
on the positive semi-axis $\xi\geq 0$, because at $\xi=0$ function $U(\xi)$ is
regular. The profiles of $s(\xi)$ are shown in figure 3(b) for different
values of $\xi_{0}$. From the known dependences of
$c_{1,2}(\xi)=\sqrt{gH(\xi)}$, we immediately derive the dependence of water
depth $H_{1,2}(\xi)=c^{2}_{1,2}(\xi)/g$. Then, from equation (1) with the
fixed current flow $U_{0}$, we can calculate the dependences of the duct width
$W(\xi)$ and bottom profile $z_{B}=B(\xi)$:

$W(\xi)\sim H^{-1}(\xi)\sim c^{-2}(\xi),\qquad B(\xi)=-H(\xi)+\mbox{const.}$
(34)

In the next examples, we consider flows in which both velocities are unlimited
in general.

###  4.3 Currents with functionally related velocity profiles

Consider now such class of fluid flows in which $U(\xi)$ and $c(\xi)$ are
related by the power type equation:

$U(\xi)=Vc^{p}(\xi),\quad V={\rm const}>0,\ \ p={\rm const}.$ (35)

We assume that $p\neq\pm 1$. When $p=-1$, we arrive at the equation
$c(x)U(x)=\mbox{const.}$ which corresponds to the case $D=0$ (see the end of
Section 2). When $p=1$, we obtain a flow with the constant Froude number,
$\mbox{Fr}\equiv U(\xi)/c(\xi)=V$, but with the varying $c(\xi)$ and $U(\xi)$:

$c(\xi)=C{\rm e}^{\mu\xi},\quad U(\xi)=Vc(\xi),\ \
\mu=\displaystyle\frac{V^{1/2}}{1-V^{2}}\,.$ (36)

As will be shown below, these are precisely such solutions that serve as the
separatrices, separating solutions with the fundamentally different
behaviours.

Substituting $U(\xi)$ from (35) into equation (19) and integrate it, we
obtain:

$\displaystyle\frac{2}{1-p}\,c^{(1-p)/2}+\displaystyle\frac{2V^{2}}{3(1-p)}\,c^{3(p-1)/2}=\displaystyle\frac{2V^{1/2}}{1+p}\Bigl{(}\xi-\xi_{0}\Bigr{)}.$

Introduction of new variable $u>0$ such that $u^{2}\equiv U/c=Vc^{p-1}$,
yields:

$u^{4}+3D_{p}(\xi-\xi_{0})\,u+3=0,\qquad
D_{p}=\displaystyle\frac{p-1}{p+1}\,.$ (37)

In terms of the variable $\zeta=D_{p}\xi$, this equation takes the same form
as equation (27), therefore, its solutions are the same, $u_{1,2}(\zeta)\equiv
v_{1,2}(\zeta)$ (see figure 3(a)). All other variables $c$, $U$ and $a$
expressed in terms of $u$ are:

$c=c_{*}\,u^{\frac{2}{p-1}},\qquad U=u^{2}\,c=c_{*}\,u^{\frac{2p}{p-1}},\qquad
a=u\,c=c_{*}\,u^{\frac{p+1}{p-1}},$ (38)

where $c_{*}=V^{\frac{1}{1-p}}$ is the value of velocities at the critical
point $\xi=\xi_{*}$, $U(\xi_{*})=c(\xi_{*})=c_{*}$. As one can see, the
behaviour of these functions depend on the parameter $p$. Indeed, if we set
$\zeta_{0}\equiv D_{p}\xi_{0}=4/3$, then we find with the help of equations
(29) and (38) that when $\zeta\to-\infty$, the asymptotic behaviour of the
functions of interest are:

$\begin{array}[]{lll}c_{1}\approx
c_{*}\Bigl{(}-\zeta\Bigr{)}^{\frac{2}{1-p}},&\ U_{1}\approx
c_{*}\Bigl{(}-\zeta\Bigr{)}^{\frac{2p}{1-p}},&\ a_{1}\approx
c_{*}\Bigl{(}-\zeta\Bigr{)}^{\frac{1+p}{1-p}},\\\ \\\ c_{2}\approx
c_{*}\Bigl{(}-3\zeta\Bigr{)}^{\frac{2}{3(p-1)}},&\ U_{2}\approx
c_{*}\Bigl{(}-3\zeta\Bigr{)}^{\frac{2p}{3(p-1)}},&\ a_{2}\approx
c_{*}\Bigl{(}-3\zeta\Bigr{)}^{\frac{p+1}{3(p-1)}}.\end{array}$ (39)

From this expressions we derive:

$\begin{array}[]{lll}c_{1},\ U_{1},\ a_{1}\ \to 0,&\ c_{2},\ U_{2},\ a_{2}\
\to\infty,&\ \ {\rm for}\ p>1,\\\ c_{1},\ U_{1},\ a_{1}\ \to\infty,&\ c_{2},\
U_{2},\ a_{2}\ \to 0,&\ \ {\rm for}\ 0<p<1,\\\ c_{1},\ a_{1}\to\infty,\ U_{1}\
\to 0,&\ c_{2},\ a_{2}\to 0,\ U_{2}\ \to\infty,&\ \ {\rm for}\ -1<p<0,\\\
c_{1}\to\infty,\ U_{1},\ a_{1}\ \to 0,&\ c_{2}\to 0,\ U_{2},\ a_{2}\
\to\infty,&\ \ {\rm for}\ p<-1.\end{array}$ (40)

In the neighbourhood of the critical point ($\zeta\to 0_{-}$) we have:

$c_{1,2}(\zeta)=c_{*}\left[1\mp\displaystyle\frac{(-2\zeta)^{1/2}}{p-1}+\dots\right],\qquad
U_{1,2}(\zeta)=c_{*}\left[1\mp\displaystyle\frac{p}{p-1}\Bigl{(}-2\zeta\Bigr{)}^{1/2}+\dots\right].$

As an example, let us consider the flow in a duct of constant width $W$. In
accordance with the flux conservation law (1), such a flow corresponds to
$p=-2$, and formulae (38) yield:

$U(\xi)=c_{*}\,u^{4/3}(\xi),\qquad c(\xi)=c_{*}\,u^{-2/3}(\xi),\qquad
H(\xi)=\displaystyle\frac{c^{2}_{*}}{g}\,u^{-4/3}(\xi).$

The profiles of velocities for $\xi_{*}=0$ and $c_{*}=1$ are shown in figure
4.

Figure 4: (Color online.) The $c(\xi)$ and $U(\xi)$ profiles of the RL flows
in a duct of a constant width: (a) subcritical flow and (b) supercritical
flow.

It should be noted that in all models with $D\neq 0$ considered above,
solutions are defined only on the semi-axis $\xi$, and the domains of their
definition are limited because a singularity with $c=U$ occurs at a finite
value of $\xi=\xi_{*}$. As the velocity $U(\xi_{*})\neq 0$, it is evident that
the flow goes over the point $\xi=\xi_{*}$ into the domain where
$\xi>\xi_{*}$. One of the versions of such flow model is discussed below in
Section 6.2.

###  4.4 A special class of flows with the exponential variation of wave speed
$c(\xi)$

The flows of this class differ significantly from those considered above. Let
$c(\xi)=C\exp(\mu\xi)$. Then equation (20) takes the form:

$\displaystyle\frac{{\rm d}u}{{\rm
d}\xi}=\displaystyle\frac{u^{2}}{1-u^{4}}-\mu u.$ (41)

This equation, in contrast to the considered above, has a stationary point
(the null-isocline) $u(\xi)=u_{0}=\mbox{const.}$ in which its right-hand side
vanishes. The value of $u_{0}$ is a positive root of the following equation
(cf. with the last equation (36) which reduces to (42) if we set
$V=u_{0}^{2}$):

$u_{0}^{4}+\displaystyle\frac{u_{0}}{\mu}-1=0.$ (42)

It is easy to see that for $\mu>0$, the value of $u_{0}<1$, i.e. corresponds
to the subcritical flows, whereas for $\mu<0$, the value of $u_{0}>1$, which
corresponds to supercritical flows.

Substituting $u=u_{0}+v$, where $|v|\ll u_{0}$, in equation (41), we obtain
after linearisation with respect to $v$

$\displaystyle\frac{{\rm d}v}{{\rm
d}\xi}=\left[\displaystyle\frac{2u_{0}}{1-u_{0}^{4}}+\displaystyle\frac{4u_{0}^{5}}{(1-u_{0}^{4})^{2}}-\mu\right]v=\mu(1+4\mu
u_{0}^{3})v.$

Solutions to this equation,

$v(\xi)=v_{0}\exp(\gamma\xi),\quad\gamma=\mu(1+4\mu u_{0}^{3}),\quad
v_{0}=\mbox{const},$ (43)

vanish when $\xi\to-\infty$. This is obvious when $\mu>0$ and, therefore,
$\gamma>0$. When $\mu<0$, we obtain with the help of equation (42) that
$\gamma$ is still positive:

$\gamma=\mu\Bigl{[}1+4(\mu/u_{0}-1)\Bigr{]}=4\mu^{2}/u_{0}-3\mu>0.$

This means that the stationary point $u_{0}$ is unstable due to small
perturbation $v(\xi)$ grows with $\xi$.

As a result, the upper half-plane $(\xi,\,u)$ is split into three strips
(separated from each other by the straight lines $u=u_{0}$ and $u=1$), in
which the behavior of solutions to equation (41) is significantly different
(see figure 5).

Figure 5: (Color online.) Solutions of equation (41). Frame (a) pertains to
positive $\mu=0.4$ ($u_{0}=0.3907$); frame (b) pertains to negative $\mu=-0.4$
($u_{0}=1.4705$). Global solutions are depicted by blue lines.

If $\mu>0$, then $0<u_{0}<1$, and solutions to equation (41) are such as shown
in figure 5(a). In the lower strip, $0<u<u_{0}$, any solution $u(\xi)$
decreases monotonically approaching $u_{0}$ from below when $\xi\to-\infty$
and zero from the top when $\xi\to+\infty$, according to $u\approx
u_{m+}\exp(-\mu\xi)$. Respectively, in the same limit we have:

$U(\xi)\equiv u^{2}(\xi)\,c(\xi)\approx Cu^{2}_{m+}e^{-\mu\xi},\quad
a(\xi)=\Bigl{[}c(\xi)\,U(\xi)\Bigr{]}^{1/2}\to C\,u_{m+}={\rm const}.$ (44)

In the middle strip, $u_{0}<u\leq 1$, any solution $u(\xi)$ grows
monotonically from $u_{0}$ when $\xi\to-\infty$ to $u=1$ at some $\xi_{*}$.
Near this limiting coordinate $\xi_{*}$ the asymptotic solution is

$u(\xi)=1-\left(\frac{\xi_{*}-\xi}{2}\right)^{1/2}-\frac{1-8\mu}{12}\Bigl{(}\xi_{*}-\xi\Bigr{)}+\dots$
(45)

In the upper strip any solution tends to infinity as $u\approx
u_{m-}\exp(-\mu\xi)$ when $\xi\to-\infty$ and decreases with $\xi$ approaching
unity, when $\xi$ tends to some $\xi_{+}$ as per the formula:

$u(\xi)=1+\left(\frac{\xi_{+}-\xi}{2}\right)^{1/2}-\frac{1-8\mu}{12}\Bigl{(}\xi_{+}-\xi\Bigr{)}+\dots$
(46)

If $\mu<0$, then $u_{0}>1$. Solutions to equation (41) are shown in figure
5(b). In the lower strip, $0<u<1$, all solutions $u(\xi)$ monotonically
increase from $u\approx u_{M-}\exp(-\mu\xi)$ when $\xi\to-\infty$ attaining
$u=1$ at some finite $\xi$ according to the dependence similar to (45). In the
middle strip, $1<u<u_{0}$, solutions $u(\xi)$ monotonically decrease from
$u_{0}$ when $\xi\to-\infty$ to $u=1$ achieving this value at some finite
$\xi$ as per the dependence similar to (45). In the upper strip, $u>u_{0}$,
solutions $u(\xi)$ monotonically increase from $u_{0}$ when $\xi\to-\infty$,
to $u\approx u_{M+}\exp(-\mu\xi)$ when $\xi\to+\infty$; the dependences
similar to (44) are held in this limit.

It should be noted that in the lower strip for $\mu>0$ and in the upper strip
for $\mu<0$ solutions describing flows are global, i.e., they are defined on
the entire $\xi$-axis, whereas in other strips, solutions are defined only to
the left of some finite point $\xi_{*}$ which is different for each particular
realization.

##  5 Global solutions and conditions of their existence

As follows from the analysis presented above for the specific flow models, in
the majority of cases solutions for the RL current profile are applicable only
on the limited spatial interval, because the profiles of the duct depth,
width, or current become either singular in certain points, or diverge at the
infinity. Therefore, one of the important questions is whether it is possible
to find such conditions when solutions for the profiles are globally defined
on the entire $x$-axis. Such a problem was solved in the case when there is no
current (Pelinovsky et al., 2017b). We provide the solution for the case when
the current is taken into account.

Let us recall that in the case of the exponential variation of the wave speed
$c(\xi)$ considered above, the null-isoclines are horizontal
($u_{0}=\mbox{const.}$). Therefore, they serve as separatrices separating the
global solutions from those that are defined on the bounded $x$-interval.
Under a non-exponential variation of $c(\xi)$, the null-isocline
$u=u_{0}(\xi)$, on which the right-hand side of equation (20) vanishes, is
still described by equation (42) but now

$\mu(\xi)\equiv\frac{{\rm d}}{{\rm d}\xi}\ln{c(\xi)}\neq\mbox{const}.$ (47)

Therefore, it is not horizontal and is intersected on the $(\xi,\,u)$-plane by
some solutions of equation (20). In what follows, we will assume that $c(\xi)$
either increases monotonically or decreases monotonically determining
accordingly the sign of $\mu(\xi)\neq 0$. As follows from equation (20), when
$\mu(\xi)>0$, then only subcritical solutions ($u(\xi)<1$) can be global,
whereas when $\mu(\xi)<0$, only supercritical solutions ($u(\xi)>1$) can be
global. Let us consider first flows with $\mu(\xi)>0$.

If function $c(\xi)$ grows faster than exponentially, then $\mu(\xi)$
monotonically increases with no limit (see figure 6(a)). The null-isocline
$u=u_{0}(\xi)$ also increases monotonically from zero when $\xi\to-\infty$ to
unity when $\xi\to+\infty$ (see line 1 in figure 7(a)). Solutions of equation
(20) either intersect the null-isocline and are global, or lay entirely above
it and are bounded from the right by some $\xi=\xi_{b}$. (Note that the null-
isocline has a positive derivative, whereas solutions below it have negative
derivatives. Therefore, all such solutions intersect the null-isocline when
$\xi$ decreases.)

Figure 6: The qualitative dependence of $\mu(\xi)$ for the flows in which
$c(\xi)$ grows faster than exponentially (a) and slower than exponentially
(b).

If function $c(\xi)$ grows slower than exponentially, then $\mu(\xi)$ has a
maximum (possibly, even more than one, but it does not matter, in principle)
vanishing when $\xi\to\pm\infty$ (see figure 6(b)). The null-isocline
$u=u_{0}(\xi)$ has qualitatively the same shape (see line 1 in figure 7(b)).
All solutions lying above or crossing its right (descending) slope (for
example, like line 3 in figure 7(b)) are, obviously, bounded from the right by
some $\xi=\xi_{b}$. Let us consider now at what conditions global solutions
such as shown by line 4 in figure 7(b) can exist.

Figure 7: (Color online.) Solutions of equation (20) in the case when $c(\xi)$
grows faster than exponentially (a) and slower than exponentially (b). Line 1
depicts the null-isoclines in both frames. Lines 2 in frame (a) illustrate
examples of global solutions; other non-numbered blue lines illustrate bounded
solutions which exist for $\xi\leq\xi_{b}$ with the individual value of
$\xi_{b}$ for each line. In frame (b) one can see several non-numbered blue
lines depicting bounded solutions; line 2 depicts a solution tangent to the
null-isocline; line 3 is an $\xi$-bounded solution partially passing below the
null-isocline; and line 4 shows a global solution.

Global solutions must be below $u_{0}(\xi)$ for sufficiently big
$\xi\geq\xi_{M}$. Let $\mu(\xi_{M})\ll 1$, then, as follows from equation
(42), $u_{0}(\xi)\approx\mu(\xi)\ll 1$ and it decreases with $\xi$ when
$\xi\geq\xi_{M}$. Therefore, equation (20) simplifies and becomes:

$\displaystyle\frac{{\rm d}u(\xi)}{{\rm d}\xi}\approx
u^{2}(\xi)-\mu(\xi)\,u(\xi)\equiv
u^{2}(\xi)-\displaystyle\frac{u(\xi)}{c(\xi)}\,\displaystyle\frac{{\rm
d}c(\xi)}{{\rm d}\xi}\,.$

This equation reduces to:

$\displaystyle\frac{1}{(cu)^{2}}\,\displaystyle\frac{{\rm d}}{{\rm
d}\xi}(cu)=\displaystyle\frac{1}{c(\xi)}\,,$

and then, can be readily solved:

$\displaystyle\frac{1}{c(\xi)\,u(\xi)}=\displaystyle\frac{1}{c(\xi_{M})\,u(\xi_{M})}-\displaystyle\int\limits^{\infty}_{\xi_{M}}\displaystyle\frac{{\rm
d}y}{c(y)}+\displaystyle\int\limits^{\infty}_{\xi}\displaystyle\frac{{\rm
d}y}{c(y)}.$ (48)

For the convergence of the integrals, function $c(\xi)$ must grow faster than
linearly, for example, as $\xi^{1+\varepsilon}$ where $\varepsilon>0$.
Therefore, when $\xi\to+\infty$, then $c(\xi)u_{0}(\xi)\approx
c^{\prime}(\xi)\to+\infty$. Thus, the existence of global solutions is secured
(i) by the convergence of the integrals in equation (48) and (ii) by the
inequality:

$a_{0}^{-1}=\displaystyle\frac{1}{c(\xi_{M})\,u(\xi_{M})}-\displaystyle\int\limits^{\infty}_{\xi_{M}}\displaystyle\frac{{\rm
d}y}{c(y)}>0.$ (49)

The latter condition can be easily satisfied by the choice of sufficiently
small $u(\xi_{M})$. As the result, on the solution which passes through the
point $\Bigl{(}\xi_{M},\,u(\xi_{M})\Bigr{)}$, the product $c(\xi)u(\xi)$ goes
to a finite limit $a_{0}>0$. Due to this, $u(\xi)<u_{0}(\xi)$ when
$\xi\geq\xi_{M}$, i.e. this solution is global. Thus, global subcritical flows
with $D\neq 0$ and smooth profiles $c(\xi)$ and $U(\xi)$ do exist under the
conditions formulated above. Figure 8 demonstrates a qualitative variation of
the flow parameters along the canal.

Figure 8: (Color online.) Qualitative dependences of typical variation of
parameters in the case of a subcritical global flow along the canal. All
variables are normalised to some reference values and presented in
dimensionless form.

To study flows with $\mu(\xi)<0$, it is convenient to use the variable
$w(\xi)=1/u(\xi)$ (see equation (21) with $a(\xi)=c(\xi)/w(\xi)$). In this
case, all the reasoning presented above remains almost the same. In
particular, if $c(\xi)$ decreases faster than exponentially, the null-isocline
and solutions on the half-plane $(\xi,\,w)$ look qualitatively the same as in
figure 7(a) (with the replace $u$ by $w$). All solutions intersecting the
null-isocline are global, whereas those which are not intersecting are bounded
on the right by some value $\xi_{b}$. If function $c(\xi)$ decreases slower
than exponentially, the solutions are similar to those shown in figure 7(b).
For the existence of global solutions, it is sufficient that $c(\xi)$
decreases with $\xi$ as $\xi^{-\varepsilon-1/3}$ when $\xi\to+\infty$ and for
some point on the solution $\Bigl{(}\xi_{M},\,w(\xi_{M})\Bigr{)}$, where
$\xi_{M}\gg 1$, the following inequality holds (cf. (49)):

$a^{3}(\xi_{M})\equiv\left[\displaystyle\frac{c(\xi_{M})}{w(\xi_{M})}\right]^{3}\equiv\left[c(\xi_{M})u(\xi_{M})\right]^{3}>\displaystyle\int\limits^{\infty}_{\xi_{M}}\\!c^{3}(y){\rm
d}y.$ (50)

This inequality can be easily satisfied by the choice of a sufficiently small
$w(\xi_{M})$ (big $u(\xi_{M})$). In the conclusion, we remind that when
$\mu(\xi)<0$, all global flows are supercritical ($U>c$).

##  6 Matching solutions for reflectionless flows

One can show that global solutions can exist only for rather artificial
velocity profiles, $c(\xi)$ and $U(\xi)$, whereas in majority of cases these
profiles are defined only on the semi-axis, either on the left or on the right
of some point $\xi_{*}$. Using solutions for $D\neq 0$ defined on the
different sides of the selected common point, one can try to construct a
global solution by matching two particular solutions. However, the matching
procedure depends on the character of the common point, whether it is regular
($U\neq c$) or critical, where $U=c$. Below we consider both these
possibilities.

###  6.1 Matching solutions in the regular point

Let us assume that there are two solutions for the fluid flow satisfying
equation (9). One of them describing velocities $c_{-}(x)$ and $U_{-}(x)$ and
defined for $x\leq x_{m}$, and another solution describing velocities
$c_{+}(x)$ and $U_{+}(x)$ and defined for $x\geq x_{m}$. Assume that
$c_{+}(x_{m})=c_{-}(x_{m})=c_{m}$ and $U_{+}(x_{m})=U_{-}(x_{m})=U_{m}$ where
$x=x_{m}$ is a regular point, i.e. such that $U_{m}\neq c_{m}$. As follows
from equation (9), transition through the point $x=x_{m}$ is reflectionless if
$D_{+}=D_{-}$. If $D_{+}\neq D_{-}$, then function $a^{\prime}(x)$ is
discontinuous, and the coefficient $Z\sim\delta(x-x_{m})$ in equation (7),
where $\delta(x)$ is the Dirac delta-function. In such a case, as will be
shown in Appendix, additional waves will appear on left or right of point
$x_{m}$.

To illustrate the possibility to construct a global solution, let us match two
solutions at a regular point. To this end, we chose two solutions with equal
constants $D_{\pm}$ obeying the equation (35) at different parameters $p$ and
$V$. In such a case, the profiles $c(\xi)$ and $U(\xi)$ depend on the same
coordinate $\xi$. Let us set

$\begin{array}[]{lll}c_{-}(\xi)=c_{a}(\xi),&\
U_{-}(\xi)=U_{a}(\xi)=V_{a}c_{a}^{p_{a}}(\xi),&\ \xi\leq\xi_{m};\\\ \\\
c_{+}(\xi)=c_{b}(\xi),&\ U_{+}(\xi)=U_{b}(\xi)=V_{b}c_{b}^{p_{b}}(\xi),&\
\xi\geq\xi_{m};\\\ \\\ c_{a}(\xi_{m})=c_{b}(\xi_{m}),&\
U_{a}(\xi_{m})=U_{b}(\xi_{m}).&\end{array}$ (51)

Then, functions $u_{a,b}(\xi)=\Bigl{[}U_{a,b}(\xi)/c_{a,b}(\xi)\Bigr{]}^{1/2}$
satisfy equations similar to equation (37):

$u_{a,b}^{4}+3D_{a,b}(\xi-\xi_{a,b})\,u_{a,b}+3=0,\qquad
D_{a,b}=\displaystyle\frac{p_{a,b}-1}{p_{a,b}+1}\,.$ (52)

Functions $U_{a,b}(\xi)$ and $c_{a,b}(\xi)$ can be presented in terms of
$u_{a,b}(\xi)$ using the expressions similar to equations (38). The existence
of the functions $u_{a}(\xi)$ and $u_{b}(\xi)$, as well as continuity of
$u(\xi)$ (i.e., $u_{a}(\xi_{m})=u_{b}(\xi_{m})=u_{m}$) are secured by the
following relations:

$D_{a}(\xi_{m}-\xi_{a})=D_{b}(\xi_{m}-\xi_{b})<-\frac{4}{3},\qquad\left(\displaystyle\frac{V_{a}}{u_{m}^{2}}\right)^{p_{b}-1}=\left(\displaystyle\frac{V_{b}}{u_{m}^{2}}\right)^{p_{a}-1}.$
(53)

For the resulting composite flow to be global, i.e. covering the entire
$\xi$-axis, the parameters $p_{a,b}$ should be chosen such that $D_{a}$ and
$D_{b}$ have different signs, namely,

$D_{a}>0\ \ \Rightarrow\ \ |p_{a}|>1,\qquad D_{b}<0\ \ \Rightarrow\ \
|p_{b}|<1.$ (54)

It is important to remember that the value and sign of $p$ significantly
impact the behavior of functions $c(\xi)$ and $U(\xi)$ when $|\xi|\to\infty$
(see equations (39) and (40)). To find the profiles of the flow and wave
velocities, in addition to $p_{a}$, $p_{b}$ and $\xi_{m}$, we must set, for
example, the flow parameters $V_{a}$ and $\xi_{a}>\xi_{m}+4/(3D_{a})$ (see
equation (53)), select from the roots of equation (52) the one that
corresponds to the nature of the flow (subcritical or supercritical), and
using equations (53), find $V_{b}$ and $\xi_{b}$,

$V_{b}=\Bigl{[}u_{m}^{2(p_{a}-p_{b})}\,V_{a}^{p_{b}-1}\Bigr{]}^{\frac{1}{p_{a}-1}},\qquad\xi_{b}=\xi_{m}-\displaystyle\frac{D_{a}}{D_{b}}\Bigl{(}\xi_{m}-\xi_{a}\Bigr{)}.$

It should be emphasized that, with the exception of the trivial case
$p_{a}=p_{b}$, the velocity profiles have a kink at $x=x_{m}$, i.e. their
derivatives are discontinuous. Indeed, calculating with the help of (38) the
logarithmic derivatives of $c(\xi)$ and $U(\xi)$, we find:

Figure 9: The result of matching of RL flow (35) for different parameters: (a)
– $p_{a}=5$, $V_{a}=1$, $p_{b}=1/5$; (b) – $p_{a}=-5$, $V_{a}=1$,
$p_{b}=-1/5$. Solid lines pertain to subcritical flows ($U<c$), dashed line –
to supercritical flows ($U>c$).

$\displaystyle\frac{c^{\prime}_{a,b}}{c_{a,b}}=\displaystyle\frac{2}{p_{a,b}-1}\,\displaystyle\frac{u^{\prime}_{a,b}}{u_{a,b}},\qquad\displaystyle\frac{U^{\prime}_{a,b}}{U_{a,b}}=\displaystyle\frac{2p_{a,b}}{p_{a,b}-1}\,\displaystyle\frac{u^{\prime}_{a,b}}{u_{a,b}},\qquad\left(\displaystyle\frac{a^{\prime}(\xi)}{a(\xi)}\right)_{a,b}=\displaystyle\frac{p_{a,b}+1}{p_{a,b}-1}\,\displaystyle\frac{u^{\prime}_{a,b}}{u_{a,b}}\,.$

On the other hand, from equation (19) we get:

$\left(\displaystyle\frac{a^{\prime}(\xi)}{a(\xi)}\right)_{a,b}=\displaystyle\frac{c_{a,b}^{3/2}\,U_{a,b}^{1/2}}{c_{a,b}^{2}-U_{a,b}^{2}}=\displaystyle\frac{u_{a,b}}{1-u_{a,b}^{4}}\,,$

so that

$\left(\displaystyle\frac{u^{\prime}_{a,b}}{u_{a,b}}\right)_{\xi_{m}}=\displaystyle\frac{p_{a,b}-1}{p_{a,b}+1}\,\displaystyle\frac{u_{m}}{1-u_{m}^{4}}\equiv\displaystyle\frac{p_{a,b}-1}{p_{a,b}+1}\,\displaystyle\frac{(U_{m}/c_{m})^{1/2}}{1-(U_{m}/c_{m})^{2}}\,,$

and then

$\begin{array}[]{l}\left(\displaystyle\frac{u^{\prime}_{b}}{u_{b}}-\displaystyle\frac{u^{\prime}_{a}}{u_{a}}\right)_{\xi_{m}}=\left(\displaystyle\frac{U^{\prime}_{b}}{U_{b}}-\displaystyle\frac{U^{\prime}_{a}}{U_{a}}\right)_{\xi_{m}}=-\left(\displaystyle\frac{c^{\prime}_{b}}{c_{b}}-\displaystyle\frac{c^{\prime}_{a}}{c_{a}}\right)_{\xi_{m}}\\\
\\\
\phantom{\left(\displaystyle\frac{u^{\prime}_{b}}{u_{b}}-\displaystyle\frac{u^{\prime}_{a}}{u_{a}}\right)_{\xi_{m}}}=\displaystyle\frac{2(p_{b}-p_{a})}{(1+p_{a})(1+p_{b})}\,\displaystyle\frac{(U_{m}/c_{m})^{1/2}}{1-(U_{m}/c_{m})^{2}}\neq
0\,.\end{array}$ (55)

In figure 9 one can see the dependences $u$, $c$ and $U$ on $\xi$ for two
versions of matching of RL flows in the regular point (the velocities $c$ and
$U$ are normalised to some reference values and presented in the dimensionless
form).

###  6.2 Matching solutions in the critical point

Let us consider now the possibility of constructing a global solution by
matching solutions with $D\neq 0$ defined on different sides of a common
critical point assuming that $c(\xi)\equiv 1$. Let us set for certainty
$\xi_{0}=4/3$ in equation (27), then the condition of no reflection (9) is
satisfied only for $\xi<0$, and a corresponding RL flow is defined only on
this semi-axis. However, on the physical axis $x=\xi/D$ such a flow is defined
for $x\leq 0$ if $D=D^{(-)}>0$, and for $x\geq 0$ if $D=D^{(+)}<0$. From these
solutions defined on the different semi-axes it is possible to construct a
composite solution defined on the entire $x$-axis and describing the
transition of the sub-critical flow into supercritical flow (see Fig. 10a), or
supercritical flow to the subcritical flow (see Fig. 10(b). If
$D^{(+)}=-D^{(-)}$, then, given the asymptotic expansions (28), the velocity
profile $U(x)$ looks quite ‘‘smooth’’ (if the infinite derivative at $x=0$ is
ignored). The theory of wave propagation on shallow water flows is well-
developed including the cases when the current speed passes through the
critical point having finite derivatives of the velocity profiles $U(x)$ and
$c(x)$ (see, for example Churilov et al., 2017; Churilov & Stepanyants, 2021,
and references therein). However, in the discussed case, a question about the
contribution of the singular (sic!) point $x=0$ to the possible generation of
additional waves still remains open. Potentially some other waves than the
incident and transmitted waves can emerge from the critical point. In
addition, at least partial absorption of an incident wave can occur in the
critical point.

In a similar way, one can construct a matched solution for a flow with a
constant velocity with the only difference being that in this case the
solution on the left ($x\leq 0$) must have $D=D^{(-)}<0$, whereas the solution
on the right ($x\geq 0$) must have $D=D^{(+)}>0$.

Figure 10: Composite solutions of equation (27) describing the transition from
the subcritical to supercritical flow (frame a) and in the opposite direction
(frame b).

##  7 Conclusion

In this paper, we have shown that reflectionless (RL) propagation of long
surface waves in a duct or in a channel with the variable width $W(x)$ and
depth $H(x)$ in the presence of a spatially varying flow is quite possible if
the velocities of the flow $U(x)$ and waves $c(x)=\sqrt{gH(x)}$ are
interrelated by equation (15). It is important to emphasize that this single
equation cannot define both functions $c(x)$ and $U(x)$ simultaneously.
Therefore, it is necessary either to define one of them or set some additional
relation between them. In the result, we arrive at the two infinite classes of
RL flows distinguishing in that whether or not the constant $D$ in equation
(15) is equal to zero.

If $D=0$, the product $c(x)U(x)$ is constant along the canal (see condition
(18)). All RL flows of this class have an interesting feature, the wider is
the canal, the higher is the flow speed $U(x)$. Indeed, condition (18) is
equivalent to $U(x)H^{1/2}(x)=\mbox{const}$. Then, from the flux conservation
equation (1) it follows that $W(x)H^{1/2}(x)=\mbox{const}$, so that $U(x)\sim
W(x)$. Note that the same relation between the water depth and duct width was
obtained by Pelinovsky et al. (2017b) to describe global RL solutions for the
case when there is no current.

This class of RL flows has an advantage that we can attribute arbitrarily the
$x$-dependence to any of two velocities $c(x)$ or $U(x)$, as well as the value
to their product. Doing so, we can also attribute to $c(x)$ and $U(x)$ such
important properties as continuity, differentiability, boundedness, and so on.
Moreover, we can define the flow on the entire $x$-axis, i.e. globally. As a
result, the considered class contains currents with a wide variety of
properties, including subcritical ($U<c$) and supercritical ($U>c$) flows,
flows with transition from the subcritical to the supercritical regime and/or
vice versa, as well as currents with other characteristics.

In contrast, when $D\neq 0$, the properties of RL flows can vary in a much
more restricted range. First of all, the currents of this class can be either
subcritical, or supercritical, because a transition through the critical point
(where $U=c$) most likely violates the RL property of the flow. Respectively,
the majority of RL flows are defined only on a certain interval of the
$x$-axis, namely, on a ray bounded by the critical point $x_{*}$ at which
$c(x_{*})=U(x_{*})$ (see Section 4). Solutions are defined either on the right
or on the left of $x_{*}$, depending on the parameters of the problem, e.g.,
on the sign of $D$.

In global flows of this class, as shown in Section 5, the profiles of $c(x)$
and $U(x)$ can be smooth functions of $x$ only if one of them, say $c(x)$, is
varying (increasing or decreasing) fast enough. Global RL flows can also be
constructed by matching bounded flows with equal values of $D$ and overlapping
domains of definition in some regular point (where $c\neq U$) common to both
flows. However, at this point the profiles of $c(x)$ and $U(x)$ have
necessarily kinks (jumps in their derivatives – see Section 6.1). With this in
mind, one can construct a piece-wise smooth flow by matching solutions many
times in different points. Matching the currents with different values of $D$
is also possible, but this will inevitably lead to the occurrence of
additional waves, i.e. to the loss of the RL property of the flow (see
Appendix).

The importance of the RL flows is that long surface waves of arbitrary form in
the linear approximation can independently propagate in the opposite
directions of the $x$-axis. This provides the most efficient energy
transmission over a long distance through the inhomogeneous environment. The
results obtained in this paper generalise earlier derived results (see Choi et
al., 2008; Didenkulova et al., 2009; Didenkulova & Pelinovsky, 2009;
Didenkulova, 2009; Grimshaw et al., 2010a; Didenkulova & Pelinovsky, 2011) for
the cases when a water flow plays an important role and must be taken into
consideration. However, we did not consider here two- or three-dimensional
effects which were studied to a degree in some of the cited papers. This
interesting issue can be investigated elsewhere later. The results obtained
can be of interest to mitigate the possible impact of waves on ships and
marine engineering constructions in the coastal zones.

Acknowledgements. S.C. was financially supported by the Ministry of Science
and Higher Education of the Russian Federation. Y.S. acknowledges the funding
of this study provided by the grant No. FSWE-2020-0007 through the State task
program in the sphere of scientific activity of the Ministry of Science and
Higher Education of the Russian Federation, and the grant No. NSH-2485.2020.5
provided by the President of Russian Federation for the State support of
leading Scientific Schools of the Russian Federation.

##  Appendix A Wave scattering in the matching point

In this Appendix, we show that if $D_{+}\neq D_{-}$, then a composite flow
will not be reflectionless. The ensemble of waves in the vicinity of the
regular matching point is determined by whether the matching flows are
subcritical ($U_{m}<c_{m}$) or supercritical ($U_{m}>c_{m}$). Solution of Eq.
(7) obviously is continuous,
$\psi_{+}(x_{m},t)=\psi_{-}(x_{m},t)\equiv\psi_{m}(t)$, and its time
derivatives are continuous too. By integrating equation (7) over $x$ within
the interval from $x_{m}-\varepsilon$ to $x_{m}+\varepsilon$ and denoting
$[f]=f(x_{m}+\varepsilon)-f(x_{m}-\varepsilon)$ at $\varepsilon\to+0$, we
obtain:

$\left[\displaystyle\frac{\partial\psi}{\partial
x}\right]+\displaystyle\frac{1}{a_{m}}\left[\displaystyle\frac{{\rm d}a}{{\rm
d}x}\right]\psi_{m}(t)=0,\qquad
a_{m}=a(x_{m})\equiv\Bigl{(}c_{m}U_{m}\Bigr{)}^{1/2}.$

Using Eq. (9), we arrive at the matching conditions:

$\Bigl{[}\psi\Bigr{]}=0,\qquad\left[\displaystyle\frac{\partial\psi}{\partial
x}\right]+\displaystyle\frac{D_{+}-D_{-}}{c_{m}^{2}-U_{m}^{2}}\,a_{m}c_{m}\psi_{m}(t)=0.$
(56)

i) Let us assume that the flows are subcritical in the vicinity of point
$x_{m}$, ($U_{m}<c_{m}$), then solution to the left of point $x_{m}$ can
contain the incident and reflected waves, whereas solution to the right of
this point contains only a transmitted wave (see Eq. (17)):

$\begin{array}[]{l}\psi_{-}(x,t)=\psi_{1}\left(t-\displaystyle\int\limits_{x_{m}}^{x}\displaystyle\frac{{\rm
d}z}{U_{-}(z)+c_{-}(z)}\right)+\psi_{2}\left(t+\displaystyle\int\limits_{x_{m}}^{x}\displaystyle\frac{{\rm
d}z}{c_{-}(z)-U_{-}(z)}\right),\\\ \\\
\psi_{+}(x,t)=\psi_{3}\left(t-\displaystyle\int\limits_{x_{m}}^{x}\displaystyle\frac{{\rm
d}z}{U_{+}(z)+c_{+}(z)}\right).\end{array}$

Matching these solutions in accordance with the conditions (56) leads to the
equation:

$\psi^{\prime}_{2}(t)=\mu\Bigl{(}\psi_{1}(t)+\psi_{2}(t)\Bigr{)},\qquad\mu=\displaystyle\frac{a_{m}}{2}\Bigl{(}D_{+}-D_{-}\Bigr{)}.$

Solving this equation under assumption that function $\psi_{1}(t)$ vanishes
sufficiently quickly when $t\to-\infty$, we derive:

$\psi_{2}(t)=\mu\\!\displaystyle\int\limits_{-\infty}^{t}\\!\psi_{1}(\tau){\rm
e}^{\mu(t-\tau)}\,{\rm d}\tau,\qquad\psi_{3}(t)=\psi_{1}(t)+\psi_{2}(t).$ (57)

It is easy to see that $\psi_{2}(t)\equiv 0$ only if $\psi_{1}(t)\equiv 0$.

ii) For the matching of supercritical flows ($U_{m}>c_{m}$), we need to
consider separately two cases.

If a wave of a positive energy arrives from the left to the point $x_{m}$,
then we have:

$\begin{array}[]{l}\psi_{-}(x,t)=\psi_{1}\left(t-\displaystyle\int\limits_{x_{m}}^{x}\displaystyle\frac{{\rm
d}z}{U_{-}(z)+c_{-}(z)}\right)\,,\\\ \\\
\psi_{+}(x,t)=\psi_{3}\left(t-\\!\displaystyle\int\limits_{x_{m}}^{x}\\!\displaystyle\frac{{\rm
d}z}{U_{+}(z)+c_{+}(z)}\right)+\psi_{4}\left(t-\\!\displaystyle\int\limits_{x_{m}}^{x}\\!\displaystyle\frac{{\rm
d}z}{U_{+}(z)-c_{+}(z)}\right),\end{array}$

Matching solutions in the point $x_{m}$ leads to the equations
$\psi^{\prime}_{4}(t)+\mu\psi_{1}(t)=0$, so that we obtain:

$\psi_{4}(t)=-\mu\\!\\!\displaystyle\int\limits_{-\infty}^{t}\\!\psi_{1}(\tau)\,{\rm
d}\tau,\qquad\psi_{3}(t)=\psi_{1}(t)-\psi_{4}(t).$ (58)

If a wave of a negative energy (Fabrikant & Stepanyants, 1998) arrives from
the left to the point $x_{m}$, then we have:

$\begin{array}[]{l}\psi_{-}(x,t)=\psi_{2}\left(t-\displaystyle\int\limits_{x_{m}}^{x}\displaystyle\frac{{\rm
d}z}{U_{-}(z)-c_{-}(z)}\right)\,,\\\ \\\
\psi_{+}(x,t)=\psi_{3}\left(t-\displaystyle\int\limits_{x_{m}}^{x}\displaystyle\frac{{\rm
d}z}{U_{+}(z)+c_{+}(z)}\right)+\psi_{4}\left(t-\displaystyle\int\limits_{x_{m}}^{x}\displaystyle\frac{{\rm
d}z}{U_{+}(z)-c_{+}(z)}\right),\end{array}$

Matching of these solutions leads to the equation
$\psi^{\prime}_{3}(t)=\mu\,\psi_{2}(t)$, so that we obtain:

$\psi_{3}(t)=\mu\\!\\!\displaystyle\int\limits_{-\infty}^{t}\\!\psi_{2}(\tau)\,{\rm
d}\tau,\qquad\psi_{4}(t)=\psi_{2}(t)-\psi_{3}(t).$ (59)

Let us note in passing that if a superposition of waves with the positive and
negative energies arrives from the left to the point $x_{m}$,
$\psi_{-}=\psi_{1}+\psi_{2}$, then

$\displaystyle\psi_{3}(t)$ $\displaystyle=$
$\displaystyle\psi_{1}(t)+\mu\\!\\!\displaystyle\int\limits_{-\infty}^{t}\\!\Bigl{(}\psi_{1}(\tau)+\psi_{2}(\tau)\Bigr{)}\,{\rm
d}\tau,$ (60) $\displaystyle\psi_{4}(t)$ $\displaystyle=$
$\displaystyle\psi_{2}(t)-\mu\\!\\!\displaystyle\int\limits_{-\infty}^{t}\\!\Bigl{(}\psi_{1}(\tau)+\psi_{2}(\tau)\Bigr{)}\,{\rm
d}\tau.$ (61)

Thus, we have seen that when $D_{+}\neq D_{-}$, the composite flow can not be
a RL flow in any regime, subcritical or supercritical.

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correspondence between the gradient and $\boldsymbol{E}^{(B)}$ and between the
Hessian and $\boldsymbol{E}^{(C,D)}$ is not preserved for multiple
frequencies. Instead, the uni- and bivariate Fourier coefficients of $E$ form
the model parameters and the derivative quantities are contractions with the
frequencies thereof. There are multiple ways in which we could generalize QAD
to multiple frequencies.

The first way to generalize QAD is to compute the gradient and Hessian with
the generalized parameter-shift rule Eq. (24) and the shift rule for Hessian
entries Eq. (36) and to construct a single-frequency model as in original QAD.
Even though we know the original energy function to contain multiple
frequencies, this would yield a local model with the correct second-order
expansion at $\boldsymbol{x}_{0}$ that exploits the evaluations savings shown
in this work. As QAD is supposed to use the model only in the neighbourhood of
$\boldsymbol{x}_{0}$, this might be sufficient for the optimization.

As a second generalization we propose a full trigonometric interpolation of
$E$ up to second order, similar to the univariate reconstruction in Sec. 3.1.
First we consider the univariate part of the model: Start by evaluating $E$ at
positions shifted in the $k$th coordinate by equidistant points and subtract
$E(\boldsymbol{x}_{0})$,

$\displaystyle E_{\mu}^{(k)}$ $\displaystyle\coloneqq
E(\boldsymbol{x}_{0}+x_{\mu}\boldsymbol{v}_{k})-E(\boldsymbol{x}_{0})$ (102)
$\displaystyle x_{\mu}$
$\displaystyle\coloneqq\frac{2\mu\pi}{2R_{k}+1},\quad\mu\in[2R_{k}].$ (103)

Then consider the (shifted) Dirichlet kernels

$\displaystyle D_{\mu}^{(k)}(x)$
$\displaystyle=\frac{1}{2R_{k}+1}\left(1+2\sum_{\ell=1}^{R_{k}}\cos(\ell(x-x_{\mu}))\right)$
(104)
$\displaystyle=\frac{\sin\left(\frac{1}{2}(2R_{k}+1)(x-x_{\mu})\right)}{(2R_{k}+1)\sin\left(\frac{1}{2}(x-x_{\mu})\right)}$
(105)

which satisfy $D^{(k)}_{\mu}(x_{\mu^{\prime}})=\delta_{\mu\mu^{\prime}}$ and
are Fourier series with integer frequencies up to $R_{k}$. Therefore, the
function222222One might be wondering why to subtract $E(\boldsymbol{x}_{0})$
just to add it manually back into the reconstruction now. This is because we
need to avoid duplicating this term when adding up the univariate and
bivariate terms of all parameters later on.

$\displaystyle\hat{E}^{(k)}(x)=\sum_{\mu=1}^{2R_{k}}E_{\mu}^{(k)}D^{(k)}_{\mu}(x)$
(106)

coincides with
$E(\boldsymbol{x}_{0}+x\boldsymbol{v}_{k})-E(\boldsymbol{x}_{0})$ at
$2R_{k}+1$ points and is a trigonometric polynomial with the same $R_{k}$
frequencies.

Similarly, the product kernels
$D_{\mu\mu^{\prime}}^{(km)}(x_{k},x_{m})=D_{\mu}^{(k)}(x_{k})D_{\mu^{\prime}}^{(m)}(x_{m})$
can be used to reconstruct the bivariate restriction of $E$ to the
$x_{k}-x_{m}$ plane. For this, evaluate the function at doubly shifted
positions and subtract both, $E(\boldsymbol{x}_{0})$ and the univariate parts:

$\displaystyle E_{\mu\mu^{\prime}}^{(km)}$ $\displaystyle\coloneqq
E(\boldsymbol{x}_{0}+x_{\mu}\boldsymbol{v}_{k}+x_{\mu^{\prime}}\boldsymbol{v}_{m})$
(107)
$\displaystyle-\hat{E}^{(k)}(x_{\mu})-\hat{E}^{(m)}(x_{\mu^{\prime}})-E(\boldsymbol{x}_{0})$
(108)

Then, the bivariate Fourier series

$\displaystyle\hat{E}^{(km)}(x_{k},x_{m})=\sum_{\mu,\mu^{\prime}=1}^{2R_{k},2R_{m}}E_{\mu\mu^{\prime}}^{(km)}D_{\mu\mu^{\prime}}^{(km)}(x_{k},x_{m})$
(109)

coincides with
$E(\boldsymbol{x}_{0}+x_{k}\boldsymbol{v}_{k}+x_{m}\boldsymbol{v}_{m})-E(\boldsymbol{x}_{0})-\hat{E}^{(k)}(x_{k})-\hat{E}^{(m)}(x_{m})$
on the entire coordinate plane spanned by $\boldsymbol{v}_{k}$ and
$\boldsymbol{v}_{m}$.

As we constructed the terms such that they do not contain the respective lower
order terms, we finally can combine them to the full trigonometric
interpolation:

$\displaystyle\hat{E}_{\text{interp}}(\boldsymbol{x})=E(\boldsymbol{x}_{0})$
$\displaystyle+\sum_{k=1}^{n}\hat{E}^{(k)}(x_{k})$ (110)
$\displaystyle+\sum_{k<m}\hat{E}^{(km)}(x_{k},x_{m}).$

This model has as many parameters as function evaluations, namely
$2(\lVert\boldsymbol{R}\rVert_{1}^{2}-\lVert\boldsymbol{R}\rVert_{2}^{2}+\lVert\boldsymbol{R}\rVert_{1})+1$,
and therefore, the trigonometric interpolation is the generalization of the
extended QAD model in App. D.1. Indeed, for $R_{k}=1$ for all $k$ we get back
$2(n^{2}-n+n)+1=2n^{2}+1$ evaluations and model parameters.

We note that the trigonometric interpolation can be implemented for non-
equidistant evaluation points in a similar manner and with the same number of
evaluations, although the elementary functions are no longer Dirichlet kernels
but take the form

$\displaystyle\mathring{D}_{\mu}^{(k)}(x)=\frac{\sin\left(\frac{1}{2}x\right)}{\sin\left(\frac{1}{2}x_{\mu}\right)}\prod_{\mu^{\prime}=1}^{2R_{k}}\frac{\sin\left(\frac{1}{2}(x-x_{\mu^{\prime}})\right)}{\sin\left(\frac{1}{2}(x_{\mu}-x_{\mu^{\prime}})\right)}.$
(111)