The KdV equation can be considered as a special case of the general equation ut + f(u)x − δg(uxx)x = 0, δ > 0, (0.1) where f is non-linear and g is linear, namely f(u) = u/2 and g(v) = v. As the parameter δ tends to 0, the dispersive behavior of the KdV equation has been throughly investigated (see, e.g., [11], [6], [2] and the references therein). We show through numerical evidence that a comp...

The unsmooth boundary will greatly affect motion morphology of a shallow water wave, and a fractal space is introduced to establish a generalized KdV-Burgers equation with fractal derivatives. The semi-inverse method is used to establish a fractal variational formulation of the problem, which provides conservation laws in an energy form in the fractal space and possible solution structures of t...

This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg-de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgĕım-Klibanov method.

In this article, we use the homotopy perturbation method and Adomian decomposition with Yang transformation to discover analytical solution time-fractional coupled Schrödinger–KdV equation. Caputo sense, fractional derivatives are described. A convergent series is used calculate solutions of PDEs. Analytical results achieved applying techniques numerically calculated represented in form tables ...

Abstract Steady plane turbulent free-surface flow over a slightly wavy bottom is considered for very large Reynolds numbers, small slopes, and Froude numbers close to the critical value 1. As in previous works, slope deviation from number are assumed be coupled such that turbulence modeling not required. The amplitudes of periodic elevations, however, half an order magnitude larger than case bu...

We present a simplified one-dimensional model for pulse wave propagation through fluid-filled tubes with elastic walls, which takes into account the elasticity of the wall as well as the tapering effect. The spatial dynamics in this model is governed by a variable coefficient KdV equation with conditions given at the inflow site. We discuss an existence theory for the associated evolution equat...

The N = 2 supersymmetric KdV equations are studied within the framework of Hirota’s bilinear method. For two such equations, namely N = 2, a = 4 and N = 2, a = 1 supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear Bäcklund transformation is given for the N = 2, a = 1 supersymmetr...

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...

In this work, we develop two new (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation and negative-order KdV-CBS (nKdV-nCBS) equation. The newly developed equations pass the Painlevé integrability test via examining compatibility conditions for each model. We examine dispersion relation derive multiple soliton solutions