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 $φ^4$ Field Theory

TL;DR

This work establishes a precise link between traveling-wave solutions of the negative-order KdV ($n$KdV) and nmKdV equations and static solutions of $1+1$-D $\phi^4$ field theory via the Lou form. By mapping the traveling-wave ODEs to $\phi^4$-type equations, the authors generate a large family of traveling-wave solutions (including numerous rational and PT-invariant forms) for both $n$KdV-1 and nmKdV-1, with parallel results for nmKdV-2. They provide an action principle for the $n$KdV system, derive infinite conservation laws, and derive systematic symmetries that yield decoupled $x$- and $t$-dependent solutions across focusing and defocusing regimes. A major contribution is the explicit construction of many new solutions, supported by an extensive Appendix with additional traveling-wave and complex solutions, and the demonstration that nmKdV-1 and nmKdV-2 share large classes of solutions. The results deepen the connection between integrable negative-order equations and scalar field theory, offering new analytical tools and potential experimental relevance for non-smooth solitons and related phenomena.

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(ξ) $, the equation for $u(ξ)$ can be mapped to the equation for the static solutions of the 1+1 dimensional $φ^4$ 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.

Figures (8)

• Figure 1: The solution $v(\xi)$ of Eq. (\ref{['5.3']}) vs $\xi$ in case $m = 1/2, c = \beta = 1$ .
• Figure 2: The solution $u(\xi)$ of Eq. (\ref{['5.4']}) vs $\xi$ in case $m = 1/2, \beta = 1$ .
• Figure 3: The solution $u(x)$ of (\ref{['5.13']}) vs $\xi$ for $\beta=1, m=1/2$.
• Figure 4: The solution $v(\xi)$ of Eq. (\ref{['5.14']}) vs $\xi$ for $\beta = m=1$.
• Figure 5: The solution $u(\xi)$ of Eq. (\ref{['5.15']}) vs $\xi$ for $\beta = m=1$.