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

TL;DR

This paper demonstrates that many real nonlinear equations admit PT-invariant kink and periodic solutions, which are linearly stable and have reduced width, expanding the scope of solutions beyond traditional real solutions.

Contribution

It introduces PT-invariant solutions for a wide class of nonlinear equations, showing their stability and providing explicit forms, including for coupled systems with mixed PT eigenvalues.

Findings

PT-invariant kink solutions have half the width of real kinks.

Both kink and PT-invariant kink solutions are linearly stable.

Existence of PT-invariant solutions with mixed PT eigenvalues in coupled equations.

Abstract

For a large number of real nonlinear equations, either continuous or discrete, integrable or nonintegrable, uncoupled or coupled, we show that whenever a real nonlinear equation admits kink solutions in terms of $\tanh \beta x$, where $\beta$ is the inverse of the kink width, it also admits solutions in terms of the PT-invariant combinations $\tanh 2\beta x \pm i \sech 2 \beta x$, i.e. the kink width is reduced by half to that of the real kink solution. We show that both the kink and the PT-invariant kink are linearly stable and obtain expressions for the zero mode in the case of several PT-invariant kink solutions. Further, for a number of real nonlinear equations we show that whenever a nonlinear equation admits periodic kink solutions in terms of $\sn(x,m)$, it also admits periodic solutions in terms of the PT-invariant combinations $\sn(x,m) \pm i \cn(x,m)$ as well as $\sn(x,m)\pm i \dn(x,m)$. Finally, for coupled equations we show that one cannot only have complex PT-invariant solutions with PT eigenvalue $+1$ or $-1$ in both the fields but one can also have solutions with PT eigenvalue $+1$ in one field and $-1$ in the other field.