Related papers: On numerical integration of coupled Korteweg-de Vr…
We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution $X_t$ by…
A recursion operator is constructed for a new integrable system of coupled Korteweg - de Vries equations by the method of gauge-invariant description of zero-curvature representations. This second-order recursion operator is characterized…
In this paper we consider some dissipative versions of the modified Korteweg de Vries equation $u_t+u_{xxx}+|D_x|^{\alpha}u+u^2u_x=0$ with $0<\alpha\leq 3$. We prove some well-posedness results on the associated Cauchy problem in the…
We analyze the multisymplectic Preissman scheme for the KdV equation with the periodic boundary condition and show that the unconvergence of the widely-used iterative methods to solve the resulting nonlinear algebra system of the Preissman…
In this paper, numerical and solitonic solutions of Korteweg de Vries(KdV) and Korteweg de Vries-Burger's (KdVB) equations with initial and boundary conditions are calculated by sinc-collocation method. The basis of method is sinc…
The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well-known equations such as the Korteweg--de Vries, coupled KdV, Harry…
In the present work, firstly, we use a minimax equality to prove the existence of a solution of certain system of varitional equations and we provide a numerical approximation of such a solution. Then, we propose a numerical method to solve…
We consider the Cauchy problem associated to the fourth-order nonlinear Schr\"{o}dinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise…
The problem of controlling and stabilising solutions to the Kuramoto-Sivashinsky equation is studied in this paper. We consider a generalised form of the equation in which the effects of an electric field and dispersion are included. Both…
The numerical simulation of nonlinear dispersive waves is a central research topic of many investigations in the nonlinear wave community. Simple and robust solvers are needed for numerical studies of water waves as well. The main…
A family of generalized Korteweg-de Vries-Burgers equations in one space dimension with a nonlinear source is considered. The purpose of this contribution is twofold. On one hand, the local well-posedness of the Cauchy problem on periodic…
This paper represents a new perspective in understanding the controllability of the Korteweg-de Vries (KdV) equation on unbounded domains. By studying the equation on both the right and left half-line with a single control input, we show…
In this work, we present a general technique for establishing the strong convergence of numerical methods for stochastic delay differential equations (SDDEs) in the infinite horizon. This technique can also be extended to analyze certain…
We propose a numerical method for finding solitary wave solutions of generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue problem on an unbounded domain. The artificial boundary conditions are obtained to make the…
In this article we develop a numerical scheme to deal with interfaces between touching numerical grids when solving Schr\"o{}dinger equation. In order to pass the information among grids we use the values of the fields only at the contact…
This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the…
We investigate the integrable structure and soliton dynamics of a coupled modified Korteweg-de Vries (cmKdV) system with a real symmetric coupling matrix. We introduce a vector reformulation of Hirota's bilinear formalism in which both the…
This article is dedicated to improve the controllability results obtained by Cerpa et al. in Commun. Contemp. Math 13 (2011) and by Micu et al. in Commun. Contemp. Math 11 (5) (2009) for a nonlinear coupled system of two Korteweg-de Vries…
We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation $ \partial_t v+v\partial_x v+\partial_x^3 v+\delta(\partial_x^2 v +\partial_x^4 v)=0$, $\delta>0$, in the Korteweg-de…
We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k\geq5$ is an integer number and $\mu=\pm1$. In the focusing case ($\mu=1$), we show that if the initial data $u_0$…