Generalized PT-symmetric nonlinear Dirac equation: exact solitary waves solutions, stability and conservation laws

TL;DR

This paper derives exact solitary wave solutions for a $ ext{PT}$-symmetric nonlinear Dirac equation with power-law nonlinearity, analyzing their stability, conservation laws, and the effects of gain-loss mechanisms.

Contribution

It introduces explicit solitary wave solutions for the $ ext{PT}$-symmetric nonlinear Dirac equation with scalar-scalar interaction and explores their stability and conservation properties.

Findings

Energy is conserved despite gain-loss terms.

The $ ext{PT}$-transition point depends on solution existence, not nonlinearity exponent.

Moving solitons can have zero momentum at specific velocities.

Abstract

We derive an exact solitary wave solution for the $\PTb$-symmetric nonlinear Dirac equation with a scalar-scalar interaction. We consider a power-law nonlinearity of the form $|\bar{\Psi}\,\Psi|^{k}\,\Psi$ for positive values of $k$. The system's energy is conserved despite the presence of a gain-loss term, which is quantified by the parameter $\Lambda$. We show that the $\PTb$-transition point is defined by the solution's existence condition and is independent of the nonlinearity exponent $k$. Furthermore, momentum is conserved, although neither the canonical momentum nor the charge is a conserved quantity. A notable result is that the stationary solution, obtained from the continuity equations, exhibits nonzero momentum in its rest frame. We also derive a moving soliton solution, where the gain-loss parameter allows the soliton's velocity to be precisely chosen so that the moving soliton possesses zero momentum. Finally, we establish that the presence of a gain-loss mechanism and higher-order nonlinearity restrict the stability domain of the solutions.