An abstract functional framework is developed for the dual Petrov-Galerkin formulation of the initial-boundary-value problems with a third-order spatial derivative. This framework is then applied to study the wellposedness and decay properties of the KdV equation in a finite interval.

The KdV equation models the propagation of long waves in dispersive media, while the NLS equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A system that couples the two equations to model the interaction of long and short waves is mathematically attractive and such a system has been studied over the last decades. We evaluate the validity of this...

Abstract We study asymptotic reductions and solitary waves of a weakly nonlocal defocusing nonlinear Schrödinger (NLS) model. The hydrodynamic form the latter is analyzed by means multiscale expansion methods. To leading-order approximation (where only first moments response function present), we show that waves, in dark solitons, are governed an effective Boussinesq/Benney–Luke (BBL) equation,...

This paper proposes a new class of arbitrarily high-order conservative numerical schemes for the generalized Korteweg--de Vries (KdV) equation. approach is based on scalar auxiliary variable method. The equation reformulated into an equivalent system by introducing variable, and energy sum two quadratic terms. Therefore, preserving Runge--Kutta method will preserve all three invariants (momentu...

in this paper, we have studied on the solutions of modied kdv equation andalso on the stability of them. we use the tanh method for this investigationand given solutions are good-behavior. the solution is shock wave and can beused in the physical investigations

We propose integrable discretizations of derivative nonlinear Schrödinger (DNLS) equations such as the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reduc...

We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV equation to other NPDEs. The method is based upon a linearization principle which can be applied on nonlinearities which have a polynomial form. We illustrate the ...

In this research we discussed about the solution of KdV equation using Homotopy Perturbation method. The that describing water wave solved by mixing method between and was built with embedding parameter p∈[0,1] which undergoes a deformation process from linear problems to nonlinear assumed is expressed in form power series p up third order. result show each order obtained resonance term. for ha...