Integrable nonlinear Klein-Gordon systems with $\mathcal{PT}$ nonlocality and/or space-time exchange nonlocality

Abstract

In additional to the parity ($\mathcal{P}$) symmetric, time reversal ($\mathcal{T}$) symmetric, and $\mathcal{PT}$ symmetric nonlocal integrable systems, some other types of nonlocal integrable Klein-Gordon models with the space-time exchange nonlocality and the moving nolocality are proposed. The Lax pairs of the established nonlinear nonlocal Klein-Gordon equations are explicitly given. A special soliton solution, composed of $\mathcal{PT}$-symmetric part and $\mathcal{PT}$-antisymmetric part, is illustrated with the shape change.