# Unconditional convergence of conservative spectral Galerkin methods for the
coupled fractional nonlinear Klein-Gordon-Schrödinger equations

Dongdong Hu1, Yayun Fu2, Wenjun Cai3, Yushun Wang3111Correspondence author.
Email addresses<EMAIL_ADDRESS>(Dongdong Hu<EMAIL_ADDRESS>(Yayun
Fu<EMAIL_ADDRESS>(Wenjun Cai<EMAIL_ADDRESS>(Yushun Wang).
1 School of Mathematics and Statistics, Jiangxi Normal University, Nanchang
330022, China
2 School of Science, Xuchang University, Xuchang 461000, China
3 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023,
China

###### Abstract

In this work, two novel classes of structure-preserving spectral Galerkin
methods are proposed which based on the Crank-Nicolson scheme and the
exponential scalar auxiliary variable method respectively, for solving the
coupled fractional nonlinear Klein-Gordon-Schrödinger equation. The paper
focuses on the theoretical analyses and computational efficiency of the
proposed schemes, the Crank-Nicoloson scheme is proved to be unconditionally
convergent and has the maximum-norm boundness of numerical solutions. The
exponential scalar auxiliary variable scheme is linearly implicit and
decoupled, but lack of the maximum-norm boundness, also, the energy structure
has been modified. Subsequently, the efficient implementations of the proposed
schemes are introduced in detail. Both the theoretical analyses and the
numerical comparisons show that the proposed spectral Galerkin methods have
high efficiency in long-time computations.
AMS subject classification: 26A33, 70H0, 65M125, 65N35
Keywords: Riesz fractional derivative; Spectral Galerkin method; Structure-
preserving algorithm; Unique solvability; Convergence

##  1 Introduction

Fractional differential equations have been widely applied in many fields due
to the nonlocal properties of fractional calculus, they can be employed to
simulate some complex anomalous (super) diffusion phenomena which cannot be
simulated by classical differential equations, and gained many researches and
considerations by physicists and mathematicians, such that there has created
extensive excellent theoretical achievements and realistic value. For
instance, Laskin [19] proposed a class of space-fractional nonlinear
Schrödinger equation by adopting $\alpha$-stable Lévy process instead of
Brownian process, which plays an essential role in quantum mechanics, water
wave dynamics and optics [17, 20] _etc._ Almost simultaneously, Alfimov and
Vázquez _et al._ [1] established a kind of space-fractional nonlinear wave
equation, which is utilized to describe the propagation of fluxons in
Josephson junctions. Recently, the study on the well-posedness of solutions of
nonlinear partial differential equations are complex and essential. In
particular, due to the nonlocal properties of fractional operators, there
exist many challenging problems on the study of analytical solutions of
fractional differential equations [24, 9]. Therefore, it is essential to
develop some efficient and stable numerical methods for fractional
differential equations [32, 36, 37, 2, 25, 10]. Interestingly, a special
dynamical ststem, called as the Klein-Gordon-Schrödinger equation which
consists of the Schrödinger equation and the wave equation, has attracted
considerable attentation in recent years [38, 30, 21]. In this paper, we focus
on effective spectral Galerkin methods for the following strong coupled
fractional nonlinear Klein-Gordon-Schrödinger (FNKGS) equation:

$\displaystyle\textrm{i}u_{t}-\frac{\lambda}{2}(-\Delta)^{\frac{\alpha}{2}}u+\kappa_{1}u\phi+2\kappa_{2}|u|^{2}u\phi=0\quad\textrm{in}\quad\Omega\times(0,T],$
(1.1)
$\displaystyle\phi_{tt}+\gamma(-\Delta)^{\frac{\alpha}{2}}\phi+\eta^{2}\phi-\kappa_{1}|u|^{2}-\kappa_{2}|u|^{4}=0\quad\textrm{in}\quad\Omega\times(0,T],$
(1.2)

subject to the initial-boundary conditions

$\displaystyle
u({x},0)=u_{0}({x}),\quad\phi({x},0)=\phi_{0}({x}),\quad\phi_{t}(x,0)=\phi_{1}(x)\quad\textrm{in}\quad\Omega,$
(1.3) $\displaystyle
u(x,t)=0,\quad\phi(x,t)=0\quad\textrm{on}\quad\partial\Omega\times(0,T].$
(1.4)

The system (1.1)–(1.4) is the fractional model describing the dynamics of
conserved complex nucleon fields $u$ interacting with neutral real scalar
meson fields $\phi$ in the bounded domain $\Omega=(a,b)$, where $1<\alpha<2$,
$\textrm{i}^{2}=-1$, $\lambda,\kappa_{1},\kappa_{2},\gamma\geq 0$ and
$\eta\in\mathbb{R}$ are any given constants, $u_{0}(x),\phi_{1}(x)$ and
$\phi_{1}(x)$ are given smooth functions. The Riesz fractional derivative is
represented by

$\displaystyle(-\Delta)^{\frac{\alpha}{2}}u=\frac{1}{2\cos(\frac{\alpha\pi}{2})}\Big{(}{}_{a}^{RL}\\!D^{{\alpha}}_{x}+{}_{x}^{RL}\\!D^{{\alpha}}_{b}\Big{)}u,$

where the left and right Riemann-Liouville fractional derivatives are defined
as

$\displaystyle{}_{a}^{RL}\\!D^{{\alpha}}_{x}u=\frac{1}{\Gamma(2-{\alpha})}\frac{\partial^{2}}{\partial
x^{2}}\int_{a}^{x}\frac{u(\xi,t)d\xi}{(x-\xi)^{{\alpha}-1}},\qquad{}_{x}^{RL}\\!D^{{\alpha}}_{b}u=\frac{1}{\Gamma(2-{\alpha})}\frac{\partial^{2}}{\partial
x^{2}}\int_{x}^{b}\frac{u(\xi,t)d\xi}{(\xi-x)^{{\alpha}-1}},$

in which $\Gamma(z)=\int^{\infty}_{0}x^{z-1}e^{-x}dx$.

It is worth noticing that the solutions of FNKGS equation conserve the mass of
the nucleon field as follows:

$\displaystyle\mathcal{M}(t)=\int_{\Omega}|u|^{2}dx=\mathcal{M}(0),\quad
t\in(0,T],$

and the total energy as follows:

$\displaystyle\mathcal{E}(t)=\int_{\Omega}(\phi_{t})^{2}+\gamma{\phi}(-\Delta)^{\frac{\alpha}{2}}\phi+\eta^{2}\phi^{2}+\lambda\bar{u}(-\Delta)^{\frac{\alpha}{2}}u-(2\kappa_{1}|u|^{2}+2\kappa_{2}|u|^{4})\phi
dx=\mathcal{E}(0),\quad t\in(0,T],$

where $\bar{u}$ is the complex conjugate of $u$.

Our object is to investigate efficient numerical schemes and the associated
convergence analyses of the FNKGS equation. The main difficulties of the
studies on numerical schemes for the FNKGS equation can be summarized as
follows: $(1)$ The proposed numerical algorithms should preserve the inherent
physical invariants. $(2)$ The unconditional convergence of numerical methods
should be obtained without any Lipschitz continuity restrictions on nonlinear
terms of original system. $(3)$ The numerical scheme should be linearly
implicit and decoupled, also, the convergence can be obtained. Based on the
aforementioned issues, we try to construct a numerical scheme which enjoys
high-order accuracy in space and high efficiency in time, for solving the
FNKGS equation to overcome some of the above challenges.

As is known to us, structure-preserving algorithm is famous for preserving the
inherent invariants of nonlinear conservative systems. In generally,
traditional numerical algorithms are proposed to solve the conservative
system, which often ignore the intrinsic conservative structure of the system.
Although they has the same stable computational ability as the structure-
preserving algorithms in a short time, the latter are superior in long-time
simulations because they can inherit the intrinsic physical characteristics of
a given dynamical system, please refer to [8, 31] and references therein.
Undoubtedly, time-stepping method is the key of the construction of structure-
preserving algorithms. For classical examples, the averaged vector field
method [26], the symplectic Runge-Kutta method [12], the Hamiltonian boundary
value method [6] and so on have catched much attention in recent years.
However, the above methods share a common drawback that the numerical
implementations are usually fully implicit and the convergence analyses are
hard to obtain. In the past decades, some novel tools are developed to provide
the possibilities for constructing the high-order linearly implicit structure-
preserving methods. For instances, readers can refer to the invariant energy
quadratization (IEQ) method [34], the scalar auxiliary variable (SAV) method
[28], the exponential scalar auxiliary variable (ESAV) method [23] and so
forth, which possess a modified energy function. In recent years, there exist
some numerical researches on the FNKGS equation. For examples, Fu and Cai _et
al._ [11] proposed a conservative scheme by combining the partitioned averaged
vector field method and pseudo spectral method. The numerical scheme is proved
to be linearly implicit and decoupled, but the convergence analysis is
unacquirable. Li and Huang _et al._ [22] derived a class of structure-
preserving scheme with the leapfrog method and standard finite element method.
Although the convergence is obtained, the energy structure of the proposed
method is modified and it suffers a low-order accuracy in space.
Simultaneously, Shi and Ma _et al._ [29] constructed a kind of numerical
scheme by using the leapfrog method and center difference method, which is
linearly implicit and unconditionally convergent, but the energy structure is
still modified and the long-time conservation of the proposed scheme is less
efficient.

The purpose of this paper is to construct a conservative Crank-Nicolson
spectral Galerkin method (CN-SGM) which enjoys the original mass and energy
structures as well as spectral accuracy in space for the FNKGS equation.
Meanwhile, we obtain the unconditional convergence analysis and unique
solvability. It is worth to noticing that the existing theoretical result [5]
shows that the solution of the FNKGS equation blows up in finite time if the
initial energy $\mathcal{E}(0)<0$. For numerical comparisons, we propose
another conservative numerical scheme by combining the exponential scalar
auxiliary variable method and spectral Galerkin method (ESAV-SGM). Extensive
numerical results illustrate two numerical phenomena: (1) The ESAV-SGM is
linearly implicit and decoupled, it possesses the higher computational
efficiency than CN-SGM. (2) Although the CN-SGM is fully implicit and coupled,
it is more accurate and enjoys the superior capabillty of capturing blow-up
than CN-SGM when $\mathcal{E}(0)<0$.

The rest of this paper are arranged as follows: In Section 2, some essential
definitions and lemmas of the Galerkin method are introduced. In Section 3, a
class of structure-preserving numerical method is constructed by combining the
spectral Galerkin method and the Crank-Nicolson method. Then the invariant
conservations and unconditional convergence are proved. In section 4, a
linearly implicit and decoupled ESAV-SGM is derived for numerical comparison.
Subsequently, the detailed numerical implementations of the proposed schemes
are offered in Section 5. Next, numerical results are reported to illustrate
the high efficiency of the proposed schemes in Section 6. Finally, some
conclusions are drawed in Section 7.

##  2 Preliminaries

In this section, we introduce some essential definitions and lemmas of
Galerkin methods which can be found in Ref. [7].

######  Definition 2.1.

Define the inner product, $L^{p}$ norm $(1\leq p<\infty)$ and $L^{\infty}$
norm as

$\displaystyle(u,v):=\int_{\Omega}\bar{v}udx,\quad\Big{\|}u\Big{\|}:=\sqrt{\Big{(}u,u\Big{)}},\quad\Big{\|}u\Big{\|}_{p}:=\sqrt[p]{\int_{\Omega}|u|^{p}dx},\quad\Big{\|}u\Big{\|}_{\infty}:=\operatorname{ess}\underset{x\in\Omega}{\sup}\Big{\\{}|u(x)|\Big{\\}}.$

######  Definition 2.2 (Symmetric fractional derivative space ).

For $\mu>0$, where $\mu\neq n-1/2,n\in\mathbb{N}$, define the semi-norm and
the norm as

$\displaystyle\Big{|}u\Big{|}_{J_{S}^{\mu}(\Omega)}:=\Big{|}\Big{(}{}_{a}^{RL}\\!D^{\mu}_{x}u,{}_{x}^{RL}\\!D^{\mu}_{b}u\Big{)}\Big{|}^{1/2},\quad\Big{\|}u\Big{\|}_{J_{S}^{\mu}(\Omega)}:=\Big{(}\Big{\|}u\Big{\|}^{2}+\Big{|}u\Big{|}^{2}_{J_{S}^{\mu}(\Omega)}\Big{)}^{1/2}.$

$J_{S}^{\mu}(\Omega)$ (or $J_{S,0}^{\mu}(\Omega)$) denotes the closure of
$C^{\infty}(\Omega)$ (or $C_{0}^{\infty}(\Omega)$) with respect to
$\Big{\|}\cdot\Big{\|}_{J_{S}^{\mu}(\Omega)}$.

######  Definition 2.3 (Fractional Sobolev space ).

For $\mu>0$, define the semi-norm and the norm as

$\displaystyle\Big{|}u\Big{|}_{H^{\mu}(\Omega)}:=\Big{\|}|\xi|^{\mu}\mathcal{F}[u,\xi]\Big{\|},\quad\Big{\|}u\Big{\|}_{H^{\mu}(\Omega)}:=\Big{(}\Big{\|}u\Big{\|}^{2}+\Big{|}u\Big{|}^{2}_{H^{\mu}(\Omega)}\Big{)}^{1/2}.$

$H^{\mu}(\Omega)$ (or $H_{0}^{\mu}(\Omega)$) denotes the closure of
$C^{\infty}(\Omega)$ (or $C_{0}^{\infty}(\Omega)$) with respect to
$\Big{\|}\cdot\Big{\|}_{H^{\mu}(\Omega)}$, and $\mathcal{F}$ represents the
Fourier transform.

######  Lemma 2.1.

Suppose $\mu>0$ and $\mu\neq n-1/2,n\in\mathbb{N}$. Then the spaces
$J_{S}^{\mu}(\Omega)$ and $H^{\mu}(\Omega)$ are equal with equivalent semi-
norms and norms, and the spaces $J_{S,0}^{\mu}(\Omega)$ and
$H_{0}^{\mu}(\Omega)$ are equal with equivalent semi-norms and norms.

######  Lemma 2.2.

Suppose $1<\mu<2$. For any $u\in H_{0}^{\mu}(\Omega)$ and $v\in
H_{0}^{\mu/2}(\Omega)$, we have

$\displaystyle\Big{(}{}_{a}^{RL}\\!D^{\mu}_{x}u,v\Big{)}=\Big{(}{}_{a}^{RL}\\!D^{\mu/2}_{x}u,{}_{x}^{RL}\\!D^{\mu/2}_{b}v\Big{)},\quad\Big{(}{}_{x}^{RL}\\!D^{\mu}_{b}u,v\Big{)}=\Big{(}{}_{x}^{RL}\\!D^{\mu/2}_{b}u,{}_{a}^{RL}\\!D^{\mu/2}_{x}v\Big{)}.$

##  3 Crank–Nicolson spectral Galerkin method

Define the following bilinear forms

$\displaystyle\mathcal{B}(u,w)=$
$\displaystyle\frac{1}{2\cos(\frac{\alpha\pi}{2})}\Big{(}({}_{a}^{RL}\\!D^{\alpha/2}_{x}u,{}_{x}^{RL}\\!D^{\alpha/2}_{b}w)+({}_{x}^{RL}\\!D^{\alpha/2}_{b}u,{}_{a}^{RL}\\!D^{\alpha/2}_{x}w)\Big{)},\quad\forall
u,w\in H_{0}^{\alpha/2}(\Omega).$

For simplicity, we introduce the semi-norms and the norms as

$\displaystyle\Big{|}u\Big{|}_{\alpha/2}:=\sqrt{\mathcal{B}(u,u)},\qquad\Big{\|}u\Big{\|}_{\alpha/2}:=\sqrt{\Big{\|}u\Big{\|}^{2}+\Big{|}u\Big{|}^{2}_{\alpha/2}}.$

We observe from Lemma 2.1 that $|\cdot|_{\alpha/2}$ and $\|\cdot\|_{\alpha/2}$
are equivalent with the semi-norms and norms of $J_{S}^{\mu}(\Omega)$ and
$H^{\mu}(\Omega)$. From Ref. [7], the bilinear form $\mathcal{B}(u,w)$ has the
continuous and coercive properties, i.e., there exist positive constants
$C_{1}$ and $C_{2}$, for any $u,w\in H_{0}^{{\alpha/2}}(\Omega)$, such that

$\displaystyle|\mathcal{B}(u,w)|\leq
C_{1}\Big{\|}u\Big{\|}_{\alpha/2}\Big{\|}w\Big{\|}_{\alpha/2},\qquad|\mathcal{B}(u,u)|\geq
C_{2}\Big{\|}u\Big{\|}_{\alpha/2}^{2}.$

Based on Lemma 2.2 and the definition of Riesz fractional derivative, the weak
form of (1.1) reads: finding $u,\phi\in H_{0}^{{\alpha/2}}(\Omega)$, such that

$\displaystyle\textrm{i}(u_{t},w)-\frac{\lambda}{2}\mathcal{B}(u,w)+\kappa_{1}(u\phi,w)+2\kappa_{2}(|u|^{2}u\phi,w)=0,\quad\forall
w\in H_{0}^{{\alpha/2}}(\Omega),$ (3.1)
$\displaystyle(\phi_{tt},w)+\gamma\mathcal{B}(\phi,w)+\eta^{2}(\phi,w)-\kappa_{1}(|u|^{2},w)-\kappa_{2}(|u|^{4},w)=0,\quad\forall
w\in H_{0}^{{\alpha/2}}(\Omega),$ (3.2)

with the initial conditions given by

$\displaystyle
u(x,0)=u_{0}(x),\quad\phi(x,0)=\phi_{0}(x),\quad\phi_{t}(x,0)=\phi_{1}(x).$

###  3.1 Fully discrete scheme

Introduce $\psi=\phi_{t}$ to reformulate (3.1)–(3.2) into an equivalent system
as follows, i.e., finding $u,\phi,\psi\in H_{0}^{{\alpha/2}}(\Omega)$, for any
$w\in H_{0}^{{\alpha/2}}(\Omega)$, such that

$\displaystyle\textrm{i}(u_{t},w)-\frac{\lambda}{2}\mathcal{B}(u,w)+\kappa_{1}(u\phi,w)+2\kappa_{2}(|u|^{2}u\phi,w)=0,$
(3.3) $\displaystyle(\phi_{t},w)=(\psi,w),$ (3.4)
$\displaystyle(\psi_{t},w)+\gamma\mathcal{B}(\phi,w)+\eta^{2}(\phi,w)-\kappa_{1}(|u|^{2},w)-\kappa_{2}(|u|^{4},w)=0.$
(3.5)

For simplicity, we introduce the following notations for $n=0,1,\cdots,N_{t}$,

$\displaystyle
u^{n}=u(x,t_{n}),\quad\delta_{t}u^{n+\frac{1}{2}}=\frac{u^{n+1}-u^{n}}{\tau},\quad
u^{n+\frac{1}{2}}=\frac{u^{n+1}+u^{n}}{2},\quad\widetilde{u}^{n+\frac{1}{2}}=\frac{3u^{n}-u^{n-1}}{2},$

where $t_{n}=n\tau$, and $\tau=T/N_{t}$ is the time step size. We denote
$u^{n}$ and $U_{N}^{n}$ as the exact solution and numerical approximation at
$t=t_{n}$, respectively. For theoretical analyses, assume that

$\displaystyle\underset{0\leq t\leq
T}{\max}\Big{\\{}\Big{\|}u\Big{\|},\Big{|}u\Big{|}_{\alpha/2},\Big{\|}u\Big{\|}_{\infty},\Big{\|}\phi\Big{\|},\Big{|}\phi\Big{|}_{\alpha/2},\Big{\|}\phi\Big{\|}_{\infty},\Big{\|}\psi\Big{\|}\Big{\\}}\leq\mathcal{C}.$
(3.6)

Without loss of generality, let $\mathcal{C}$ be a general positive constant
which is independent of $\tau$ and $N$. For a fixed positive integer $N$,
denote $P_{N}(\Omega)$ to be the polynomial space with the degree no more than
$N$ in interval $\Omega$. The approximation space $X_{N}^{0}(\Omega)$ is
defined as

$\displaystyle X_{N}^{0}(\Omega)=P_{N}(\Omega)\cap
H_{0}^{{\alpha/2}}(\Omega).$

It is clear that $X_{N}^{0}(\Omega)$ is a subspace of
$H_{0}^{{\alpha/2}}(\Omega)$. By using the modified Crank–Nicolson scheme for
temporal derivative in (3.3)-(3.5), the Crank–Nicolson scheme with truncation
errors reads: finding $u^{n},\psi^{n},\phi^{n},\in
H_{0}^{{\alpha/2}}(\Omega)$, for any $w\in H_{0}^{{\alpha/2}}(\Omega)$, such
that

$\displaystyle\textrm{i}(\delta_{t}u^{n+1/2},w)-\frac{\lambda}{2}\mathcal{B}(u^{n+1/2},w)+\kappa_{1}(u^{n+1/2}\phi^{n+1/2},w)$
(3.7)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\kappa_{2}\Big{(}(|u^{n+1}|^{2}+|u^{n}|^{2})u^{n+1/2}\phi^{n+1/2},w\Big{)}=(\widehat{\mathscr{R}}_{1}^{n},w),$
$\displaystyle(\delta_{t}\phi^{n+1/2},w)=(\psi^{n+1/2},w)+(\widehat{\mathscr{R}}_{2}^{n},w),$
(3.8)
$\displaystyle(\delta_{t}\psi^{n+1/2},w)+\gamma\mathcal{B}(\phi^{n+1/2},w)+\eta^{2}(\phi^{n+1/2},w)-\frac{\kappa_{1}}{2}\Big{(}(|u^{n+1}|^{2}+|u^{n}|^{2}),w\Big{)}$
(3.9)
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\frac{\kappa_{2}}{2}\Big{(}(|u^{n+1}|^{4}+|u^{n}|^{4}),w\Big{)}=(\widehat{\mathscr{R}}_{3}^{n},w).$

Using the Taylor’s expansion, suppose $u(\cdot,t),\phi(\cdot,t)\in
C^{3}([0,T])$. One easily arrives at the following truncation errors

$\displaystyle\big{|}(\widehat{\mathscr{R}}_{1}^{n})\big{|}\leq\mathcal{C}\tau^{2},\qquad\big{|}(\widehat{\mathscr{R}}_{2}^{n})\big{|}\leq\mathcal{C}\tau^{2},\qquad\big{|}(\widehat{\mathscr{R}}_{3}^{n})\big{|}\leq\mathcal{C}\tau^{2}.$
(3.10)

Similar to the proof of [16], assume that $\phi(\cdot,t)\in C^{4}([0,T])$, one
has

$\displaystyle\big{|}\delta_{t}(\widehat{\mathscr{R}}_{2}^{n+1/2})\big{|}\leq\mathcal{C}\tau^{2}.$
(3.11)

Omitting the small terms in (3.7)-(3.9) and replacing the exact solutions
$u^{n}$, $\psi^{n}$ and $\phi^{n}$ with the numerical solutions $U_{N}^{n}$,
$\Psi_{N}^{n}$ and $\Phi_{N}^{n}$, we propose the fully discrete CN–SGM scheme
as follows: finding $U_{N}^{n},\Psi_{N}^{n},\Phi_{N}^{n}\in
X_{N}^{0}(\Omega)$, for any $w_{N}\in X_{N}^{0}(\Omega)$, such that

$\displaystyle\textrm{i}(\delta_{t}U_{N}^{n+1/2},w_{N})-\frac{\lambda}{2}\mathcal{B}(U_{N}^{n+1/2},w_{N})+\kappa_{1}(U_{N}^{n+1/2}\Phi_{N}^{n+1/2},w_{N})$
(3.12)
$\displaystyle\qquad\qquad\qquad\qquad+\kappa_{2}\Big{(}(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})U_{N}^{n+1/2}\Phi_{N}^{n+1/2},w_{N}\Big{)}=0,$
$\displaystyle(\delta_{t}\Phi_{N}^{n+1/2},w_{N})=(\Psi_{N}^{n+1/2},w_{N}),$
(3.13)
$\displaystyle(\delta_{t}\Psi_{N}^{n+1/2},w_{N})+\gamma\mathcal{B}(\Phi_{N}^{n+1/2},w_{N})+\eta^{2}(\Phi_{N}^{n+1/2},w_{N})-\frac{\kappa_{1}}{2}\Big{(}(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2}),w_{N}\Big{)}$
(3.14)
$\displaystyle\qquad\qquad\qquad\qquad-\frac{\kappa_{2}}{2}\Big{(}(|U_{N}^{n+1}|^{4}+|U_{N}^{n}|^{4}),w_{N}\Big{)}=0,$

with the initial conditions

$\displaystyle
U^{0}_{N}=\Pi_{N}^{{\alpha/2},0}u_{0}(x),\quad\Phi^{0}_{N}=\Pi_{N}^{{\alpha/2},0}\phi_{0}(x),\quad\Psi^{0}_{N}=\Pi_{N}^{{\alpha/2},0}\phi_{1}(x),$
(3.15)

where the orthogonal projection operator $\Pi_{N}^{{\alpha/2},0}$:
$H_{0}^{{\alpha/2}}(\Omega)\rightarrow X_{N}^{0}(\Omega)$ is defined as

$\displaystyle\mathcal{B}(u-\Pi_{N}^{\alpha/2,0}u,w_{N})=0,\quad\forall
w_{N}\in X_{N}^{0}(\Omega).$

###  3.2 Conservations and boundness

######  Theorem 3.1 (Mass and energy conservative laws).

The spectral Galerkin scheme (3.12)–(3.14) possesses the mass and energy
conservative laws in the following discrete sense that

$\displaystyle\mathcal{M}^{n}=\cdots=\mathcal{M}^{0},\qquad\mathcal{E}^{n}=\cdots=\mathcal{E}^{0},\quad
n=1,2,\cdots,N_{t},$ (3.16)

in which

$\displaystyle\mathcal{M}^{n}=\Big{\|}U_{N}^{n}\Big{\|}^{2}$

and

$\displaystyle\mathcal{E}^{n}=\Big{\|}\Psi_{N}^{n}\Big{\|}^{2}+\gamma\Big{|}\Phi^{n}_{N}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\Phi^{n}_{N}\Big{\|}^{2}+\lambda\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}-2\int_{\Omega}\Big{(}\kappa_{1}\Big{|}U_{N}^{n}\Big{|}^{2}+\kappa_{2}\Big{|}U_{N}^{n}\Big{|}^{4}\Big{)}\Phi_{N}^{n}dx.$

###### Proof.

Choosing $w_{N}=U_{N}^{n+1/2}$ in (3.12) and taking the imaginary part, we
obtain $\Big{\|}U_{N}^{n+1}\Big{\|}^{2}=\Big{\|}U_{N}^{n}\Big{\|}^{2}$ which
implies the mass conservation.

Selecting $w_{N}=\delta_{t}U_{N}^{n+1/2}$ in (3.12) and taking the real part,
we have

$\displaystyle\textrm{Re}(\textrm{i}\delta_{t}U_{N}^{n+1/2},\delta_{t}U_{N}^{n+1/2})-\frac{\lambda}{2}\textrm{Re}\Big{\\{}\mathcal{B}(U_{N}^{n+1/2},\delta_{t}U_{N}^{n+1/2})\Big{\\}}+\kappa_{1}\textrm{Re}(U_{N}^{n+1/2}\Phi_{N}^{n+1/2},\delta_{t}U_{N}^{n+1/2})$
$\displaystyle\qquad\qquad\qquad\qquad+\kappa_{2}\textrm{Re}\Big{(}(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})U_{N}^{n+1/2}\Phi_{N}^{n+1/2},\delta_{t}U_{N}^{n+1/2}\Big{)}=0,$
(3.17)

where “Re” represents the real part of a complex number.

By some calculations, we have

$\displaystyle\textrm{Re}(\textrm{i}\delta_{t}U_{N}^{n+1/2},\delta_{t}U_{N}^{n+1/2})=0,\qquad\textrm{Re}\Big{\\{}\mathcal{B}(U_{N}^{n+1/2},\delta_{t}U_{N}^{n+1/2})\Big{\\}}=\frac{\Big{|}U^{n+1}_{N}\Big{|}^{2}_{\alpha/2}-\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}}{2\tau},$
(3.18)
$\displaystyle\textrm{Re}(U_{N}^{n+1/2}\Phi_{N}^{n+1/2},\delta_{t}U_{N}^{n+1/2})$
$\displaystyle=\frac{1}{2\tau}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{2}-\Big{|}U_{N}^{n}\Big{|}^{2}\Big{)}\Phi_{N}^{n+1/2}dx$
$\displaystyle=\frac{1}{4\tau}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{2}-\Big{|}U_{N}^{n}\Big{|}^{2}\Big{)}\Big{(}\Phi_{N}^{n+1}+\Phi_{N}^{n}\Big{)}dx$
(3.19)

and

$\displaystyle\textrm{Re}\Big{(}(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})U_{N}^{n+1/2}\Phi_{N}^{n+1/2},\delta_{t}U_{N}^{n+1/2}\Big{)}$
$\displaystyle=\frac{1}{2\tau}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{4}-\Big{|}U_{N}^{n}\Big{|}^{4}\Big{)}\Phi_{N}^{n+1/2}dx$
$\displaystyle=\frac{1}{4\tau}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{4}-\Big{|}U_{N}^{n}\Big{|}^{4}\Big{)}\Big{(}\Phi_{N}^{n+1}+\Phi_{N}^{n}\Big{)}dx.$
(3.20)

Substituting (3.18)–(3.1) into (3.17), we conclude

$\displaystyle\lambda\Big{(}\Big{|}U^{n+1}_{N}\Big{|}^{2}_{\alpha/2}-\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}\Big{)}$
$\displaystyle-2\kappa_{1}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{2}\Phi_{N}^{n+1}-\Big{|}U_{N}^{n}\Big{|}^{2}\Phi_{N}^{n}+\Big{|}U_{N}^{n+1}\Big{|}^{2}\Phi_{N}^{n}-\Big{|}U_{N}^{n}\Big{|}^{2}\Phi_{N}^{n+1}\Big{)}dx$
$\displaystyle-2\kappa_{2}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{4}\Phi_{N}^{n+1}-\Big{|}U_{N}^{n}\Big{|}^{4}\Phi_{N}^{n}+\Big{|}U_{N}^{n+1}\Big{|}^{4}\Phi_{N}^{n}-\Big{|}U_{N}^{n}\Big{|}^{4}\Phi_{N}^{n+1}\Big{)}dx=0.$
(3.21)

Setting $w_{N}=\delta_{t}\Phi_{N}^{n+1/2}$ in (3.14), we get

$\displaystyle(\delta_{t}\Psi_{N}^{n+1/2},\delta_{t}\Phi_{N}^{n+1/2})+\gamma\mathcal{B}(\Phi_{N}^{n+1/2},\delta_{t}\Phi_{N}^{n+1/2})+\eta^{2}(\Phi_{N}^{n+1/2},\delta_{t}\Phi_{N}^{n+1/2})$
$\displaystyle\qquad\qquad\qquad-\frac{\kappa_{1}}{2}\Big{(}(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2}),\delta_{t}\Phi_{N}^{n+1/2}\Big{)}-\frac{\kappa_{2}}{2}\Big{(}(|U_{N}^{n+1}|^{4}+|U_{N}^{n}|^{4}),\delta_{t}\Phi_{N}^{n+1/2}\Big{)}=0,$
(3.22)

in which

$\displaystyle(\delta_{t}\Psi_{N}^{n+1/2},\delta_{t}\Phi_{N}^{n+1/2})=(\Psi_{N}^{n+1/2},\delta_{t}\Psi_{N}^{n+1/2})=\frac{\Big{\|}\Psi_{N}^{n+1}\Big{\|}^{2}-\Big{\|}\Psi_{N}^{n}\Big{\|}^{2}}{2\tau},$
(3.23)
$\displaystyle\mathcal{B}(\Phi_{N}^{n+1/2},\delta_{t}\Phi_{N}^{n+1/2})=\frac{\Big{|}\Phi^{n+1}_{N}\Big{|}^{2}_{\alpha/2}-\Big{|}\Phi^{n}_{N},\Big{|}^{2}_{\alpha/2}}{2\tau},$
(3.24)
$\displaystyle(\Phi_{N}^{n+1/2},\delta_{t}\Phi_{N}^{n+1/2})=\frac{\Big{\|}\Phi_{N}^{n+1}\Big{\|}^{2}-\Big{\|}\Phi_{N}^{n}\Big{\|}^{2}}{2\tau},$
(3.25)
$\displaystyle\Big{(}(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2}),\delta_{t}\Phi_{N}^{n+1/2}\Big{)}=\frac{1}{\tau}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{2}+\Big{|}U_{N}^{n}\Big{|}^{2}\Big{)}\Big{(}\Phi_{N}^{n+1}-\Phi_{N}^{n}\Big{)}dx$
(3.26)

and

$\displaystyle\Big{(}(|U_{N}^{n+1}|^{4}+|U_{N}^{n}|^{4}),\delta_{t}\Phi_{N}^{n+1/2}\Big{)}=\frac{1}{\tau}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{4}+\Big{|}U_{N}^{n}\Big{|}^{4}\Big{)}\Big{(}\Phi_{N}^{n+1}-\Phi_{N}^{n}\Big{)}dx.$
(3.27)

Substituting (3.23)–(3.26) into (3.1), we obtain

$\displaystyle\Big{(}\Big{\|}\Psi_{N}^{n+1}\Big{\|}^{2}$
$\displaystyle-\Big{\|}\Psi_{N}^{n}\Big{\|}^{2}\Big{)}+\gamma\Big{(}\Big{\|}\Phi_{N}^{n+1}\Big{\|}^{2}-\Big{\|}\Phi_{N}^{n}\Big{\|}^{2}\Big{)}+\eta^{2}\Big{(}\Big{|}\Phi^{n+1}_{N}\Big{|}^{2}_{\alpha/2}-\Big{|}\Phi^{n}_{N},\Big{|}^{2}_{\alpha/2}\Big{)}$
$\displaystyle-2\kappa_{1}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{2}\Phi_{N}^{n+1}-\Big{|}U_{N}^{n}\Big{|}^{2}\Phi_{N}^{n}+\Big{|}U_{N}^{n}\Big{|}^{2}\Phi_{N}^{n+1}-\Big{|}U_{N}^{n+1}\Big{|}^{2}\Phi_{N}^{n}\Big{)}dx$
(3.28)
$\displaystyle-2\kappa_{2}\int_{\Omega}\Big{(}\Big{|}U_{N}^{n+1}\Big{|}^{4}\Phi_{N}^{n+1}-\Big{|}U_{N}^{n}\Big{|}^{4}\Phi_{N}^{n}+\Big{|}U_{N}^{n}\Big{|}^{4}\Phi_{N}^{n+1}-\Big{|}U_{N}^{n+1}\Big{|}^{4}\Phi_{N}^{n}\Big{)}dx=0.$

Summing up (3.1) and (3.1), the energy conservation (3.16) is immediate. This
ends the proof. ∎

######  Lemma 3.1 ([37, 14]).

Let $\mu$ and $r$ be arbitrary real numbers satisfying $1<\alpha<2$, $r\geq
1$. Then there exists a positive constant $\mathcal{C}$ independent of $N$,
such that, for any function $u\in H_{0}^{\frac{\alpha}{2}}(\Omega)\cap
H^{r}(\Omega)$, the following estimates hold:

$\left\\{\begin{array}[]{ll}&\Big{\|}u-\Pi_{N}^{\frac{\alpha}{2},0}u\Big{\|}\leq\mathcal{C}N^{-r}\Big{\|}u\Big{\|}_{H^{r}(\Omega)},\quad\alpha\neq\frac{3}{2}\\\
&\Big{\|}u-\Pi_{N}^{\frac{\alpha}{2},0}u\Big{\|}\leq\mathcal{C}N^{\epsilon-r}\Big{\|}u\Big{\|}_{H^{r}(\Omega)},\quad\alpha=\frac{3}{2},\quad
0<\epsilon<\frac{1}{2}\end{array}\right.$

and

$\displaystyle\Big{|}u-\Pi_{N}^{\frac{\alpha}{2},0}u\Big{\|}_{\frac{\alpha}{2}}\leq\mathcal{C}N^{\frac{\alpha}{2}-r}\Big{\|}u\Big{\|}_{H^{r}(\Omega)}.$

######  Lemma 3.2 (Fractional Gagliardo-Nirenberg inequality [13, 33]).

For $1<\alpha<2$, there exist positive constants $\mathcal{C}_{4}$ and
$\mathcal{C}_{8}$, such that

$\displaystyle\Big{\|}u\Big{\|}^{4}_{4}\leq\mathcal{C}_{4}\Big{|}u\Big{|}_{\alpha/2}^{2/\alpha}\Big{\|}u\Big{\|}^{4-2/\alpha},\qquad\Big{\|}u\Big{\|}^{8}_{8}\leq\mathcal{C}_{8}\Big{|}u\Big{|}_{\alpha/2}^{2/\alpha}\Big{\|}u\Big{\|}^{8-2/\alpha}.$

######  Lemma 3.3 (Sobolev inequality [18]).

For $\frac{1}{2}<\mu<1$, then there exists a positive constant $\mathcal{C}$,
such that

$\displaystyle\Big{\|}u\Big{\|}_{\infty}\leq\mathcal{C}\Big{\|}u\Big{\|}_{\mu},\quad\forall
u\in H^{\mu}_{0}(\Omega).$

In theoretical analyses, we will frequently use the following Young’s
inequality

$\displaystyle ab\leq\varepsilon
a^{2}+\frac{1}{4\varepsilon}b^{2}\quad\textrm{for}\quad a,b\geq
0,\quad\varepsilon>0.$

######  Theorem 3.2 (Boundness of numerical solution).

The numerical solution of spectral Galerkin scheme (3.12)–(3.14) is long-time
bounded in the following sense

$\displaystyle\Big{\|}U_{N}^{n}\Big{\|}\leq\mathcal{C},\quad\Big{|}U^{n}_{N}\Big{|}_{\alpha/2}\leq\mathcal{C},\quad\Big{\|}U^{n}_{N}\Big{\|}_{\infty}\leq\mathcal{C},\qquad
n=0,1,\cdots,N_{t}$

and

$\displaystyle\Big{\|}\Psi^{n}_{N}\Big{\|}^{2}\leq\mathcal{C},\quad\Big{\|}\Phi_{N}^{n}\Big{\|}\leq\mathcal{C},\quad\Big{|}\Phi^{n}_{N}\Big{|}_{\alpha/2}\leq\mathcal{C},\quad\Big{\|}\Phi^{n}_{N}\Big{\|}_{\infty}\leq\mathcal{C},\qquad
n=0,1,\cdots,N_{t}.$

###### Proof.

From the triangle inequality, the assumptions on exact solution (3.6) and
Lemma 3.1, for sufficient small $N^{-1}$, we derive

$\displaystyle\Big{\|}U_{N}^{0}\Big{\|}=\Big{\|}\Pi_{N}^{{\alpha/2},0}u_{0}\Big{\|}\leq\Big{\|}u_{0}\Big{\|}+\Big{\|}u_{0}-\Pi_{N}^{{\alpha/2},0}u_{0}\Big{\|}\leq\mathcal{C}.$

Thus, we have $\mathcal{M}^{0}\leq\mathcal{C}$. By similar procedure, we
deduce $\mathcal{E}^{0}\leq\mathcal{C}$.

According to the mass conservation (3.16), we have

$\displaystyle\Big{\|}U_{N}^{n}\Big{\|}^{2}=\mathcal{M}^{n}=\mathcal{M}^{0}=\Big{\|}U_{N}^{0}\Big{\|}^{2}\leq\mathcal{C},\qquad
n=0,1,\cdots,N_{t}.$

It follows from energy conservation (3.16) and the Young’s inequality that

$\displaystyle\Big{\|}\Psi_{N}^{n}\Big{\|}^{2}$
$\displaystyle+\gamma\Big{|}\Phi^{n}_{N}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\Phi^{n}_{N}\Big{\|}^{2}+\lambda\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}=\mathcal{E}^{0}+2\int_{\Omega}\Big{(}\kappa_{1}\Big{|}U_{N}^{n}\Big{|}^{2}+\kappa_{2}\Big{|}U_{N}^{n}\Big{|}^{4}\Big{)}\Phi_{N}^{n}dx$
$\displaystyle\leq\mathcal{E}^{0}+\frac{\eta^{2}}{2}\Big{\|}\Phi_{N}^{n}\Big{\|}^{2}+\frac{2}{\eta^{2}}\int_{\Omega}\Big{(}\kappa_{1}\Big{|}U_{N}^{n}\Big{|}^{2}+\kappa_{2}\Big{|}U_{N}^{n}\Big{|}^{4}\Big{)}^{2}dx$
(3.29)
$\displaystyle\leq\mathcal{C}+\frac{\eta^{2}}{2}\Big{\|}\Phi_{N}^{n}\Big{\|}^{2}+\frac{4\kappa_{1}^{2}}{\eta^{2}}\Big{\|}U_{N}^{n}\Big{\|}_{4}^{4}+\frac{4\kappa_{2}^{2}}{\eta^{2}}\Big{\|}U_{N}^{n}\Big{\|}^{8}_{8}.$

By using Lemma 3.2 and the Holder inequality, we derive

$\displaystyle\Big{\|}U_{N}^{n}\Big{\|}_{4}^{4}\leq\mathcal{C}_{4}\Big{|}U_{N}^{n}\Big{|}_{\alpha/2}^{2/\alpha}\Big{\|}U_{N}^{n}\Big{\|}^{4-2/\alpha}\leq\mathcal{C}_{4}\epsilon_{1}\Big{|}U_{N}^{n}\Big{|}_{\alpha/2}^{2}+\mathcal{C}_{4}\mathcal{C}(\epsilon_{1})\Big{\|}U_{N}^{n}\Big{\|}^{\frac{4\alpha-2}{\alpha-1}}\leq\mathcal{C}_{4}\epsilon_{1}\Big{|}U_{N}^{n}\Big{|}_{\alpha/2}^{2}+\mathcal{C},$
(3.30)

where $\epsilon_{1}$ is a positive constant, and $\mathcal{C}(\epsilon_{1})>0$
is a constant dependent of the positive constant $\epsilon_{1}$. Similarly, we
obtain

$\displaystyle\Big{\|}U_{N}^{n}\Big{\|}_{8}^{8}\leq\mathcal{C}_{8}\epsilon_{2}\Big{|}U_{N}^{n}\Big{|}_{\alpha/2}^{2}+\mathcal{C}.$
(3.31)

Substituting (3.30) and (3.31) into (3.2), we conclude

$\displaystyle\Big{\|}\Psi_{N}^{n}\Big{\|}^{2}$
$\displaystyle+\gamma\Big{|}\Phi^{n}_{N}\Big{|}^{2}_{\alpha/2}+\frac{\eta^{2}}{2}\Big{\|}\Phi^{n}_{N}\Big{\|}^{2}+\lambda\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}\leq\mathcal{C}+\Big{(}\frac{4\kappa_{1}^{2}\mathcal{C}_{4}\epsilon_{1}}{\eta^{2}}+\frac{4\kappa_{2}^{2}\mathcal{C}_{8}\epsilon_{2}}{\eta^{2}}\Big{)}\Big{|}U_{N}^{n}\Big{|}_{\alpha/2}^{2}.$
(3.32)

Taking appropriate $\epsilon_{1}$ and $\epsilon_{2}$, such that

$\displaystyle\frac{4\kappa_{1}^{2}\mathcal{C}_{4}\epsilon_{1}}{\eta^{2}}+\frac{4\kappa_{2}^{2}\mathcal{C}_{8}\epsilon_{2}}{\eta^{2}}=\frac{\lambda}{2}.$
(3.33)

Substituting (3.33) into (3.32), we derive

$\displaystyle\Big{\|}\Psi_{N}^{n}\Big{\|}^{2}$
$\displaystyle+\gamma\Big{|}\Phi^{n}_{N}\Big{|}^{2}_{\alpha/2}+\frac{\eta^{2}}{2}\Big{\|}\Phi^{n}_{N}\Big{\|}^{2}+\frac{\lambda}{2}\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}\leq\mathcal{C}.$

Therefore, we have

$\displaystyle\Big{\|}\Psi^{n}_{N}\Big{\|}^{2}\leq\mathcal{C},\quad\Big{|}\Phi^{n}_{N}\Big{|}^{2}_{\alpha/2}\leq\mathcal{C},\quad\Big{\|}\Phi^{n}_{N}\Big{\|}^{2}\leq\mathcal{C},\quad\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}\leq\mathcal{C},\qquad
n=0,1,\cdots,N_{t}.$

By admitting Lemma 3.3, we obtain

$\displaystyle\Big{\|}U^{n}_{N}\Big{\|}_{\infty}\leq\mathcal{C}\sqrt{\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}+\Big{\|}U^{n}_{N}\Big{\|}^{2}}\leq\mathcal{C},\quad\Big{\|}\Phi^{n}_{N}\Big{\|}_{\infty}\leq\mathcal{C}\sqrt{\Big{|}\Phi^{n}_{N}\Big{|}^{2}_{\alpha/2}+\Big{\|}\Phi^{n}_{N}\Big{\|}^{2}}\leq\mathcal{C}.$

This completes the proof. ∎

###  3.3 Unique solvability

######  Lemma 3.4.

Suppose that $X_{i}$, $Y_{i}$ (i=1,2,3,4) are complex functions. Then the
following inequalities hold

$\displaystyle(1)~{}~{}\Big{|}X_{1}X_{2}-Y_{1}Y_{2}\Big{|}\leq\max\Big{\\{}|X_{1}|,|Y_{2}|\Big{\\}}\Big{(}\Big{|}X_{1}-Y_{1}\Big{|}+\Big{|}X_{2}-Y_{2}\Big{|}\Big{)},$
(3.34) $\displaystyle(2)~{}~{}\Big{|}|X_{1}|^{2}-|Y_{1}|^{2}\Big{|}\leq
2\max\Big{\\{}|X_{1}|,|Y_{1}|\Big{\\}}\Big{|}X_{1}-Y_{1}\Big{|},$ (3.35)
$\displaystyle(3)~{}~{}\Big{|}|X_{1}|^{4}-|Y_{1}|^{4}\Big{|}\leq
2(|X_{1}|^{2}+|Y_{1}|^{2})\max\Big{\\{}|X_{1}|,|Y_{1}|\Big{\\}}\Big{|}X_{1}-Y_{1}\Big{|},$
(3.36)
$\displaystyle(4)~{}~{}\Big{|}(|X_{3}|^{2}+|X_{4}|^{2})X_{1}X_{2}-(|Y_{3}|^{2}+|Y_{4}|^{2})Y_{1}Y_{2}\Big{|}$
$\displaystyle\qquad\qquad\leq
2\max\Big{\\{}|X_{3}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\max\Big{\\{}|X_{3}|,|Y_{3}|\Big{\\}}\Big{|}X_{3}-Y_{3}\Big{|}$
$\displaystyle\qquad\qquad\qquad+2\max\Big{\\{}|X_{4}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\max\Big{\\{}|X_{4}|,|Y_{4}|\Big{\\}}\Big{|}X_{4}-Y_{4}\Big{|}$
(3.37)
$\displaystyle\qquad\qquad\qquad+\max\Big{\\{}|X_{3}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\max\Big{\\{}|X_{1}|,|Y_{2}|\Big{\\}}\Big{(}\Big{|}X_{1}-Y_{1}\Big{|}+\Big{|}X_{2}-Y_{2}\Big{|}\Big{)}$
$\displaystyle\qquad\qquad\qquad+\max\Big{\\{}|X_{4}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\max\Big{\\{}|X_{1}|,|Y_{2}|\Big{\\}}\Big{(}\Big{|}X_{1}-Y_{1}\Big{|}+\Big{|}X_{2}-Y_{2}\Big{|}\Big{)}.$

###### Proof.

From the triangle inequality and some careful calculations, one derives

$\displaystyle\Big{|}X_{1}X_{2}-Y_{1}Y_{2}\Big{|}$
$\displaystyle=\Big{|}X_{1}X_{2}+(X_{1}-Y_{1})Y_{2}-X_{1}Y_{2}\Big{|}=\Big{|}(X_{1}-Y_{1})Y_{2}+X_{1}(X_{2}-Y_{2})\Big{|}$
$\displaystyle\leq\Big{|}X_{1}-Y_{1}\Big{|}\Big{|}Y_{2}\Big{|}+\Big{|}X_{1}\Big{|}\Big{|}X_{2}-Y_{2}\Big{|}$
$\displaystyle\leq\max\Big{\\{}|X_{1}|,|Y_{2}|\Big{\\}}\Big{(}\Big{|}X_{1}-Y_{1}\Big{|}+\Big{|}X_{2}-Y_{2}\Big{|}\Big{)}.$

Following the inequality (3.34), one arrives at

$\displaystyle\Big{|}|X_{1}|^{2}-|Y_{1}|^{2}\Big{|}$
$\displaystyle=\Big{|}X_{1}\overline{X_{1}}-Y_{1}\overline{Y_{1}}\Big{|}$
$\displaystyle\leq\max\Big{\\{}|X_{1}|,|\overline{Y_{1}}|\Big{\\}}\Big{(}\Big{|}X_{1}-Y_{1}\Big{|}+\Big{|}\overline{X_{1}}-\overline{Y_{1}}\Big{|}\Big{)}.$
$\displaystyle=2\max\Big{\\{}|X_{1}|,|Y_{1}|\Big{\\}}\Big{|}X_{1}-Y_{1}\Big{|}.$

Similarly, we conclude

$\displaystyle\Big{|}|X_{1}|^{4}-|Y_{1}|^{4}\Big{|}$
$\displaystyle=\Big{|}(|X_{1}|^{2}+|Y_{1}|^{2})(|X_{1}|^{2}-|Y_{1}|^{2})\Big{|}$
$\displaystyle\leq(|X_{1}|^{2}+|Y_{1}|^{2})\Big{|}|X_{1}|^{2}-|Y_{1}|^{2}\Big{|}$
$\displaystyle\leq
2(|X_{1}|^{2}+|Y_{1}|^{2})\max\Big{\\{}|X_{1}|,|Y_{1}|\Big{\\}}\Big{|}X_{1}-Y_{1}\Big{|}.$

It follows from the relation (3.34) and (3.35) that

$\displaystyle\Big{|}(|X_{3}|^{2}+|X_{4}|^{2})X_{1}X_{2}-(|Y_{3}|^{2}+|Y_{4}|^{2})Y_{1}Y_{2}\Big{|}$
$\displaystyle=\Big{|}|X_{3}|^{2}X_{1}X_{2}-|Y_{3}|^{2}Y_{1}Y_{2}\Big{|}+\Big{|}|X_{4}|^{2}X_{1}X_{2}-|Y_{4}|^{2}Y_{1}Y_{2}\Big{|}$
$\displaystyle\leq\max\Big{\\{}|X_{3}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\Big{(}\Big{|}|X_{3}|^{2}-|Y_{3}|^{2}\Big{|}+\Big{|}X_{1}X_{2}-Y_{1}Y_{2}\Big{|}\Big{)}$
$\displaystyle\qquad+\max\Big{\\{}|X_{4}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\Big{(}\Big{|}|X_{4}|^{2}-|Y_{4}|^{2}\Big{|}+\Big{|}X_{1}X_{2}-Y_{1}Y_{2}\Big{|}\Big{)}$
$\displaystyle\leq
2\max\Big{\\{}|X_{3}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\max\Big{\\{}|X_{3}|,|Y_{3}|\Big{\\}}\Big{|}X_{3}-Y_{3}\Big{|}$
$\displaystyle\qquad+2\max\Big{\\{}|X_{4}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\max\Big{\\{}|X_{4}|,|Y_{4}|\Big{\\}}\Big{|}X_{4}-Y_{4}\Big{|}$
$\displaystyle\qquad+\max\Big{\\{}|X_{3}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\max\Big{\\{}|X_{1}|,|Y_{2}|\Big{\\}}\Big{(}\Big{|}X_{1}-Y_{1}\Big{|}+\Big{|}X_{2}-Y_{2}\Big{|}\Big{)}$
$\displaystyle\qquad+\max\Big{\\{}|X_{4}|^{2},|Y_{1}||Y_{2}|\Big{\\}}\max\Big{\\{}|X_{1}|,|Y_{2}|\Big{\\}}\Big{(}\Big{|}X_{1}-Y_{1}\Big{|}+\Big{|}X_{2}-Y_{2}\Big{|}\Big{)}.$

This completes the proof. ∎

######  Lemma 3.5 (Browder fixed point theorem [4]).

Let $(\mathcal{H},(\cdot,\cdot))$ be a finite dimensional inner product space,
$||\cdot||$ the associated norm, and
$\mathscr{F}:\mathcal{H}\rightarrow\mathcal{H}$ be continuous, such that

$\displaystyle\exists\ \delta>0,\ \forall\ z\in\mathcal{H},\ \|z\|=\delta,\
s.t.\ \text{Re}(\mathscr{F}(z),z)\geq 0,$

there exists $z^{*}\in\mathcal{H},\|z^{*}\|\leq\delta$ such that
$\mathscr{F}(z^{*})=0$.

######  Theorem 3.3.

For the given initial values and sufficient small $\tau$, the numerical
solutions of spectral Galerkin scheme (3.12)–(3.14) are uniquely solvable.

###### Proof.

For the sake of readability, we leave the proof of this theorem to Appendix A.
∎

###  3.4 Convergence analysis

######  Lemma 3.6 ([15]).

For time sequences $w=\\{w^{0},w^{1},\cdots,w^{n},w^{n+1}\\}$ and
$g=\\{g^{0},g^{1},\cdots,g^{n},g^{n+1}\\}$, there is

$\mid
2\tau\sum\limits^{n}_{k=0}g^{k}\delta_{t}w^{k+1/2}\mid\leq\tau\sum\limits^{n}_{k=1}\mid
w^{k}\mid^{2}+\tau\sum\limits^{n-1}_{k=0}\mid\delta_{t}g^{k+1/2}\mid^{2}+\frac{1}{2}\mid
w^{n+1}\mid^{2}+2\mid g^{n}\mid^{2}+\mid w^{0}\mid^{2}+\mid g^{0}\mid^{2}.$

######  Lemma 3.7 (Gronwall inequality I [35]).

Suppose that the discrete grid function
$\\{w^{n}\mid~{}n=0,1,2,\cdots,N_{t};\\\ N_{t}\tau=T\\}$ satisfies the
following inequality

$w^{n}-w^{n-1}\leq A\tau w^{n}+B\tau w^{n-1}+C_{n}\tau,$

where $A$, $B$ and $C_{n}$ are non-negative constants, then

$\underset{1\leq n\leq N_{t}}{\max}\mid
w^{n}\mid\leq\Big{(}w^{0}+\tau\sum\limits^{N_{t}}_{k=1}C_{k}\Big{)}e^{2(A+B)T},$

where $\tau$ is sufficiently small, such that $(A+B)\tau<\frac{1}{2}$,
$(N_{t}>1)$.

######  Lemma 3.8 (Gronwall inequality II [35]).

Suppose that the discrete grid function
$\\{w^{n}\mid~{}n=0,1,2,\cdots,N_{t};\\\ N_{t}\tau=T\\}$ satisfies the
following inequality

$w^{n}\leq A+\tau\sum\limits^{n}_{k=1}B_{k}w^{k},$

where $A$ and $B_{k}~{}(k=0,1,2,\cdots,N_{t})$ are non-negative constants,
then

$\underset{1\leq n\leq N_{t}}{\max}\mid w^{n}\mid\leq
A\exp({2\tau\sum\limits^{N_{t}}_{k=1}B_{k}}),$

where $\tau$ is sufficiently small, such that $\tau\underset{1\leq k\leq
N_{t}}{\max}B_{k}\leq 1/2$.

######  Theorem 3.4.

Suppose $u\in C^{3}(H_{0}^{{\frac{\alpha}{2}}(\Omega)}(\Omega)\cap
H^{r}(\Omega),[0,T])$ and $\phi\in
C^{4}(H_{0}^{{\frac{\alpha}{2}}(\Omega)}(\Omega)\cap H^{r}(\Omega),[0,T])$
$(r>1)$ are the solutions of (1.1)–(1.2), $U_{N}^{n}$ and $\Phi_{N}^{n}$ are
the solutions of (3.12)–(3.14). For sufficiently small $\tau$ and $N^{-1}$,
the CN–SGM scheme is unconditionally convergent in the sense that

$\left\\{\begin{array}[]{ll}\Big{\|}u^{n}-U^{n}_{N}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),\quad\Big{\|}\phi^{n}-\Phi^{n}_{N}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),\quad\alpha\neq\frac{3}{2},\\\
\Big{\|}u^{n}-U^{n}_{N}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{\epsilon-r}),\quad\Big{\|}\phi^{n}-\Phi^{n}_{N}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{\epsilon-r}),\quad\alpha=\frac{3}{2},\quad
0<\epsilon<\frac{1}{2}\end{array}\right.$

and

$\displaystyle\Big{\|}\phi^{n}-\Phi^{n}_{N}\Big{\|}_{\infty}\leq\mathcal{C}(\tau^{2}+N^{\frac{\alpha}{2}-r}).$

###### Proof.

Denote the error functions

$\displaystyle\varepsilon_{u}^{n}=u^{n}-U_{N}^{n}=(u^{n}-\Pi_{N}^{\alpha/2,0}u^{n})+(\Pi_{N}^{\alpha/2,0}u^{n}-U_{N}^{n})=\eta_{u}^{n}+\xi_{u}^{n},$
$\displaystyle\varepsilon_{\phi}^{n}=\phi^{n}-\Phi_{N}^{n}=(\phi^{n}-\Pi_{N}^{\alpha/2,0}\phi^{n})+(\Pi_{N}^{\alpha/2,0}\phi^{n}-\Phi_{N}^{n})=\eta_{\phi}^{n}+\xi_{\phi}^{n},$
$\displaystyle\varepsilon_{\psi}^{n}=\psi^{n}-\Psi_{N}^{n}=(\psi^{n}-\Pi_{N}^{\alpha/2,0}\psi^{n})+(\Pi_{N}^{\alpha/2,0}\psi^{n}-\Phi_{N}^{n})=\eta_{\psi}^{n}+\xi_{\psi}^{n},$
$\displaystyle\varepsilon_{u}^{0}=u_{0}-\Pi_{N}^{\alpha/2,0}u_{0},\qquad\varepsilon_{\phi}^{0}=\phi_{0}-\Pi_{N}^{\alpha/2,0}\phi_{0},\qquad\varepsilon_{\psi}^{0}=\phi_{1}-\Pi_{N}^{\alpha/2,0}\phi_{1}.$

Noticing that

$\displaystyle\xi_{u}^{0}=\Pi_{N}^{\alpha/2,0}u_{0}-U_{N}^{0}=\Pi_{N}^{\alpha/2,0}u_{0}-\Pi_{N}^{\alpha/2,0}u_{0}=0,$
$\displaystyle\xi_{\phi}^{0}=\Pi_{N}^{\alpha/2,0}\phi_{0}-\Phi_{N}^{0}=\Pi_{N}^{\alpha/2,0}\phi_{0}-\Pi_{N}^{\alpha/2,0}\phi_{0}=0,$
$\displaystyle\xi_{\psi}^{0}=\Pi_{N}^{\alpha/2,0}\phi_{1}-\Psi_{N}^{0}=\Pi_{N}^{\alpha/2,0}\phi_{1}-\Pi_{N}^{\alpha/2,0}\phi_{1}=0.$

Subtracting (3.12)–(3.14) from (3.7)–(3.9), in view of the definition of
orthogonal projection operator $\Pi_{N}^{{\alpha/2},0}$, we have

$\displaystyle\textrm{i}(\delta_{t}\xi_{u}^{n+1/2},w_{N})-\frac{\lambda}{2}\mathcal{B}(\xi_{u}^{n+1/2},w_{N})+\kappa_{1}(u^{n+1/2}\phi^{n+1/2}-U_{N}^{n+1/2}\Phi_{N}^{n+1/2},w_{N})$
$\displaystyle\quad+\kappa_{2}\Big{(}(|u^{n+1}|^{2}+|u^{n}|^{2})u^{n+1/2}\phi^{n+1/2}-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})U_{N}^{n+1/2}\Phi_{N}^{n+1/2},w_{N}\Big{)}$
(3.38) $\displaystyle\quad=(\mathscr{R}_{1}^{n},w_{N}),\quad\forall w_{N}\in
X_{N}^{0}(\Omega),$
$\displaystyle(\delta_{t}\xi_{\phi}^{n+1/2},w_{N})=(\xi_{\psi}^{n+1/2},w_{N})+(\mathscr{R}_{2}^{n},w_{N}),\quad\forall
w_{N}\in X_{N}^{0}(\Omega),$ (3.39)
$\displaystyle(\delta_{t}\xi_{\psi}^{n+1/2},w_{N})+\gamma\mathcal{B}(\xi_{\phi}^{n+1/2},w_{N})+\eta^{2}(\xi_{\phi}^{n+1/2},w_{N})$
$\displaystyle\qquad\qquad-\frac{\kappa_{1}}{2}\Big{(}(|u^{n+1}|^{2}+|u^{n}|^{2})-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2}),w_{N}\Big{)}$
(3.40)
$\displaystyle\qquad\qquad-\frac{\kappa_{2}}{2}\Big{(}(|u^{n+1}|^{4}+|u^{n}|^{4})-(|U_{N}^{n+1}|^{4}+|U_{N}^{n}|^{4}),w_{N}\Big{)}=(\mathscr{R}_{3}^{n},w_{N}),\quad\forall
w_{N}\in X_{N}^{0}(\Omega),$

where

$\displaystyle\mathscr{R}_{1}^{n}=\widehat{\mathscr{R}}_{1}^{n}-\textrm{i}\delta_{t}\eta_{u}^{n+1/2},$
$\displaystyle\mathscr{R}_{2}^{n}=\widehat{\mathscr{R}}_{2}^{n}+\eta_{\psi}^{n+1/2}-\delta_{t}\eta_{\phi}^{n+1/2},$
$\displaystyle\mathscr{R}_{3}^{n}=\widehat{\mathscr{R}}_{3}^{n}-\delta_{t}\eta_{\psi}^{n+1/2}-\eta^{2}\eta_{\phi}^{n+1/2}.$

For the case $\alpha\neq\frac{3}{2}$, according to Taylor’s expansion and
Lemma 3.1, we have

$\displaystyle\Big{\|}\eta_{\psi}^{n+\frac{1}{2}}\Big{\|}$
$\displaystyle\leq\Big{\|}\eta_{\psi}^{n+\frac{1}{2}}-\eta_{\psi}(\cdot,t_{n+1/2})\Big{\|}+\Big{\|}\eta_{\psi}(\cdot,t_{n+1/2})\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),$
$\displaystyle\Big{\|}\delta_{t}\eta_{\phi}^{n+\frac{1}{2}}\Big{\|}$
$\displaystyle\leq\Big{\|}\delta_{t}\eta_{\phi}^{n+\frac{1}{2}}-\partial_{t}\eta_{\phi}(\cdot,t_{n+1/2})\Big{\|}+\Big{\|}\partial_{t}\eta_{\phi}(\cdot,t_{n+1/2})\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}).$

Similarly, we have

$\displaystyle\Big{\|}\eta_{\phi}^{n+\frac{1}{2}}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),\quad\Big{\|}\delta_{t}\eta_{\psi}^{n+\frac{1}{2}}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),\quad\Big{\|}\delta_{t}\eta_{u}^{n+\frac{1}{2}}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}).$

Thus, we obtain

$\displaystyle\Big{\|}\mathscr{R}_{1}^{n}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),\quad\Big{\|}\mathscr{R}_{2}^{n}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),\quad\Big{\|}\mathscr{R}_{3}^{n}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}).$

By admitting the above relations and (3.11), we deduce

$\displaystyle\Big{\|}\delta_{t}\mathscr{R}_{2}^{n+1/2}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}).$

Selecting $w_{N}=\xi_{u}^{n+1/2}$ in (3.4) and taking the imaginary part, we
deduce

$\displaystyle\textrm{Im}\Big{\\{}\textrm{i}(\delta_{t}\xi_{u}^{n+1/2},\xi_{u}^{n+1/2})\Big{\\}}=-\kappa_{1}\textrm{Im}(u^{n+1/2}\phi^{n+1/2}-U_{N}^{n+1/2}\Phi_{N}^{n+1/2},\xi_{u}^{n+1/2})$
$\displaystyle\quad\quad-\kappa_{2}\textrm{Im}\Big{(}(|u^{n+1}|^{2}+|u^{n}|^{2})u^{n+1/2}\phi^{n+1/2}-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})U_{N}^{n+1/2}\Phi_{N}^{n+1/2},\xi_{u}^{n+1/2}\Big{)}$
(3.41)
$\displaystyle\quad\quad+\textrm{Im}(\mathscr{R}_{1}^{n},\xi_{u}^{n+1/2}).$

By some careful calculations, we obtain

$\displaystyle\textrm{Im}\Big{\\{}\textrm{i}(\delta_{t}\xi_{u}^{n+1/2},\xi_{u}^{n+1/2})\Big{\\}}=\frac{1}{2\tau}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}-\Big{\|}\xi_{u}^{n}\Big{\|}^{2}\Big{)}.$
(3.42)

Employing Lemma 3.4, Lemma 3.1 and the Cauchy-Schwarz inequality, we conclude

$\displaystyle-\kappa_{1}$
$\displaystyle\textrm{Im}(u^{n+1/2}\phi^{n+1/2}-U_{N}^{n+1/2}\Phi_{N}^{n+1/2},\xi_{u}^{n+1/2})\leq\kappa_{1}\Big{\|}u^{n+1/2}\phi^{n+1/2}-U_{N}^{n+1/2}\Phi_{N}^{n+1/2}\Big{\|}\Big{\|}\xi_{u}^{n+1/2}\Big{\|}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}u^{n+1/2}\phi^{n+1/2}-U_{N}^{n+1/2}\Phi_{N}^{n+1/2}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n+1/2}\Big{\|}^{2}\Big{)}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}u^{n+1/2}-U_{N}^{n+1/2}\Big{\|}^{2}+\Big{\|}\phi^{n+1/2}-\Phi_{N}^{n+1/2}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n+1/2}\Big{\|}^{2}\Big{)}$
(3.43)
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1/2}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n+1/2}\Big{\|}^{2}+N^{-2r}\Big{)}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}+N^{-2r}\Big{)},$
$\displaystyle-\kappa_{2}\textrm{Im}$
$\displaystyle\Big{(}(|u^{n+1}|^{2}+|u^{n}|^{2})u^{n+1/2}\phi^{n+1/2}-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})U_{N}^{n+1/2}\Phi_{N}^{n+1/2},\xi_{u}^{n+1/2}\Big{)}$
$\displaystyle\leq\kappa_{2}\Big{\|}(|u^{n+1}|^{2}+|u^{n}|^{2})u^{n+1/2}\phi^{n+1/2}-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})U_{N}^{n+1/2}\Phi_{N}^{n+1/2}\Big{\|}\Big{\|}\xi_{u}^{n+1/2}\Big{\|}$
(3.44)
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}+N^{-2r}\Big{)}$

and

$\displaystyle\textrm{Im}(\mathscr{R}_{1}^{n},\xi_{u}^{n+1/2})\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\tau^{4}\Big{)}.$
(3.45)

Substituting (3.42)–(3.45) into (3.4), one has

$\displaystyle\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}\leq\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\mathcal{C}\tau\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}+\tau^{4}+N^{-2r}\Big{)}.$
(3.46)

Choosing $w_{N}=\xi_{\phi}^{n+1/2}$ in (3.39), we have

$\displaystyle\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}-\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}$
$\displaystyle=2\tau(\xi_{\psi}^{n+1/2},\xi_{\phi}^{n+1/2})+2\tau(\mathscr{R}_{2}^{n},\xi_{\phi}^{n+1/2})$
(3.47)
$\displaystyle\leq\mathcal{C}\tau\Big{(}\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\tau^{4}+N^{-2r}\Big{)}.$

Employing the Gronwall inequality I (Lemma 3.7), one obtains

$\displaystyle\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}\leq\mathcal{C}\tau\sum\limits^{n}_{k=0}\Big{\|}\xi_{\psi}^{k+1}\Big{\|}^{2}+\mathcal{C}(\tau^{4}+N^{-2r}),$
(3.48)

which implies

$\displaystyle\tau\sum\limits^{n}_{k=0}\Big{\|}\xi_{\phi}^{k+1}\Big{\|}^{2}$
$\displaystyle\leq\mathcal{C}\tau^{2}\sum\limits^{n}_{k=0}\sum\limits^{k}_{j=0}\Big{\|}\xi_{\psi}^{j+1}\Big{\|}^{2}+\mathcal{C}(\tau^{4}+N^{-2r})$
$\displaystyle\leq\mathcal{C}\tau^{2}\sum\limits^{n}_{k=0}\sum\limits^{n}_{j=0}\Big{\|}\xi_{\psi}^{j+1}\Big{\|}^{2}+\mathcal{C}(\tau^{4}+N^{-2r})$
(3.49)
$\displaystyle\leq\mathcal{C}\tau\sum\limits^{n}_{k=0}\Big{\|}\xi_{\psi}^{k+1}\Big{\|}^{2}+\mathcal{C}(\tau^{4}+N^{-2r}).$

Taking $w_{N}=\delta_{t}\xi_{\phi}^{n+1/2}$ in (3.40), we have

$\displaystyle(\delta_{t}\xi_{\psi}^{n+1/2},\delta_{t}\xi_{\phi}^{n+1/2})+\gamma\frac{\Big{|}\xi_{\phi}^{n+1}\Big{|}^{2}_{\alpha/2}-\Big{|}\xi_{\phi}^{n}\Big{|}^{2}_{\alpha/2}}{2\tau}+\eta^{2}\frac{\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}-\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}}{2\tau}$
$\displaystyle\qquad\qquad-\frac{\kappa_{1}}{2}\Big{(}(|u^{n+1}|^{2}+|u^{n}|^{2})-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2}),\delta_{t}\xi_{\phi}^{n+1/2}\Big{)}$
(3.50)
$\displaystyle\qquad\qquad-\frac{\kappa_{2}}{2}\Big{(}(|u^{n+1}|^{4}+|u^{n}|^{4})-(|U_{N}^{n+1}|^{4}+|U_{N}^{n}|^{4}),\delta_{t}\xi_{\phi}^{n+1/2}\Big{)}=(\mathscr{R}_{3}^{n},\delta_{t}\xi_{\phi}^{n+1/2}).$

It follows from (3.39) that

$\displaystyle\Big{(}\delta_{t}\xi_{\psi}^{n+1/2},\delta_{t}\xi_{\phi}^{n+1/2}\Big{)}$
$\displaystyle=(\xi_{\psi}^{n+\frac{1}{2}},\delta_{t}\xi_{\psi}^{n+1/2})+(\mathscr{R}_{2}^{n},\delta_{t}\xi_{\psi}^{n+1/2})$
$\displaystyle=\frac{\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}-\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}}{2\tau}+(\mathscr{R}_{2}^{n},\delta_{t}\xi_{\psi}^{n+1/2}).$
(3.51)

Setting $w_{N}=\delta_{t}\Phi_{N}^{n+\frac{1}{2}}-\Psi_{N}^{n+\frac{1}{2}}$ in
(3.13), we deduce

$\displaystyle\delta_{t}\Phi_{N}^{n+\frac{1}{2}}=\Psi_{N}^{n+\frac{1}{2}}.$

It is trivial to check that

$\displaystyle(\xi_{\psi}^{n+1/2},w_{N})=(\delta_{t}\xi_{\phi}^{n+1/2}+\Pi_{N}^{\alpha/2,0}\psi^{n+1/2}-\Pi_{N}^{\alpha/2,0}\delta_{t}\phi^{n+1/2},w_{N}),\quad\forall
w_{N}\in X_{N}^{0}(\Omega),$

which implies

$\displaystyle\delta_{t}\xi_{\phi}^{n+1/2}=\xi_{\psi}^{n+1/2}+\Pi_{N}^{\alpha/2,0}\delta_{t}\phi^{n+1/2}-\Pi_{N}^{\alpha/2,0}\psi^{n+1/2}.$

By employing Lemma 3.4, Lemma 3.1 and the Cauchy-Schwarz inequality that

$\displaystyle-$
$\displaystyle\frac{\kappa_{1}}{2}\Big{(}(|u^{n+1}|^{2}+|u^{n}|^{2})-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2}),\delta_{t}\xi_{\phi}^{n+1/2}\Big{)}$
$\displaystyle\leq\frac{\kappa_{1}}{2}\Big{\|}(|u^{n+1}|^{2}+|u^{n}|^{2})-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})\Big{\|}\Big{\|}\delta_{t}\xi_{\phi}^{n+1/2}\Big{\|}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}(|u^{n+1}|^{2}+|u^{n}|^{2})-(|U_{N}^{n+1}|^{2}+|U_{N}^{n}|^{2})\Big{\|}^{2}+\Big{\|}\delta_{t}\xi_{\phi}^{n+1/2}\Big{\|}^{2}\Big{)}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}u^{n+1}-U_{N}^{n+1}\Big{\|}^{2}+\Big{\|}u^{n}-U_{N}^{n}\Big{\|}^{2}+\Big{\|}\delta_{t}\xi_{\phi}^{n+1/2}\Big{\|}^{2}\Big{)}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1/2}+\Pi_{N}^{\alpha/2,0}\delta_{t}\phi^{n+1/2}-\Pi_{N}^{\alpha/2,0}\psi^{n+1/2}\Big{\|}^{2}+N^{-2r}\Big{)}$
$\displaystyle=\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1/2}+\Pi_{N}^{\alpha/2,0}\delta_{t}\phi^{n+1/2}-\delta_{t}\phi^{n+1/2}$
(3.52)
$\displaystyle\quad+\delta_{t}\phi^{n+1/2}-\psi^{n+1/2}+\psi^{n+1/2}-\Pi_{N}^{\alpha/2,0}\psi^{n+1/2}\Big{\|}^{2}+N^{-2r}\Big{)}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1/2}\Big{\|}^{2}+\Big{\|}\Pi_{N}^{\alpha/2,0}\delta_{t}\phi^{n+1/2}-\delta_{t}\phi^{n+1/2}\Big{\|}^{2}$
$\displaystyle\quad+\Big{\|}\delta_{t}\phi^{n+1/2}-\psi^{n+1/2}\Big{\|}^{2}+\Big{\|}\psi^{n+1/2}-\Pi_{N}^{\alpha/2,0}\psi^{n+1/2}\Big{\|}^{2}+N^{-2r}\Big{)}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\Big{\|}\delta_{t}\phi^{n+1/2}-\psi^{n+1/2}\Big{\|}^{2}+N^{-2r}\Big{)}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\tau^{4}+N^{-2r}\Big{)},$

where the last inequality has used the following estimate: by taking
$w_{N}=\delta_{t}{\phi}^{n+1/2}-{\psi}^{n+1/2}$ in (3.8), one has

$\displaystyle\Big{\|}\delta_{t}{\phi}^{n+1/2}-{\psi}^{n+1/2}\Big{\|}\leq\Big{\|}\mathscr{R}_{2}^{n}\Big{\|}\leq\mathcal{C}(\tau^{4}+N^{-2r}).$

Similar estimates, we obtain

$\displaystyle-\frac{\kappa_{2}}{2}\Big{(}(|u^{n+1}|^{4}$
$\displaystyle+|u^{n}|^{4})-(|U_{N}^{n+1}|^{4}+|U_{N}^{n}|^{4}),\delta_{t}\xi_{\phi}^{n+1/2}\Big{)}$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\tau^{4}+N^{-2r}\Big{)}$
(3.53)

and

$\displaystyle(\mathscr{R}_{3}^{n},\delta_{t}\xi_{\phi}^{n+1/2})$
$\displaystyle\leq\mathcal{C}\Big{(}\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\tau^{4}+N^{-2r}\Big{)}.$
(3.54)

Substituting (3.4)–(3.54) into (3.4), we arrive at

$\displaystyle\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\gamma\Big{|}\xi_{\phi}^{n+1}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}$
$\displaystyle\leq\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\gamma\Big{|}\xi_{\phi}^{n}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}+\mathcal{C}\tau\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\tau^{4}+N^{-2r}\Big{)}$
(3.55)
$\displaystyle\qquad-2\tau(\mathscr{R}_{2}^{n},\delta_{t}\xi_{\psi}^{n+1/2}).$

Summing up (3.46) and (3.4), we have

$\displaystyle\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\gamma\Big{|}\xi_{\phi}^{n+1}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}$
$\displaystyle\leq\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\gamma\Big{|}\xi_{\phi}^{n}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}+\mathcal{C}\tau\Big{(}\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{u}^{n}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{n}\Big{\|}^{2}$
(3.56)
$\displaystyle\qquad+\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n}\Big{\|}^{2}+\tau^{4}+N^{-2r}\Big{)}-2\tau(\mathscr{R}_{2}^{n},\delta_{t}\xi_{\psi}^{n+1/2})$
$\displaystyle\leq\mathcal{C}\tau\sum\limits^{n}_{k=0}\Big{(}\Big{\|}\xi_{u}^{k+1}\Big{\|}^{2}+\Big{\|}\xi_{\phi}^{k+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{k+1}\Big{\|}^{2}\Big{)}+\mathcal{C}\Big{(}\tau^{4}+N^{-2r}\Big{)}-2\tau\sum\limits^{n}_{k=0}(\mathscr{R}_{2}^{k},\delta_{t}\xi_{\psi}^{k+1/2}).$

It follows from (3.4) and Lemma 3.6 that

$\displaystyle\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\gamma\Big{|}\xi_{\phi}^{n+1}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}$
$\displaystyle\leq\mathcal{C}\tau\sum\limits^{n}_{k=0}\Big{(}\Big{\|}\xi_{u}^{k+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{k+1}\Big{\|}^{2}\Big{)}+\mathcal{C}\Big{(}\tau^{4}+N^{-2r}\Big{)}-2\tau\sum\limits^{n}_{k=0}(\mathscr{R}_{2}^{k},\delta_{t}\xi_{\psi}^{k+1/2})$
(3.57)
$\displaystyle\leq\mathcal{C}\tau\sum\limits^{n}_{k=0}\Big{(}\Big{\|}\xi_{u}^{k+1}\Big{\|}^{2}+\Big{\|}\xi_{\psi}^{k+1}\Big{\|}^{2}\Big{)}+\frac{1}{2}\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\mathcal{C}\Big{(}\tau^{4}+N^{-2r}\Big{)}.$

Then, we reformulate (3.4) as

$\displaystyle\Big{\|}\xi_{u}^{n+1}\Big{\|}^{2}+\frac{1}{2}\Big{\|}\xi_{\psi}^{n+1}\Big{\|}^{2}+\gamma\Big{|}\xi_{\phi}^{n+1}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}$
$\displaystyle\leq\mathcal{C}\tau\sum\limits^{n}_{k=0}\Big{(}\Big{\|}\xi_{u}^{k+1}\Big{\|}^{2}+\frac{1}{2}\Big{\|}\xi_{\psi}^{k+1}\Big{\|}^{2}\Big{)}+\mathcal{C}\Big{(}\tau^{4}+N^{-2r}\Big{)}$
(3.58)
$\displaystyle\leq\mathcal{C}\tau\sum\limits^{n}_{k=0}\Big{(}\Big{\|}\xi_{u}^{k+1}\Big{\|}^{2}+\frac{1}{2}\Big{\|}\xi_{\psi}^{k+1}\Big{\|}^{2}+\gamma\Big{|}\xi_{\phi}^{n+1}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\xi_{\phi}^{n+1}\Big{\|}^{2}\Big{)}+\mathcal{C}\Big{(}\tau^{4}+N^{-2r}\Big{)}.$

Utilizing the Gronwall inequality II (Lemma 3.8) for (3.4), we conclude

$\displaystyle\Big{\|}\xi_{u}^{n+1}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),\quad\Big{\|}\xi_{\phi}^{n+1}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}),\quad\Big{|}\xi_{\phi}^{n+1}\Big{|}_{\alpha/2}\leq\mathcal{C}(\tau^{2}+N^{-r}).$

By applying Lemma 3.1, we derive

$\displaystyle\Big{|}{\phi}^{n+1}-{\Phi}_{N}^{n+1}\Big{|}_{\alpha/2}\leq\Big{|}\eta_{\phi}^{n+1}\Big{|}_{\alpha/2}+\Big{|}\xi_{\phi}^{n+1}\Big{|}_{\alpha/2}\leq\mathcal{C}(\tau^{2}+N^{\alpha/2-r}),$
$\displaystyle\Big{\|}u^{n+1}-U_{N}^{n+1}\Big{\|}\leq\Big{\|}\eta_{u}^{n+1}\Big{\|}+\Big{\|}\xi_{u}^{n+1}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r})$

and

$\displaystyle\Big{\|}\phi^{n+1}-\Phi_{N}^{n+1}\Big{\|}\leq\Big{\|}\eta_{\phi}^{n+1}\Big{\|}+\Big{\|}\xi_{\phi}^{n+1}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{-r}).$

Therefore, combining the above estimates and Lemma 3.3, we get

$\displaystyle\Big{\|}\phi^{n+1}-\Phi_{N}^{n+1}\Big{\|}_{\infty}\leq\mathcal{C}\sqrt{\Big{\|}\phi^{n+1}-\Phi_{N}^{n+1}\Big{\|}^{2}+\Big{|}{\phi}^{n+1}-{\Phi}_{N}^{n+1}\Big{|}_{\alpha/2}^{2}}\leq\mathcal{C}(\tau^{2}+N^{\alpha/2-r}).$

When $\alpha=\frac{3}{2}$, the inference procedures are similar to the case
$\alpha\neq\frac{3}{2}$, here we omit it. We obtain the following consequences

$\displaystyle\Big{\|}u^{n}-U^{n}_{N}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{\epsilon-r}),\quad\Big{\|}\phi^{n}-\Phi^{n}_{N}\Big{\|}\leq\mathcal{C}(\tau^{2}+N^{\epsilon-r}),\quad\Big{\|}\phi^{n}-\Phi^{n}_{N}\Big{\|}_{\infty}\leq\mathcal{C}(\tau^{2}+N^{\frac{\alpha}{2}-r}),\quad
0<\epsilon<\frac{1}{2}.$

Thus, the proof is completed. ∎

##  4 Linearly implicit and decoupled conservative scheme

The ESAV formulation of the FNKGS equation (1.1)–(1.2) introduces the
exponential scalar auxiliary variables

$\displaystyle
p(t)=\exp\Big{(}2\int_{\Omega}(\kappa_{1}|u|^{2}+\kappa_{2}|u|^{4})\phi
dx\Big{)}.$

Taking the derivative of $\ln(p)$, we have

$\displaystyle\frac{d}{dt}\ln(p)$
$\displaystyle=2\int_{\Omega}\Big{(}2\kappa_{1}\textrm{Re}(\bar{u}u_{t})+4\kappa_{2}|u|^{2}\textrm{Re}(\bar{u}u_{t})\Big{)}\phi+\Big{(}\kappa_{1}|u|^{2}+\kappa_{2}|u|^{4}\Big{)}\phi_{t}dx$
$\displaystyle=4\textrm{Re}\Big{(}(\kappa_{1}+2\kappa_{2}|u|^{2})u\phi,u_{t}\Big{)}+2\Big{(}\kappa_{1}|u|^{2}+\kappa_{2}|u|^{4},\phi_{t}\Big{)}$
$\displaystyle=4\textrm{Re}\Big{(}p{F}(u,\phi)u\phi,u_{t}\Big{)}+2\Big{(}p{G}(u,\phi),\phi_{t}\Big{)},$

where

$\displaystyle{F}(u,\phi)=\frac{\kappa_{1}+2\kappa_{2}|u|^{2}}{\exp\Big{(}2\int_{\Omega}(\kappa_{1}|u|^{2}+\kappa_{2}|u|^{4})\phi
dx\Big{)}},\quad{G}(u,\phi)=\frac{\kappa_{1}|u|^{2}+\kappa_{2}|u|^{4}}{\exp\Big{(}2\int_{\Omega}(\kappa_{1}|u|^{2}+\kappa_{2}|u|^{4})\phi
dx\Big{)}}.$

Then, the system (3.3)–(3.5) can be reformulated into an equivalent form

$\displaystyle\textrm{i}(u_{t},w)-\frac{\lambda}{2}\mathcal{B}(u,w)+(p{F}(u,\phi)u\phi,w)=0,$
(4.1) $\displaystyle(\phi_{t},w)=(\psi,w),$ (4.2)
$\displaystyle(\psi_{t},w)+\gamma\mathcal{B}(\phi,w)+\eta^{2}(\phi,w)-(p{G}(u,\phi),w)=0,$
(4.3)
$\displaystyle\frac{d}{dt}\ln(p)=4\textrm{Re}\Big{(}p{F}(u,\phi)u\phi,u_{t}\Big{)}+2\Big{(}p{G}(u,\phi),\phi_{t}\Big{)},$
(4.4)

with

$\displaystyle
u({x},0)=u_{0}({x}),\quad\phi({x},0)=\phi_{0}({x}),\quad\phi_{t}(x,0)=\phi_{1}(x)\quad
p_{0}=\exp\Big{(}2\int_{\Omega}(\kappa_{1}|u_{0}|^{2}+\kappa_{2}|u_{0}|^{4})\phi
dx\Big{)}.$

The energy conservation in the ESAV formulation is

$\displaystyle\frac{d}{dt}\Big{(}\Big{\|}\psi\Big{\|}^{2}+\gamma\Big{|}\phi\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\phi\Big{\|}^{2}+\lambda\Big{|}u\Big{|}^{2}_{\alpha/2}-\ln(p)\Big{)}=0.$

Combining the implicit midpoint method and spectral Galerkin method, the
ESAV–SGM for (1.1)–(1.2) is constructed as: finding
$U^{n}_{N},\Phi^{n}_{N},\Psi^{n}_{N}\in X_{N}^{0}(\Omega)$, for $w_{N}\in
X_{N}^{0}(\Omega)$, such that

$\displaystyle\textrm{i}(\delta_{t}U_{N}^{n+1/2},w_{N})-\frac{\lambda}{2}\mathcal{B}(U_{N}^{n+1/2},w_{N})+\widetilde{P}^{n+1/2}\Big{(}F(\widetilde{U}_{N}^{n+1/2},\widetilde{\Phi}_{N}^{n+1/2})\widetilde{U}_{N}^{n+1/2}\widetilde{\Phi}_{N}^{n+1/2},w_{N}\Big{)}=0,$
(4.5)
$\displaystyle(\delta_{t}\Phi_{N}^{n+1/2},w_{N})=(\Psi_{N}^{n+1/2},w_{N}),$
(4.6)
$\displaystyle(\delta_{t}\Psi_{N}^{n+1/2},w_{N})+\gamma\mathcal{B}(\Phi_{N}^{n+1/2},w_{N})+\eta^{2}(\Phi_{N}^{n+1/2},w_{N})-\widetilde{P}^{n+1/2}\Big{(}G(\widetilde{U}_{N}^{n+1/2},\widetilde{\Phi}_{N}^{n+1/2}),w_{N}\Big{)}=0,$
(4.7)
$\displaystyle\frac{\ln({P}^{n+1})-\ln({P}^{n})}{\tau}=4\widetilde{P}^{n+1/2}\textrm{Re}\Big{(}{F}(\widetilde{U}_{N}^{n+1/2},\widetilde{\Phi}_{N}^{n+1/2})\widetilde{U}_{N}^{n+1/2}\widetilde{\Phi}_{N}^{n+1/2},\delta_{t}{U}_{N}^{n+1/2}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+2\widetilde{P}^{n+1/2}\Big{(}G(\widetilde{U}_{N}^{n+1/2},\widetilde{\Phi}_{N}^{n+1/2}),\delta_{t}{\Phi}_{N}^{n+1/2}\Big{)},$
(4.8)

with the startup scheme

$\displaystyle\textrm{i}(\delta_{t}U_{N}^{1/2},w_{N})-\frac{\lambda}{2}\mathcal{B}(U_{N}^{1/2},w_{N})+{P}^{1/2}\Big{(}F({U}_{N}^{1/2},\Phi_{N}^{1/2}){U}_{N}^{1/2}{\Phi}_{N}^{1/2},w_{N}\Big{)}=0,$
(4.9) $\displaystyle(\delta_{t}\Phi_{N}^{1/2},w_{N})=(\Psi_{N}^{1/2},w_{N}),$
(4.10)
$\displaystyle(\delta_{t}\Psi_{N}^{1/2},w_{N})+\gamma\mathcal{B}(\Phi_{N}^{1/2},w_{N})+\eta^{2}(\Phi_{N}^{1/2},w_{N})-{P}^{1/2}\Big{(}G({U}_{N}^{1/2},\Phi_{N}^{1/2}),w_{N}\Big{)}=0,$
(4.11)
$\displaystyle\frac{\ln({P}^{1})-\ln({P}^{0})}{\tau}=4{P}^{1/2}\textrm{Re}\Big{(}{F}({U}_{N}^{1/2},\Phi_{N}^{1/2}){U}_{N}^{1/2}{\Phi}_{N}^{1/2},\delta_{t}{U}_{N}^{1/2}\Big{)}+2{P}^{1/2}\Big{(}G({U}_{N}^{1/2},\Phi_{N}^{1/2}),\delta_{t}{\Phi}_{N}^{1/2}\Big{)}.$
(4.12)

######  Theorem 4.1.

The numerical solution of the ESAV–SGM (4.5)-(4.8) enjoys an energy
preservation in the sense that

$\displaystyle\mathcal{E}^{n}=\cdots=\mathcal{E}^{0},\qquad
n=0,1,\cdots,N_{t},$

where

$\displaystyle\mathcal{E}^{n}=\Big{\|}\Psi^{n}_{N}\Big{\|}^{2}+\gamma\Big{|}\Phi^{n}_{N}\Big{|}^{2}_{\alpha/2}+\eta^{2}\Big{\|}\Phi^{n}_{N}\Big{\|}^{2}+\lambda\Big{|}U^{n}_{N}\Big{|}^{2}_{\alpha/2}-\ln(P^{n}).$

###### Proof.

Setting $w_{N}=\delta_{t}U_{N}^{n+1/2}$ in (4.5) and taking the real part,
then choosing $w_{N}=\delta_{t}\Phi_{N}^{n+1/2}$ in (4.7), summing up the
above resulting formulae similar to Theorem 3.1, the energy preservation is
readily and immediately available. ∎

##  5 Decoupled iterative implementation

In this section, we focus on the implementation of CN–SGM (3.12)–(3.14), the
implementation of ESAV–SGM can be similarly modified. The associated numerical
solutions $U_{N}^{n}$, $\Phi_{N}^{n}$ and $\Psi_{N}^{n}$ are of the form

$\displaystyle
U_{N}^{n}=\sum\limits^{N-2}_{k=0}\widehat{U}^{n}_{k}\varsigma_{k}(x),\qquad\Phi_{N}^{n}=\sum\limits^{N-2}_{k=0}\widehat{\Phi}^{n}_{k}\varsigma_{k}(x),\qquad\Psi_{N}^{n}=\sum\limits^{N-2}_{k=0}\widehat{\Psi}^{n}_{k}\varsigma_{k}(x)$

belong to the $N$-dimensional polynomial space $X_{N}^{0}(\Omega)$ which is
given by

$\displaystyle
X_{N}^{0}(\Omega)=\textrm{span}\\{\varsigma_{k}(x):~{}k=0,1,\cdots,N-2\\},$

where $\varsigma_{k}(x)$ is determined by the following recurrence relation

$\displaystyle\varsigma_{k}(x)=L_{k}(\widehat{x})-L_{k+2}(\widehat{x}),\quad\widehat{x}\in[-1,1],\quad
x=\frac{(b-a)\widehat{x}+(a+b)}{2}\in[a,b],$

in which $L_{k}(\hat{x})$ represents Legendre orthogonal polynomial [27] which
satisfies the following three-term recurrence relation

$\left\\{\begin{array}[]{ll}L_{0}(\hat{x})=1,\quad L_{1}(\hat{x})=\hat{x},\\\
(k+1)L_{k+1}(\hat{x})=(2k+1)\hat{x}L_{k}(\hat{x})-kL_{k-1}(\hat{x}),\quad
k\geq 1.\end{array}\right.$

The associated mass and stiff matrices of CN–SGM are all symmetric and
computed by

$\displaystyle\mathbb{M}_{l,k}=\Big{(}\varsigma_{k}({x}),\varsigma_{l}({x})\Big{)},\quad\mathbb{S}_{l,k}=\frac{1}{2\cos(\frac{\alpha\pi}{2})}\Big{[}\Big{(}{}_{a}^{RL}\\!D^{\alpha/2}_{x}\varsigma_{k}({x}),{}_{x}^{RL}\\!D^{\alpha/2}_{b}\varsigma_{l}({x})\Big{)}+\Big{(}{}_{x}^{RL}\\!D^{\alpha/2}_{b}\varsigma_{k}({x}),{}_{a}^{RL}\\!D^{\alpha/2}_{x}\varsigma_{l}({x})\Big{)}\Big{]}.$

The linearized and decoupled fixed-point iterative algorithm is presented for
the CN–SGM in the following form

$\left\\{\begin{array}[]{ll}\Big{(}\textrm{i}\mathbb{M}-\frac{\lambda\tau}{4}\mathbb{S}\Big{)}\widehat{U}^{n+1,s+1}=\Big{(}\textrm{i}\mathbb{M}+\frac{\lambda\tau}{4}\mathbb{S}\Big{)}\widehat{U}^{n}-\frac{\tau}{4}\Big{(}\kappa_{1}\mathcal{N}_{1}^{n+1,s}+\kappa_{2}\mathcal{N}_{2}^{n+1,s}\Big{)},\quad
s=0,1,\cdots,\\\ \\\
\Big{(}\mathbb{M}+\frac{\tau^{2}}{4}\mathbb{S}+\frac{\eta^{2}\tau^{2}}{4}\mathbb{M}\Big{)}\widehat{\Phi}^{n+1,s+1}=\Big{(}\mathbb{M}-\frac{\tau^{2}}{4}\mathbb{S}-\frac{\eta^{2}\tau^{2}}{4}\mathbb{M}\Big{)}\widehat{\Phi}^{n}+\tau\mathbb{M}\widehat{\Psi}^{n}+\frac{\tau^{2}}{4}\Big{(}\kappa_{1}\mathcal{N}_{3}^{n+1,s}+\kappa_{2}\mathcal{N}_{4}^{n+1,s}\Big{)},\end{array}\right.$

where

$\widehat{U}^{n+1,0}=\left\\{\begin{array}[]{ll}\widehat{U}^{0},\quad n=0,\\\
2\widehat{U}^{n}-\widehat{U}^{n-1},\quad n\geq
1,\end{array}\right.\qquad\widehat{\Phi}^{n+1,0}=\left\\{\begin{array}[]{ll}\widehat{\Phi}^{0},\quad
n=0,\\\ 2\widehat{\Phi}^{n}-\widehat{\Phi}^{n-1},\quad n\geq
1\end{array}\right.$

and

$\left\\{\begin{array}[]{ll}(\mathcal{N}_{1}^{n+1,s})_{l}=\Big{(}(U_{N}^{n}+U_{N}^{n+1,s})(\Phi_{N}^{n}+\Phi_{N}^{n+1,s}),\varsigma_{l}(x)\Big{)},~{}l=0,1,\cdots,N-2,\\\
\\\
(\mathcal{N}_{2}^{n+1,s})_{l}=\Big{(}(|U_{N}^{n}|^{2}+|U_{N}^{n+1,s}|^{2})(U_{N}^{n}+U_{N}^{n+1,s})(\Phi_{N}^{n}+\Phi_{N}^{n+1,s}),\varsigma_{l}(x)\Big{)},~{}l=0,1,\cdots,N-2,\\\
\\\
(\mathcal{N}_{3}^{n+1,s})_{l}=\Big{(}|U_{N}^{n}|^{2}+|U_{N}^{n+1,s}|^{2},\varsigma_{l}(x)\Big{)},~{}l=0,1,\cdots,N-2,\\\
\\\
(\mathcal{N}_{4}^{n+1,s})_{l}=\Big{(}|U_{N}^{n}|^{4}+|U_{N}^{n+1,s}|^{4},\varsigma_{l}(x)\Big{)},~{}l=0,1,\cdots,N-2.\end{array}\right.$

Then, $U_{N}^{n+1,s+1}$ and $\Phi_{N}^{n+1,s+1}$ numerically converge to the
numerical solutions $U_{N}^{n+1}$ and $\Phi_{N}^{n+1}$, respectively, if there
satisfies
$\Big{\|}U_{N}^{n+1,s+1}-U_{N}^{n+1,s}\Big{\|}+\Big{\|}\Phi_{N}^{n+1,s+1}-\Phi_{N}^{n+1,s}\Big{\|}\leq
tol$ for the given stopping criterion $tol$ [3].

##  6 Numerical experiments

In this section, some numerical results are reported to verify the proposed
spectral Galerkin method. All the simulations are implemented by using Matlab
R2018a software on a computer with Intel Core i7 and 16 GB RAM. Without
special instructions, we always take the stopping criterion $tol=10^{-14}$.

In numerical tests, we compute the $L^{2}-$ and $L^{\infty}-$norm errors at
$t=N_{t}\tau$ by

$\displaystyle\|u^{N_{t}}-U_{N}^{N_{t}}\|=\sqrt{\int_{\Omega}\Big{|}u^{N_{t}}(x)-U_{N}^{N_{t}}(x)\Big{|}^{2}dx}\approx\sqrt{\frac{b-a}{2}\sum\limits^{M}_{j=0}\Big{|}u^{N_{t}}(x_{j})-U^{N_{t}}_{N}(x_{j})\Big{|}^{2}\varpi_{j}}$

and

$\displaystyle\Big{\|}u^{N_{t}}-U_{N}^{N_{t}}\Big{\|}_{\infty}:=\underset{0\leq
j\leq
M}{\max}\Big{|}u^{N_{t}}(x_{j})-U^{N_{t}}_{N}(x_{j})\Big{|}\quad\textrm{with}\quad
x_{j}=\frac{(b-a)\widehat{x}_{j}+(a+b)}{2},$

where $\\{\widehat{x}_{j}\\}$ and $\\{\varpi_{j}\\}$ are points and weights of
the Legendre–Gauss–Lobatto quadrature, respectively, and $M=\mu N$
$(\mu>1,\mu\in\mathbb{N}_{+})$. In convergence test, for the case that the
solution of equation is unknown, we intend to regard the more accurate
numerical solutions as the reference solutions. To illustrate the energy
preservation of the proposed numerical schemes, define the following relative
mass and energy deviations

$\displaystyle\textrm{RM}^{n}=\frac{|\mathcal{M}^{n}-\mathcal{M}^{0}|}{\mathcal{M}^{0}},\qquad\textrm{RE}^{n}=\frac{|\mathcal{E}^{n}-\mathcal{E}^{0}|}{\mathcal{E}^{0}}.$

######  Example 6.1.

_Consider the FNKGS system ( 1.1)–(1.2) with $\lambda=1$, $\kappa_{1}=1$,
$\kappa_{2}=0$, $\gamma=1$ and $\eta=1$ in domain $(-20,20)\times(0,T]$. When
$\alpha=2$, the system has the exact solitary wave solutions as follows:_

$\displaystyle
u(x,t,\nu)=\frac{3\sqrt{2}}{4\sqrt{1-\nu^{2}}}\operatorname{sech}^{2}\frac{x-\nu
t-\chi_{0}}{2\sqrt{1-\nu^{2}}}\exp\Big{(}\textrm{i}(\nu
x+\frac{1-\nu^{2}+\nu^{4}}{2(1-\nu^{2})})t\Big{)},$ (6.1)
$\displaystyle\phi(x,t,\nu)=\frac{3}{4(1-\nu^{2})}\operatorname{sech}^{2}\frac{x-\nu
t-\chi_{0}}{2\sqrt{1-\nu^{2}}},$ (6.2)
$\displaystyle\phi_{t}(x,t,\nu)=\frac{3\nu}{4(1-\nu^{2})^{\frac{3}{2}}}\operatorname{sech}^{2}\frac{x-\nu
t-\chi_{0}}{2\sqrt{1-\nu^{2}}}\tanh\frac{x-\nu
t-\chi_{0}}{2\sqrt{1-\nu^{2}}}.$ (6.3)

_Here, the initial datum $u_{0}(x)$, $\phi_{0}(x)$ and $\phi_{1}(x)$ are
determined by the exact solutions (6.1)–(6.3), where $\nu=0.8$ and
$\chi_{0}=-10$. _

Firstly, we carry out testing the convergence accuracy of CN–SGM in time and
space. For the fixed temporal step, we find from Fig. 1 that the errors are
exponentially decaying in $L^{2}$ and $L^{\infty}$ norm. For the fixed
polynomial degree $N=150$, the temporal numerical results are listed in Table
1. The numerical results show that the proposed CN–SGM possesses the second-
order accuracy in $L^{2}$ and $L^{\infty}$ norm, which are in great accordance
with the theoretical results. Secondly, we show the effect of fractional order
$\alpha$ on numerical profiles. It is obvious to observe from Fig. 2 that the
fractional order $\alpha$ will dramatically affect the shapes of the solitons,
we refer readers to [22] for more details. Finally, for the purpose of
numerical comparisons, we utilize the CN–SGM, the ESAV–SGM and the finite
difference scheme (LF-FDM [29]) to solve Example 6.1 until $t=100$. Fig. 3 and
Fig. 4 illustrates that the CN–SGM uniformly preserves the mass and energy to
machine accuracy. Fig. 4 verifies that the ESAV–SGM preserves the energy well,
but the mass preservation can not be done from Fig. 3. Actually, the mass
conservation is also an important structure for the convergence and stability
analyses of a conservative algorithm. Fig. 3 and Fig. 4 shows that the LF-FDM
is not up to the task of long-time mass and energy preservations. These
numerical pictures illustrate that the CN–SGM enjoys a great superior than the
ESAV–SGM and the LF-FDM for energy preservation. Subsequently, we plot the
numerical errors and associated CPU time of the CN–SGM and the ESAV–SGM in
Fig. 5 by setting the various temporal step $\tau$ and the same polynomial
degree $N$. From the numerical results, we observe that the ESAV–SGM performs
more effective than the CN–SGM, in the other word, the ESAV–SGM needs less
time to obtain the same errors. It is worth noticing that the CN–SGM can
obtain the more accurate numerical results than the ESAV–SGM for the same
temporal step $\tau$ and polynomial degree $N$. Therefore, the CN–SGM and the
ESAV–SGM each have their own advantages in long-time computations.

Fig. 1: Spectral accuracy tests of CN–SGM for Example 6.1 with $\tau=1/1000$
at t=1. Table. 1: Temporal accuracy tests of CN–SGM for Example 6.1 with
$N=150$ at $t=1$.

$\alpha$ | $\tau$ | $\|u^{Nt}-U_{N}^{Nt}\|$ | Conv.rate | $\|\phi^{Nt}-\Phi_{N}^{Nt}\|$ | Conv.rate | $\|\phi^{Nt}-\Phi_{N}^{Nt}\|_{\infty}$ | Conv.rate | CPU time
---|---|---|---|---|---|---|---|---
1.2 | 0.1 | 4.5813e-03 | - | 1.1683e-03 | - | 12134e-03 | - | 1.03s
| 0.05 | 1.1492e-03 | 1.9951 | 2.9386e-04 | 1.9912 | 30651e-04 | 1.9851 | 1.54s
| 0.025 | 2.8769e-04 | 1.9981 | 7.3579e-05 | 1.9978 | 76822e-05 | 1.9963 | 2.43s
| 0.0125 | 7.2149e-05 | 1.9955 | 1.8402e-05 | 1.9994 | 19218e-05 | 1.9991 | 4.00s
1.5 | 0.1 | 4.8761e-03 | - | 1.0746e-03 | - | 1.0233e-03 | - | 1.05s
| 0.05 | 1.2232e-03 | 1.9951 | 2.7114e-04 | 1.9866 | 2.6063e-04 | 1.9731 | 1.69s
| 0.025 | 3.0612e-04 | 1.9985 | 6.7948e-04 | 1.9965 | 6.5450e-05 | 1.9935 | 2.69s
| 0.0125 | 7.6661e-05 | 1.9975 | 1.6996e-04 | 1.9992 | 1.6384e-05 | 1.9981 | 4.30s
1.8 | 0.1 | 5.4219e-03 | - | 9.0221e-04 | - | 8.5874e-04 | - | 1.04s
| 0.05 | 1.3661e-03 | 1.9888 | 2.2578e-04 | 1.9985 | 2.1491e-04 | 1.9989 | 1.69s
| 0.025 | 3.4227e-04 | 1.9968 | 5.6461e-05 | 1.9996 | 5.3741e-05 | 1.9997 | 2.43s
| 0.0125 | 8.5705e-05 | 1.9977 | 1.4124e-05 | 1.9991 | 1.3429e-05 | 2.0012 | 3.99s

Fig. 2: Time evolutions of the relative energy and mass of CN–SGM for Example
6.1 with $\tau=1/10$ and $N=100$.

Fig. 3: Time evolutions of the relative mass deviations of CN–SGM (Left),
ESAV–SGM (Middle) and LF–FDM (Right) for Example 6.1 with $\tau=1/10$ and
$N=100$.

Fig. 4: Time evolutions of the relative energy deviations of CN–SGM (Left),
ESAV–SGM (Middle) and LF–FDM (Right) for Example 6.1 with $\tau=1/10$ and
$N=100$.

Fig. 5: Errors $\|u^{Nt}-U_{N}^{Nt}\|+\|\phi^{Nt}-\Phi_{N}^{Nt}\|$ and the
corresponding CPU time costs for Example 6.1 with $N=150$ and
$\tau=1/20,1/40,1/80,1/160,1/320$.

######  Example 6.2.

_Consider the FNKGS system ( 1.1)–(1.2) with $\lambda=1$, $\kappa_{1}=1$,
$\kappa_{2}\geq 0$, $\gamma=1$ and $\eta=1$ in domain $(-20,20)\times(0,T]$.
We choose the following initial datum with two solitons:_

$\displaystyle u_{0}(x)=u(x-p_{1},0,\nu_{1})+u(x-p_{2},0,\nu_{2}),$
$\displaystyle\phi_{0}(x)=\phi(x-p_{1},0,\nu_{1})+\phi(x-p_{2},0,\nu_{2}),$
$\displaystyle\phi_{1}(x)=\phi_{t}(x-p_{1},0,\nu_{1})+\phi_{t}(x-p_{2},0,\nu_{2}),$

_where $p_{1}=-p_{2}=-10$, $\nu_{1}=-\nu_{2}=0.8$ and $\chi_{0}=0$._

For the FNKGS system with Yukawa interaction, we intend to perform the effect
of fractional orders on the numerical accuracy and discrete conservation laws.
We apply the CN–SGM to solve Example 6.2, the convergence rate and the
associated CPU time for various $\kappa_{2}$ are shown in Table 2–Table 4, the
numerical results verify the second-order accuracy in time, which are in good
agreement with theoretical results. For the fixed temporal step $\tau=1/50$
and polynomial degree $N=200$, the numerical solutions $|U_{N}^{n}(x)|$ and
$\Phi_{N}^{n}(x)$ are plotted in Fig. 6, we observe that the collisions will
occur earlier as $\alpha$ increases, and the effect of fractional order
$\alpha$ on two solitons collide is distinct. By setting $\tau=1/50$, $N=150$
and $\kappa_{2}=0.01$, the time evolution of relative mass and relative energy
errors are presented in Fig. 7, the numerical results exactly confirm that the
proposed spectral Galerkin scheme preserves the mass and energy well in long-
time computations. In the end, one can also notice from Fig. 8 that the
discrete energy occurs negative if we choose a large $\kappa_{2}$, which
demonstrates that the analytical solutions of the FNKGS system (1.1)–(1.2)
will be blow up in finite time [5]. Also, the blow-up time will get smaller
when the ESAV–SGM is adopted, which implies that the CN–SGM is more suitable
than the ESAV–SGM to capture blow up.

Table. 2: Temporal accuracy of CN–SGM for Example 6.2 with $\kappa_{2}=0$ and
$N=150$ at $t=1$.

$\alpha$ | $\tau$ | $\|u^{Nt}-U_{N}^{Nt}\|$ | Conv.rate | $\|\phi^{Nt}-\Phi_{N}^{Nt}\|$ | Conv.rate | $\|\phi^{Nt}-\Phi_{N}^{Nt}\|_{\infty}$ | Conv.rate | CPU time
---|---|---|---|---|---|---|---|---
1.2 | 0.1 | 6.4901e-03 | - | 1.6517e-03 | - | 1.2104e-03 | - | 1.15s
| 0.05 | 1.6281e-03 | 1.9950 | 4.1544e-04 | 1.9912 | 3.0573e-04 | 1.9851 | 1.73s
| 0.025 | 4.0760e-04 | 1.9980 | 1.0402e-04 | 1.9978 | 7.6629e-05 | 1.9963 | 2.44s
| 0.0125 | 1.0228e-04 | 1.9946 | 2.6016e-05 | 1.9994 | 1.9169e-05 | 1.9991 | 4.15s
1.5 | 0.1 | 6.8998e-03 | - | 1.5194e-03 | - | 1.0228e-03 | - | 1.03s
| 0.05 | 1.7308e-03 | 1.9951 | 3.8338e-04 | 1.9866 | 2.6053e-04 | 1.9729 | 1.60s
| 0.025 | 4.3318e-04 | 1.9984 | 9.6079e-05 | 1.9965 | 6.5416e-05 | 1.9937 | 2.42s
| 0.0125 | 1.0851e-04 | 1.9972 | 2.4032e-05 | 1.9993 | 1.6379e-05 | 1.9978 | 4.10s
1.8 | 0.1 | 7.6685e-03 | - | 1.2759e-03 | - | 8.5880e-04 | - | 1.01s
| 0.05 | 1.9321e-03 | 1.9888 | 3.1930e-04 | 1.9985 | 2.1494e-04 | 1.9984 | 1.71s
| 0.025 | 4.8409e-04 | 1.9968 | 7.9848e-05 | 1.9996 | 5.3753e-05 | 1.9995 | 2.48s
| 0.0125 | 1.2122e-04 | 1.9976 | 1.9986e-05 | 1.9983 | 1.3424e-05 | 2.0015 | 4.18s

Table. 3: Temporal accuracy of CN–SGM for Example 6.2 with $\kappa_{2}=0.01$
and $N=150$ at $t=1$.

$\alpha$ | $\tau$ | $\|u^{Nt}-U_{N}^{Nt}\|$ | Conv.rate | $\|\phi^{Nt}-\Phi_{N}^{Nt}\|$ | Conv.rate | $\|\phi^{Nt}-\Phi_{N}^{Nt}\|_{\infty}$ | Conv.rate | CPU time
---|---|---|---|---|---|---|---|---
1.2 | 0.1 | 7.6888e-03 | - | 1.6980e-03 | - | 1.2975e-03 | - | 1.09s
| 0.05 | 1.9294e-03 | 1.9946 | 4.2726e-04 | 1.9906 | 3.2780e-04 | 1.9848 | 1.62s
| 0.025 | 4.8297e-04 | 1.9981 | 1.0699e-04 | 1.9976 | 8.2164e-05 | 1.9962 | 2.52s
| 0.0125 | 1.2105e-04 | 1.9963 | 2.6759e-05 | 1.9994 | 2.0555e-05 | 1.9990 | 4.09s
1.5 | 0.1 | 8.1416e-04 | - | 1.6199e-03 | - | 1.2303e-03 | - | 1.05s
| 0.05 | 2.0423e-04 | 1.9951 | 4.0976e-04 | 1.9830 | 3.1367e-04 | 1.9716 | 1.64s
| 0.025 | 5.1110e-04 | 1.9985 | 1.0275e-04 | 1.9956 | 7.8783e-05 | 1.9933 | 2.62s
| 0.0125 | 1.2794e-04 | 1.9981 | 2.5706e-05 | 1.9990 | 1.9725e-05 | 1.9978 | 4.18s
1.8 | 0.1 | 9.2848e-03 | - | 1.0966e-03 | - | 6.9719e-04 | - | 1.17s
| 0.05 | 2.3492e-03 | 1.9827 | 2.7501e-04 | 1.9955 | 1.7479e-04 | 1.9959 | 1.55s
| 0.025 | 5.8940e-04 | 1.9948 | 6.8813e-05 | 1.9987 | 4.3738e-05 | 1.9987 | 2.56s
| 0.0125 | 1.4756e-04 | 1.9979 | 1.7233e-05 | 1.9975 | 1.0917e-05 | 2.0024 | 4.04s

Table. 4: Temporal accuracy of CN–SGM for Example 6.2 with $\kappa_{2}=0.1$
and $N=150$ at $t=1$.

$\alpha$ | $\tau$ | $\|u^{Nt}-U_{N}^{Nt}\|$ | Conv.rate | $\|\phi^{Nt}-\Phi_{N}^{Nt}\|$ | Conv.rate | $\|\phi^{Nt}-\Phi_{N}^{Nt}\|_{\infty}$ | Conv.rate | CPU time
---|---|---|---|---|---|---|---|---
1.2 | 0.1 | 5.0647e-02 | - | 3.9455e-03 | - | 3.7289e-03 | - | 1.78s
| 0.05 | 1.2893e-02 | 1.9739 | 1.0040e-03 | 1.9745 | 9.5011e-04 | 1.9726 | 2.19s
| 0.025 | 3.2382e-03 | 1.9933 | 2.5212e-04 | 1.9935 | 2.3866e-04 | 1.9931 | 3.34s
| 0.0125 | 8.1054e-04 | 1.9982 | 6.3100e-05 | 1.9984 | 5.9737e-05 | 1.9983 | 5.28s
1.5 | 0.1 | 5.4014e-02 | - | 6.6000e-03 | - | 5.9641e-03 | - | 1.76s
| 0.05 | 1.3701e-02 | 1.9790 | 1.6773e-03 | 1.9763 | 1.5190e-03 | 1.9731 | 2.22s
| 0.025 | 3.4379e-03 | 1.9947 | 4.2105e-04 | 1.9941 | 3.8152e-04 | 1.9933 | 3.41s
| 0.0125 | 8.6037e-04 | 1.9985 | 1.0537e-04 | 1.9985 | 9.5486e-05 | 1.9984 | 5.46s
1.8 | 0.1 | 5.1174e-02 | - | 6.3376e-03 | - | 5.2555e-03 | - | 1.67s
| 0.05 | 1.2977e-02 | 1.9795 | 1.6217e-03 | 1.9664 | 1.3529e-03 | 1.9578 | 2.12s
| 0.025 | 3.2563e-03 | 1.9946 | 4.0782e-04 | 1.9915 | 3.4072e-04 | 1.9894 | 3.22s
| 0.0125 | 8.1506e-04 | 1.9982 | 1.0211e-04 | 1.9978 | 8.5323e-05 | 1.9976 | 5.18s

Fig. 6: Time evolution of numerical solutions of CN–SGM for Example 6.2 with
$\tau=1/50$ and $N=200$ for $\kappa_{2}=0$.

Fig. 7: Time evolution of the relative mass and energy deviations of CN–SGM
for Example 6.2 with $\tau=1/50$ and $N=150$ for $\kappa_{2}=0.01$.

Fig. 8: Time evolution of the mass and energy of CN–SGM and ESAV–SGM for
Example 6.2 with $\tau=1/50$ and $N=150$ for $\kappa_{2}=1$ till the solutions
blow up.

##  7 Conclusions

In this paper, we focus on the spectral Galerkin methods for the coupled
fractional nonlinear Klein-Gordon-Schrödinger equation. We have proposed two
structure-preserving schemes, also analyzed the advantages and disadvantages
of each other by numerical comparisons. In theoretical aspects, the maximum-
norm boundness of numerical solutions of the CN–SGM are proved, and the unique
solvability is obtained with the help of Browder fixed point theorem and
maximum-norm boundness of the numerical solutions. Moreover, the unconditional
convergence are analyzed without any restriction on grid ratio. Numerical
results are reported to verify that the proposed numerical schemes have
excellent capabilities in numerical accuracy and long-time conservations. In
future work, the convergence proof of the ESAV-SGM will be carried out, also,
our schemes and theoretical results will be extended to the high-dimensional
FNKGS equation.

##  8 Acknowledgements

This work is partially supported by the National Natural Science Foundation of
China (Grant Nos. 12171245, 11971242) and the Research Start-up Foundation of
Jiangxi Normal University.

##  9 Declarations

Conflict of interest The authors declare that they have no conflict of
interest.

##  Appendix A Appendix: Proof of Theorem 3.3

The proof of Theorem 3.3 is divided into two parts, including the existence
and uniqueness.

(I) Existence:

###### Proof.

It is worth noting that $U_{N}^{n+1}=2U_{N}^{n+1/2}-U_{N}^{n}$ and
$\delta_{t}\Phi_{N}^{n+1/2}=\Psi_{N}^{n+1/2}$. We reformulate the spectral
Galerkin scheme (3.12)–(3.14) into the following form

$\displaystyle(U_{N}^{n+1/2}-U_{N}^{n},w_{N})+\frac{\lambda\tau}{4}\textrm{i}\mathcal{B}(U_{N}^{n+1/2},w_{N})-\frac{\kappa_{1}\tau}{2}\textrm{i}(U_{N}^{n+1/2}\Phi_{N}^{n+1/2},w_{N})$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{\kappa_{2}\tau}{2}\textrm{i}\Big{(}(|2U_{N}^{n+1/2}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2})U_{N}^{n+1/2}\Phi_{N}^{n+1/2},w_{N}\Big{)}=0,$
(A.1)
$\displaystyle(2\Phi_{N}^{n+1/2}-2\Phi_{N}^{n}-\tau\Psi_{N}^{n},w_{N})+\frac{\gamma\tau^{2}}{2}\mathcal{B}(\Phi_{N}^{n+1/2},w_{N})+\frac{\eta^{2}\tau^{2}}{2}(\Phi_{N}^{n+1/2},w_{N})$
$\displaystyle\qquad-\frac{\kappa_{1}\tau^{2}}{4}\Big{(}(|2U_{N}^{n+1/2}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2}),w_{N}\Big{)}-\frac{\kappa_{2}\tau^{2}}{4}\Big{(}(|2U_{N}^{n+1/2}-U_{N}^{n}|^{4}+|U_{N}^{n}|^{4}),w_{N}\Big{)}=0.$
(A.2)

Now, we carry out proving the existence of $U_{N}^{n+1/2}$ and
$\Phi_{N}^{n+1/2}$. For convenience, we define the map
$\textbf{s}=(s_{1},s_{2})$, $\mathscr{F}=(\mathscr{F}_{1},\mathscr{F}_{2}):$
$(X_{N}^{0}(\Omega),X_{N}^{0}(\Omega))\rightarrow(X_{N}^{0}(\Omega),X_{N}^{0}(\Omega))$,
such that

$\displaystyle(\mathscr{F}_{1}(\textbf{s}),w_{N})=$
$\displaystyle(s_{1}-U_{N}^{n},w_{N})+\frac{\lambda\tau}{4}\textrm{i}\mathcal{B}(s_{1},w_{N})-\frac{\kappa_{1}\tau}{2}\textrm{i}(s_{1}s_{2},w_{N})$
(A.3)
$\displaystyle-\frac{\kappa_{2}\tau}{2}\textrm{i}\Big{(}(|2s_{1}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2})s_{1}s_{2},w_{N}\Big{)},\quad\forall
w_{N}\in X_{N}^{0}(\Omega)$

and

$\displaystyle(\mathscr{F}_{2}(\textbf{s}),w_{N})=$
$\displaystyle(2s_{2}-2\Phi_{N}^{n}-\tau\Psi_{N}^{n},w_{N})+\frac{\gamma\tau^{2}}{2}\mathcal{B}(s_{2},w_{N})+\frac{\eta^{2}\tau^{2}}{2}(s_{2},w_{N})$
(A.4)
$\displaystyle-\frac{\kappa_{1}\tau^{2}}{4}\Big{(}|2s_{1}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2},w_{N}\Big{)}-\frac{\kappa_{2}\tau^{2}}{4}\Big{(}|2s_{1}-U_{N}^{n}|^{4}+|U_{N}^{n}|^{4},w_{N}\Big{)},\quad\forall
w_{N}\in X_{N}^{0}(\Omega).$

Choosing $w_{N}=s_{1}$ in (A.3) and taking the real part, from the Young’s
inequality, we derive

$\displaystyle\textrm{Re}(\mathscr{F}_{1}(\textbf{s}),s_{1})=$
$\displaystyle\textrm{Re}(s_{1}-U_{N}^{n},s_{1})\\}=\|s_{1}\|^{2}-\textrm{Re}(U_{N}^{n},s_{1})\geq\frac{1}{2}\|s_{1}\|^{2}-\frac{1}{2}\|U_{N}^{n}\|^{2}\geq\frac{1}{2}\|s_{1}\|^{2}-\mathcal{C}.$

Setting $w_{N}=s_{2}$ in (A.4), by utilizing the Cauchy-Schwarz inequality, we
obtain

$\displaystyle(\mathscr{F}_{2}(\textbf{s}),s_{2})\geq(2s_{2}-2\Phi_{N}^{n}-\tau\Psi_{N}^{n},s_{2})-\frac{\kappa_{1}\tau^{2}}{4}\Big{(}|2s_{1}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2},s_{2}\Big{)}-\frac{\kappa_{2}\tau^{2}}{4}\Big{(}|2s_{1}-U_{N}^{n}|^{4}+|U_{N}^{n}|^{4},s_{2}\Big{)}$
$\displaystyle=2\Big{\|}s_{2}\Big{\|}^{2}-(2\Phi_{N}^{n}+\tau\Psi_{N}^{n},s_{2})-\frac{\kappa_{1}\tau^{2}}{4}\Big{(}|2s_{1}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2},s_{2}\Big{)}-\frac{\kappa_{2}\tau^{2}}{4}\Big{(}|2s_{1}-U_{N}^{n}|^{4}+|U_{N}^{n}|^{4},s_{2}\Big{)}.$
(A.5)

By employing the boundness of the numerical solutions (see Theorem 3.2) and
the Young’s inequality, for sufficient small $\tau$, we conclude

$\displaystyle(2\Phi_{N}^{n}+\tau\Psi_{N}^{n},s_{2})\leq\Big{\|}2\Phi_{N}^{n}+\tau\Psi_{N}^{n}\Big{\|}\Big{\|}s_{2}\Big{\|}\leq\frac{1}{2}\Big{\|}2\Phi_{N}^{n}+\tau\Psi_{N}^{n}\Big{\|}^{2}+\frac{1}{2}\Big{\|}s_{2}\Big{\|}^{2},$
(A.6)
$\displaystyle\frac{\kappa_{1}\tau^{2}}{4}\Big{(}|2s_{1}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2},s_{2}\Big{)}$
$\displaystyle\leq\frac{\kappa_{1}^{2}\tau^{4}}{32}\Big{\|}|2s_{1}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2}\Big{\|}^{2}+\frac{1}{2}\Big{\|}s_{2}\Big{\|}^{2}\leq\mathcal{C}+\frac{1}{2}\Big{\|}s_{2}\Big{\|}^{2}$
(A.7)

and

$\displaystyle\frac{\kappa_{2}\tau^{2}}{2}\Big{(}|2s_{1}-U_{N}^{n}|^{4}+|U_{N}^{n}|^{4},s_{2}\Big{)}\leq\mathcal{C}+\frac{1}{2}\Big{\|}s_{2}\Big{\|}^{2}.$
(A.8)

Substituting (A.6)–(A.8) into (A), we arrive at

$\displaystyle(\mathscr{F}_{2}(\textbf{s}),s_{2})\geq\frac{1}{2}\Big{\|}s_{2}\Big{\|}^{2}-\frac{1}{2}\Big{\|}2\Phi_{N}^{n}+\tau\Psi_{N}^{n}\Big{\|}^{2}-\mathcal{C}.$

Therefore, we have

$\displaystyle\textrm{Re}(\mathscr{F}(\textbf{s}),s_{1})$
$\displaystyle=\textrm{Re}(\mathscr{F}_{1}(\textbf{s}),s_{2})+\textrm{Re}(\mathscr{F}_{2}(\textbf{s}),s)$
$\displaystyle\geq\frac{1}{2}\Big{\|}\textbf{s}\Big{\|}^{2}-\frac{1}{2}\Big{\|}2\Phi_{N}^{n}+\tau\Psi_{N}^{n}\Big{\|}^{2}-\mathcal{C}=\frac{1}{2}\Big{(}\Big{\|}\textbf{s}\Big{\|}^{2}-\Big{\|}\Big{(}2\Phi_{N}^{n}+\tau\Psi_{N}^{n},\sqrt{2\mathcal{C}}\Big{)}\Big{\|}^{2}\Big{)}.$

Taking
$\delta=\Big{\|}\Big{(}2\Phi_{N}^{n}+\tau\Psi_{N}^{n},\sqrt{2\mathcal{C}}\Big{)}\Big{\|}$,
which satisfies the condition of Lemma 3.5, we derive

$\displaystyle\text{Re}(\mathscr{F}(\textbf{s}),\textbf{s})\geq
0,\quad\forall\textbf{s}:~{}~{}\|\textbf{s}\|=\delta.$

This proves the existence of the numerical solution of (3.12)–(3.14). ∎

Next, we prove the uniqueness of the numerical solution.

(II) Uniqueness:

###### Proof.

We prove the theorem by introduction. It is obvious to find from (3.15) that
the numerical solution
$(U_{N}^{0},\Phi^{0}_{N},\Psi_{N}^{0})\in(X_{N}^{0}(\Omega),X_{N}^{0}(\Omega),X_{N}^{0}(\Omega))$
exists and is unique. Assume that $(U_{N}^{n},\Phi^{n}_{N},\Psi_{N}^{n})$ is
the unique solution of (3.12)–(3.14) for $n=0,1,\cdots,N_{t}-1$. Next, we
prove the uniqueness of the solution $(U_{N}^{n+1/2},\Phi^{n+1/2}_{N})$.
Assume there are two solutions $X^{n+1/2}=(X_{1}^{n+1/2},X_{2}^{n+1/2})$ and
$Y^{n+1/2}=(Y_{1}^{n+1/2},Y_{2}^{n+1/2})$ for scheme (3.12)–(3.14). Then
$X_{1}^{n+1/2}-Y_{1}^{n+1/2}$ and $X_{2}^{n+1/2}-Y_{2}^{n+1/2}$ satisfy (A.3)
and (A.4) as follows

$\displaystyle(\mathscr{F}_{1}({X}^{n+1/2})-\mathscr{F}_{1}({Y}^{n+1/2}),X_{1}^{n+1/2}-Y_{1}^{n+1/2})=0,$
$\displaystyle(\mathscr{F}_{2}({X}^{n+1/2})-\mathscr{F}_{2}({Y}^{n+1/2}),X_{2}^{n+1/2}-Y_{2}^{n+1/2})=0.$

By the definition of $\mathscr{F}_{1}$, we have

$\displaystyle\Big{\|}X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{\|}^{2}$
$\displaystyle+\frac{\lambda\tau}{4}\textrm{i}\mathcal{B}(X_{1}^{n+1/2}-Y_{1}^{n+1/2},X_{1}^{n+1/2}-Y_{1}^{n+1/2})$
$\displaystyle-\frac{\kappa_{1}\tau}{2}\textrm{i}(X_{1}^{n+1/2}X_{2}^{n+1/2}-Y_{1}^{n+1/2}Y_{2}^{n+1/2},X_{1}^{n+1/2}-Y_{1}^{n+1/2})$
(A.9)
$\displaystyle-\frac{\kappa_{2}\tau}{2}\textrm{i}\Big{(}(|2X_{1}^{n+1/2}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2})X_{1}^{n+1/2}X_{2}^{n+1/2}$
$\displaystyle-(|2Y_{1}^{n+1/2}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2})Y_{1}^{n+1/2}Y_{2}^{n+1/2},X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{)}=0.$

Taking the real part of (A), by admitting the boundness of the numerical
solutions (see Theorem 3.2), Lemma 3.4 and the Cauchy-Schwarz inequality, we
obtain

$\displaystyle\Big{\|}X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{\|}^{2}$
$\displaystyle=-\frac{\kappa_{1}\tau}{2}\textrm{Im}(X_{1}^{n+1/2}X_{2}^{n+1/2}-Y_{1}^{n+1/2}Y_{2}^{n+1/2},X_{1}^{n+1/2}-Y_{1}^{n+1/2})$
$\displaystyle\quad+\frac{\kappa_{2}\tau}{2}\textrm{Im}\Big{(}(|2X_{1}^{n+1/2}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2})X_{1}^{n+1/2}X_{2}^{n+1/2}$
(A.10)
$\displaystyle\quad-(|2Y_{1}^{n+1/2}-U_{N}^{n}|^{2}+|U_{N}^{n}|^{2})Y_{1}^{n+1/2}Y_{2}^{n+1/2},X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{)}$
$\displaystyle\leq\mathcal{C}\tau\Big{(}\Big{\|}X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{\|}^{2}+\Big{\|}X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{\|}^{2}\Big{)}.$

Taking into account of the definition of $\mathscr{F}_{2}$, we get

$\displaystyle\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}+\frac{\gamma\tau^{2}}{4}\mathcal{B}(X_{2}^{n+1/2}-Y_{2}^{n+1/2},X_{2}^{n+1/2}-Y_{2}^{n+1/2})$
$\displaystyle+\frac{\eta^{2}\tau^{2}}{4}(X_{2}^{n+1/2}-Y_{2}^{n+1/2},X_{2}^{n+1/2}-Y_{2}^{n+1/2})$
$\displaystyle\qquad-\frac{\kappa_{1}\tau^{2}}{8}\Big{(}(|2X_{2}^{n+1/2}-U_{N}^{n}|^{2})-(|2Y_{2}^{n+1/2}-U_{N}^{n}|^{2}),X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{)}$
$\displaystyle-\frac{\kappa_{2}\tau^{2}}{8}\Big{(}(|2X_{2}^{n+1/2}-U_{N}^{n}|^{4})-(|2Y_{2}^{n+1/2}-U_{N}^{n}|^{4}),X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{)}=0.$

By employing Theorem 3.2, Lemma 3.4 and the Cauchy-Schwarz inequality, we
deduce

$\displaystyle\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}$
$\displaystyle\leq\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}+\frac{\gamma\tau^{2}}{4}\Big{|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{|}^{2}_{\alpha/2}+\frac{\eta^{2}\tau^{2}}{4}\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}$
$\displaystyle=\frac{\kappa_{1}\tau^{2}}{8}\Big{(}(|2X_{2}^{n+1/2}-U_{N}^{n}|^{2})-(|2Y_{2}^{n+1/2}-U_{N}^{n}|^{2}),X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{)}$
$\displaystyle\qquad+\frac{\kappa_{2}\tau^{2}}{8}\Big{(}(|2X_{2}^{n+1/2}-U_{N}^{n}|^{4})-(|2Y_{2}^{n+1/2}-U_{N}^{n}|^{4}),X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{)}$
(A.11)
$\displaystyle\leq\mathcal{C}\tau^{2}\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}.$

Summing up (A) and (A), for a sufficient small $\tau$, we arrive at

$\displaystyle\Big{\|}X^{n+1/2}-Y^{n+1/2}\Big{\|}^{2}$
$\displaystyle=\Big{\|}X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{\|}^{2}+\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}$
$\displaystyle\leq\mathcal{C}\tau\Big{\|}X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{\|}^{2}+\mathcal{C}\tau\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}+\mathcal{C}\tau^{2}\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}$
$\displaystyle\leq\mathcal{C}\tau\Big{\|}X_{1}^{n+1/2}-Y_{1}^{n+1/2}\Big{\|}^{2}+\mathcal{C}\tau\Big{\|}X_{2}^{n+1/2}-Y_{2}^{n+1/2}\Big{\|}^{2}$
$\displaystyle=\mathcal{C}\tau\Big{\|}X^{n+1/2}-Y^{n+1/2}\Big{\|}^{2}.$

By assuming $\mathcal{C}\tau<1$, it leads to
$\Big{\|}X^{n+1/2}-Y^{n+1/2}\Big{\|}=0$, which implies
$X_{1}^{n+1/2}=Y_{1}^{n+1/2}$ and $X_{2}^{n+1/2}=Y_{2}^{n+1/2}$. This ends the
proof of the uniqueness. ∎

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