The Inverse Scattering Transform for the Kdv Equation with Step-like Singular Miura Initial Profiles
نویسندگان
چکیده
We develop the inverse scattering transform for the KdV equation with real singular initial data q (x) of the form q (x) = r′ (x) + r (x), where r ∈ Lloc, r|R+ = 0. As a consequence we show that the solution q (x, t) is a meromorphic function with no real poles for any t > 0.
منابع مشابه
The Cole-hopf and Miura Transformations Revisited
An elementary yet remarkable similarity between the Cole-Hopf transformation relating the Burgers and heat equation and Miura's transformation connecting the KdV and mKdV equations is studied in detail. 1. Introduction Our aim in this note is to display the close similarity between the well-known Cole{Hopf transformation relating the Burgers and the heat equation, and the celebrated Miura trans...
متن کاملExplicit multiple singular periodic solutions and singular soliton solutions to KdV equation
Based on some stationary periodic solutions and stationary soliton solutions, one studies the general solution for the relative lax system, and a number of exact solutions to the Korteweg-de Vries (KdV) equation are first constructed by the known Darboux transformation, these solutions include double and triple singular periodic solutions as well as singular soliton solutions whose amplitude d...
متن کاملVariational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves
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...
متن کاملNew Exact Solutions of a Variable-Coefficient KdV Equation
In this paper, we apply the Miura transformation to construct the connection between a variablecoefficient KdV (vcKdV) equation and a variable-coefficient modified KdV (vcmKdV) equation under certain constraint. Solving the vcmKdV equation by use of the auxiliary equation method and using the Miura transformation, we find a rich variety of new exact solutions for the vcKdV equation, which inclu...
متن کاملA Novel Nonlinear Evolution Equation Integrable by the Inverse Scattering Method
In this report we consider the nonlinear evolution equation (ut +uux)x +u = 0 (Vakhnenko equation – VE) that can be integrated by the inverse scattering transform (IST) method. This equation arose as a result describing the high-frequency perturbations in a relaxing medium. The VE has two families of travelling wave solutions, both of which are stable to long wavelength perturbations. In partic...
متن کامل