Novel PT-invariant Solutions For a Large Number of Real Nonlinear Equations

TL;DR

This paper demonstrates that many real nonlinear equations, whether continuous or discrete, integrable or not, admit PT-invariant solutions constructed from hyperbolic and elliptic functions, expanding the solution space.

Contribution

It introduces a systematic method to generate PT-invariant solutions for a wide class of real nonlinear equations using hyperbolic and elliptic functions.

Findings

Real nonlinear equations admit PT-invariant solutions in terms of hyperbolic functions.

PT-invariant solutions also exist in the periodic case with Jacobi elliptic functions.

The approach applies to both integrable and nonintegrable equations.

Abstract

For a large number of real nonlinear equations, either continuous or discrete, integrable or nonintegrable, we show that whenever a real nonlinear equation admits a solution in terms of $\sech x$, it also admits solutions in terms of the PT-invariant combinations $\sech x \pm i \tanh x$. Further, for a number of real nonlinear equations we show that whenever a nonlinear equation admits a solution in terms $\sech^2 x$, it also admits solutions in terms of the PT-invariant combinations $\sech^2 x \pm i \sech x \tanh x$. Besides, we show that similar results are also true in the periodic case involving Jacobi elliptic functions.