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.

Using Levi-Civita's theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equation, describing velocity a shallow fluid up to first order. We use perturbation theory, and long wave, slowly varying approximations for water. The KdV equation describes nontrivial dynamics all water particles from surface bottom layer. A crucial new step made in our work is proof that natural conseq...

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 paper, we apply the Exp-function method to Rosenau-Kawahara and Rosenau-KdV equations. Rosenau-Kawahara equation is the combination of the Rosenau and standard Kawahara equations and Rosenau-KdV equation is the combination of the Rosenau and standard KdV equations. These equations are nonlinear partial differential equations (NPDE) which play an important role in mathematical physics. E...

A new approach to the problem of asymptotic integrability of physical systems is developed and applied to the KdV equation with higher-order corrections. A central object of the approach is an integrable reference equation, which is constructed by defining a proper Lie– Bäcklund group of transformations and applying it to the leading order equation. It is shown that the solitary wave solutions ...

Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard...