Unbounded Solutions of the Modified Korteweg-De Vries Equation

TL;DR

This paper establishes local existence and uniqueness of solutions for the focusing modified Korteweg-de Vries equation within unbounded function classes, using asymptotic expansions and discretization methods.

Contribution

It introduces a framework for solutions with asymptotic expansions at infinity and connects genuine solutions to Schwartz class functions solving a generalized equation.

Findings

Proves local existence and uniqueness in unbounded function classes

Shows asymptotic solutions differ from genuine solutions by Schwartz class functions

Uses discretization methods to solve the generalized mKdV equation

Abstract

We prove local existence and uniqueness of solutions of the focusing modified Korteweg - de Vries equation $u_t + u^2u_x + u_{xxx} = 0$ in classes of unbounded functions that admit an asymptotic expansion at infinity in decreasing powers of $x$. We show that an asymptotic solution differs from a genuine solution by a smooth function that is of Schwartz class with respect to $x$ and that solves a generalized version of the focusing mKdV equation. The latter equation is solved by discretization methods.