A new complex line soliton for the two-dimensional KdV–Burgers equation

Making use of a extended tanh method with symbolic computation, we find a new complex line soliton for the two-dimensional (2D) KdV–Burgers equation. Its real part is the sum of the shock wave solution of a 2D Burgers equation and the solitary wave solution of a 2D KdV (KP) equation, and its imaginary part is the product of the shock wave solution of a 2D Burgers equation and the solitary wave solution of a 2D MKdV (MKP) equation.     

A conservative numerical method for the Cahn–Hilliard equation in complex domains

We propose an efficient finite difference scheme for solving the Cahn–Hilliard equation with a variable mobility in complex domains. Our method employs a type of unconditionally gradient stable splitting discretization. We also extend the scheme to compute the Cahn–Hilliard equation in arbitrarily shaped domains. We prove the mass conservation property of the proposed discrete scheme for complex domains. The resulting discretized equations are solved using a multigrid method. Numerical simulations are presented to demonstrate that the proposed scheme can deal with complex geometries robustly. Furthermore, the multigrid efficiency is retained even if the embedded domain is present.     

Numerical simulation of the soliton solutions for a complex modified Korteweg—de Vries equation by a finite difference method

In this paper,a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modifed Korteweg de Vries(MKdV)equation(which is equivalent to the Sasa-Satsuma equation)with the vanishing boundary condition.It is proved that such a numerical scheme has the second order accuracy both in space and time,and conserves the mass in the discrete level.Meanwhile,the resuling scheme is shown to be unconditionally stable via the von Nuemann analysis.In addition,an iterative method and the Thomas algorithm are used together to enhance the computational efficiency.In numerical experiments,this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation.The numerical accuracy,mass conservation and linear stability are tested to assess the scheme's performance.     

Fractional sub-equation method for Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative

ABSTRACT

In this paper, we present the exact solutions obtained for the space–time conformable generalized Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative. The conformable sub-equation method is applied to obtain the solutions; the solutions obtained are compared with the extended tanh-function method for the special case when the fractional order takes the integer order. The analytical solutions show that the conformable sub-equation method is very effective for the conformable-coupled KdV and mKdV equations.     

Compacton-like solutions for modified KdV and nonlinear Schrödinger equation with external sources

We present new types of compacton-like solutions for modified KdV and nonlinear Schrödinger equation with external sources, using a recently developed fractional transformation. In particular, we explicate these novel compactons for the trigonometric case, and compare their properties with those of the compactons and solitons in the case of modified KdV equation. Keeping in mind the significance of nonlinear Schrödinger equation with external source, for pulse propagation through asymmetric twin-core fibres, we hope that the newly found compacton may be launched in a long-haul telecommunication network utilizing asymmetric twin-core fibres.     

Using the complex WKB method for studying the evolution of initial pulses obeying the nonlinear Schrödinger equation

Using the complex WKB method, we found an asymptotic solution of the associated Zakharov-Shabat problem in the limit of a small coefficient h 0 multiplying the derivative of the potential that has a single hump. The obtained formulas can be used in describing the evolution of optical pulses of such shape obeying the nonlinear Schrödinger equation (NSE). Several examples are considered.V. V. Kuibyshev State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 19–25, May, 1993.     

Application of the group foliation method to the complex Monge-Ampère equation

We apply the method of group foliation to the complex Monge-Ampère equation (CMA ₂) to establish a regular framework for finding its non-invariant solutions. We employ an infinite symmetry subgroup ofCMA ₂ to produce a foliation of the solution space into orbits of solutions with respect to this group and a corresponding splitting ofCMA ₂ into an automorphic system and a resolvent system. We propose a new approach to group foliation which is based on the commutator algebra of operators of invariant differentiation. This algebra together with its Jacobi identities provides the commutator representation of the resolvent system. Presented by M.B. Sheftel at the DI-CRM Workshop held in Prague, 18–21 June 2000.     

Explicit multi-symplectic method for the Zakharov—Kuznetsov equation

We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.     

Multiple complex soliton solutions for the integrable KdV,fifth-order Lax,modified KdV,Burgers, and Sharma–Tasso–Olver equations

We show that completely integrable equations give multiple complex soliton solutions in addition to multiple real real soliton solutions. We exhibit complex categories of the simplified Hirota’s method to confirm these new findings. To demonstrate the power of the new complex forms, we test it on integrable KdV, fifth-order Lax, modified KdV, fifth-order modified KdV, Burgers, and Sharma–Tasso–Olver equations.     

Multi-symplectic method for the coupled Schrdinger–KdV equations

In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schr¨odinger-KdV equations(CSKE) with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral(MSFP)scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank–Nicholson(CN) method.Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.     

An operator splitting method for the Degasperis–Procesi equation

An operator splitting method is proposed for the Degasperis–Procesi (DP) equation, by which the DP equation is decomposed into the Burgers equation and the Benjamin–Bona–Mahony (BBM) equation. Then, a second-order TVD scheme is applied for the Burgers equation, and a linearized implicit finite difference method is used for the BBM equation. Furthermore, the Strang splitting approach is used to construct the solution in one time step. The numerical solutions of the DP equation agree with exact solutions, e.g. the multipeakon solutions very well. The proposed method also captures the formation and propagation of shockpeakon solutions, and reveals wave breaking phenomena with good accuracy.     

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq–Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq–Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.     

A nonconforming finite element method for the Cahn–Hilliard equation

This paper reports a fully discretized scheme for the Cahn–Hilliard equation. The method uses a convexity-splitting scheme to discretize in the temporal variable and a nonconforming finite element method to discretize in the spatial variable. And, the scheme can preserve the mass conservation and energy dissipation properties of the original problem. Some typical phase transition phenomena are also observed through the numerical examples.     

Non-standard finite-difference time-domain method for solving the Schrödinger equation

The literature on chaos has highlighted several chaotic systems with special features. In this work, a novel chaotic jerk system with non-hyperbolic equilibrium is proposed. The dynamics of this new system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents. In addition, we investigate the time-delay effects on the proposed system. Realisation of such a system is presented to verify its feasibility.     

Direct discontinuous Galerkin method for the generalized Burgers—Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge-Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.     

Solution of the radial Schrödinger equation by a modified “shooting” method

Based on a comparative analysis of various modifications of the shooting method, used for solving the radial Schrödinger equation, a new algorithm is proposed, which allows to increase considerably its solution accuracy. This is achieved, first, by rising the order of approximation of the Schrödinger equation fromh ⁴ (in the traditional Numerov-Cooley method) up toh ⁶ and, second, by using the variable integration steph. The computer program has been written in Fortran IV to execute this algorithm for calculating vibrational wave functions and related quantities for diatomic molecules.     

Gauge and Bäcklund transformations for the generalized sine-Gordon equation and its η-dependent modified equation

We study the generalized sine-Gordon hierarchy and its associated-dependent modified sine-Gordon hierarchy. Two Bäcklund transformations for these two families are constructed. One of them is a generalization of the Bäcklund transformations of Wadatiet al. and the other one is new. Gauge transformations of a relevant AKNS system are employed to reduce the integration of these equations via the Bäcklund transformations to quadratures. Three generations of explicit solutions of the sine-Gordon equation are presented.     

Global existence inL 1 for the modified nonlinear Enskog equation in ℝ3

A global existence theorem with large initial data inL ¹ is given for the modified Enskog equation in ³. The method, which is based on the existence of a Liapunov functional (analog of theH-Boltzmann theorem), utilizes a weak compactness argument inL ¹ in a similar way to the DiPerna-Lions proof for the Boltzmann equation. The existence theorem is obtained under certain condition on the behavior of the geometric factorY. The condition onY amounts to the fact that theL ¹ norm of the collision term grows linearly when the local density tends to infinity.     

Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

We examine a numerical method to approximate to a fractional diffusion equation with the Riesz fractional derivative in a finite domain, which has second order accuracy in time and space level. In order to approximate the Riesz fractional derivative, we use the “fractional centered derivative” approach. We determine the error of the Riesz fractional derivative to the fractional centered difference. We apply the Crank–Nicolson method to a fractional diffusion equation which has the Riesz fractional derivative, and obtain that the method is unconditionally stable and convergent. Numerical results are given to demonstrate the accuracy of the Crank–Nicolson method for the fractional diffusion equation with using fractional centered difference approach.