Numerical solution of Boussinesq systems of KdV--KdV type
نویسندگان
چکیده
Abstract In this paper we consider a coupled KdV system of Boussinesq type and its symmetric version. These systems were previously shown to possess generalized solitary waves consisting of a solitary pulse that decays symmetrically to oscillations of small, constant amplitude. We solve numerically the periodic initial-value problem for these systems using a high order accurate, fully discrete, Galerkin-finite element method. (In the case of the symmetric system, it is possible to prove rigorous, optimal-order, error estimates for this scheme.) The numerical scheme is used in an exploratory fashion to study radiating solitary-wave solutions of these systems that consist, in their simplest form, of a main, solitary-wave-like pulse that decays asymmetrically to small-amplitude, outward-propagating, oscillatory wave trains (ripples). In particular, we study the generation of radiating solitary waves, the onset of ripple formation and various aspects of the interaction and long time behaviour of these solutions.
منابع مشابه
Numerical solution of coupled KdV systems of Boussinesq equations: I. The numerical scheme and existence of generalized solitary waves
We consider some Boussinesq systems of water wave theory, which are of coupled KdV type. After a brief review of the theory of existence-uniqueness of solutions of the associated initial-value problems, we turn to the numerical solution of their initialand periodic boundary-value problems by unconditionally stable, highly accurate methods that use Galerkin/finite element type schemes with perio...
متن کاملNumerical Solution of Boussinesq Systems of Kdv-kdv Type: Ii. Evolution of Radiating Solitary Waves
In this paper we consider a coupled KdV system of Boussinesq type and its symmetric version. These systems were previously shown to possess Generalized Solitary Waves consisting of a solitary pulse that decays symmetrically to oscilations of small, constant amplitude. We solve numerically the periodic initial-value problem for these systems using a high order accurate, fully discrete, Galerkin-...
متن کاملUnconditionally Stable Difference Scheme for the Numerical Solution of Nonlinear Rosenau-KdV Equation
In this paper we investigate a nonlinear evolution model described by the Rosenau-KdV equation. We propose a three-level average implicit finite difference scheme for its numerical solutions and prove that this scheme is stable and convergent in the order of O(τ2 + h2). Furthermore we show the existence and uniqueness of numerical solutions. Comparing the numerical results with other methods in...
متن کاملHigher Order Modulation Equations for a Boussinesq Equation
In order to investigate corrections to the common KdV approximation to long waves, we derive modulation equations for the evolution of long wavelength initial data for a Boussinesq equation. The equations governing the corrections to the KdV approximation are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation al...
متن کاملWell-posednesss of Strongly Dispersive Two-dimensional Surface Waves Boussinesq Systems
We consider in this paper the well-posedness for the Cauchy problem associated to two-dimensional dispersive systems of Boussinesq type which model weakly nonlinear long wave surface waves. We emphasize the case of the strongly dispersive ones with focus on the “KdV-KdV” system which possesses the strongest dispersive properties and which is a vector two-dimensional extension of the classical K...
متن کامل