Connection Between the Exact Moving Solutions of the Negative Korteweg-de Vries (nKdV) Equation and the Negative Modified Korteweg-de Vries (nmKdV) Equation and the Static Solutions of 1+1 Dimensional $\phi^4$ Field Theory

TL;DR

This paper establishes a connection between moving solutions of the negative KdV and negative mKdV equations and static solutions of the $
+1$ dimensional $
^4$ field theory, introducing many new solutions and deriving an action principle.

Contribution

It reveals a mapping between the negative KdV/mKdV equations' solutions and $
^4$ field theory, and presents new solutions including rational ones, along with an action principle derivation.

Findings

Mapped moving solutions to static $
^4$ solutions

Generated numerous new solutions for nKdV and nmKdV equations

Derived an action principle for the nKdV equation

Abstract

The negative order KdV (nKdV) and the modified KdV (nmKdV) equations have two different formulations based on different hierarchy operators. Both equations can be written in terms of a nonlinear differential equation for a field $u(x,t)$ which we call the ``Lou form" of the equation. We find that for moving solutions of the nKdV equation and the nmKdV equation written in the ``Lou form" with $u(x,t) \rightarrow u (x-ct)= u(\xi) $, the equation for $u(\xi)$ can be mapped to the equation for the static solutions of the 1+1 dimensional $\phi^4$ field theory. Using this mapping we obtain a large number of solutions of the nKdV and the nmKdV equation, most of which are new. We also show that the nKdV equation can be derived from an Action Principle for both of its formulations. Furthermore, for both forms of the nmKdV equations as well as for both focusing and defocusing cases, we show that with a suitable ansatz one can decouple the $x$ and $t$ dependence of the nmKdV field $u(x,t)$ and obtain novel solutions in all the cases. We also obtain novel rational solutions of both the nKdV and the nmKdV equations.