We consider the Cauchy problem for Klein–Gordon equation in $${\mathbb{R}}^{d},d\ge 1,$$ with random initial data. introduce a family of measures $${\mu }_{0}^{\varepsilon },\varepsilon >0$$ depending on small parameter ε. The }$$ are assumed to be locally homogeneous or slowly varying under spatial shifts order o(ε−1) and inhomogeneous ε−1. Moreover, correlation functions decrease uniformly ε ...

We have presented a new hyperbolic auxiliary function method for obtaining traveling wave solutions of nonlinear partial differential equations. Applying this, exact traveling wave solutions for the coupled Sine-Gordon equations are constructed.

Witten's non-relativistic formalism of supersymmetric quantum mechanics was based on a factorization and partnership between Schrödinger equations. We show how it accommodates a transition to the partnership between relativistic Klein-Gordon equations.

This paper presents the perturbation theory for the double–sine–Gordon equation. We received the system of differential equations that shows the soliton parameters modification under perturbation’s influence. In particular case λ = 0 the results of the research transform into well-known perturbation theory for the sine–Gordon equation.