Cauchy Problem for for some high order generalization of Korteweg - de Vries equation

TL;DR

This paper investigates the well-posedness and solution properties of a high-order generalization of the Korteweg-de Vries equation, including nonlinear and linear cases, with results on existence, uniqueness, and continuous dependence on initial data.

Contribution

It establishes well-posedness and solution regularity for a high-order KdV-type equation in various function spaces, extending previous results to more complex differential operators.

Findings

Proves well-posedness for linear and nonlinear cases.

Establishes existence and uniqueness of solutions in specified function spaces.

Demonstrates continuous dependence of solutions on initial conditions.

Abstract

In this work we study Cauchy problem for a high-order differential equation $\frac{\partial u(y,x)}{\partial y}+P(\frac{\partial}{\partial x})u(y,x)=\gamma\frac{\partial}{\partial x}(u^2(y,x))+F(y,x)$. We prove that the problem is well-posed both for linear ($\gamma =0$) and nonlinear equations on the class of rapidly decaying Schwartz functions. Furthermore, for the case when the initial condition is given on $L_2(\mathbf{R}^1)$ we prove the existence of the unique solution on the space $L_{\infty}(0,y_0; L_2(\mathbf{R}^1))\bigcap L_2(0,y_0; H^{n-1}(\mathbf{R}^1))\bigcap L_2(0,y_0;H^{n}(-r, r))$, where $r$ is an arbitrary positive number. It is also shown that the solution continuously depends on the initial conditions.