Fast evaluation for the two-dimensional nonlinear coupled time–space fractional Klein–Gordon–Zakharov equations

Abstract

In this paper, we develop a fast algorithm to solve the two-dimensional nonlinear coupled time–space fractional Klein–Gordon–Zakharov (KGZ) equations. The $L2-1_{\sigma }$ method based on an efficient sum-of-exponentials (SOE) approximation and a Fourier spectral method are used to approximate the time and space direction, respectively. And we use the previous time levels to deal with the nonlinear terms to obtain a linearized numerical scheme. Finally, a numerical example is given to show that our numerical method is of second order accuracy in time, spectral accuracy in space and the fast algorithm is effective.

Introduction

In this paper, we consider the following nonlinear coupled time–space fractional KGZ equations,

$${}_0^C{D}_t^{{\alpha }_1}\psi (X,Y,t)=-{(-\Delta )}^{\frac{\beta }{2}}\psi (X,Y,t)-{\lambda }_1\psi (X,Y,t)-\psi (X,Y,t)\varphi (X,Y,t)-{\lambda }_2{\left|\psi (X,Y,t)\right|}^2\psi (X,Y,t),(X,Y,t)\in {\Omega }^{\prime }\times I,$$

$${}_0^C{D}_t^{{\alpha }_2}\varphi (X,Y,t)=c^2\Delta \varphi (X,Y,t)+\Delta \left({\left|\psi (X,Y,t)\right|}^2\right),(X,Y,t)\in {\Omega }^{\prime }\times I,$$

with the following initial and boundary conditions,

$$\psi (X,Y,0)={\psi }_0(X,Y),\varphi (X,Y,0)={\varphi }_0(X,Y),(X,Y)\in {\Omega }^{\prime },$$

$${\psi }_t(X,Y,0)={\psi }_1(X,Y),{\varphi }_t(X,Y,0)={\varphi }_1(X,Y),(X,Y)\in {\Omega }^{\prime },$$

$$\psi (X,Y,t)=0,\varphi (X,Y,t)=0,(X,Y)\in \partial {\Omega }^{\prime },t\in I,$$

where $1<{\alpha }_1,{\alpha }_2,\beta <2,{\lambda }_1,{\lambda }_2>0,{\Omega }^{\prime }=[a,b]\times [c,d],I=(0,T]$. $\psi$ and $\varphi$ are complex-valued functions representing the fast time scale component of the electric field raised by electrons and the derivation of ion density from its equilibrium, respectively. ${\psi }_0,{\psi }_1,{\varphi }_0,{\varphi }_1$ are the given smooth functions, and $c$ is the propagation speed of wave [1], [2], [3]. The fractional Laplacian ${(-\Delta )}^{\frac{\beta }{2}}$ is defined through the Fourier transform [4]

$$\widehat{{(-\Delta )}^{\frac{\beta }{2}}}u\left(\mathit{\xi }\right)={\left|\mathit{\xi }\right|}^{\beta }\hat{u}\left(\mathit{\xi }\right),$$

with $\hat{u}$ being the Fourier transform of $u$, namely,

$$\hat{u}\left(\mathit{\xi }\right)={\int }_{{\mathbb{R}}^2}{\text{e}}^{-\text{i}\mathbf{x}\cdot \mathit{\xi }}u\left(\mathbf{x}\right)\text{d}\mathbf{x},$$

where $\mathbf{x}=(x,y),{\text{i}}^2=-1$. The time fractional derivative in Caputo sense ${}_0^C{D}_t^{\alpha }(1<\alpha <2)$ is defined in [5],

$${}_0^C{D}_t^{\alpha }u\left(t\right)=\frac{1}{\Gamma (2-\alpha )}{\int }_0^t\frac{u^{\prime \prime }\left(s\right)\text{d}s}{{(t-s)}^{\alpha -1}}.$$

To reduce the cost of the storage and the computation, we resort to the SOE method [6], [7] to speed up the evaluation of the convolution integrals in (8).

It should be noted that Eqs. (1)–(2) become the classical nonlinear coupled KGZ equations when ${\alpha }_1={\alpha }_2=\beta =2$. The KGZ equations are classical nonlinear dispersive partial differential equations for describing the mutual interaction between the Langmuir waves and ion acoustic waves in a plasma, which play an important role in the investigation of the dynamics of strong Langmuir turbulence in plasma physics [8], [9]. There has been a great deal of interest to model diverse physical phenomena by using the fractional calculus [10]. The space fractional KGZ equations were numerically studied by finite difference method in [11], [12], [13]. However, to the best of our knowledge, there is no work that takes into account the Fourier spectral method and fast $L2-1_{\sigma }$ method for the time–space fractional KGZ equations.