An elementary proof of the existence of monotone traveling waves solutions in a generalized Klein–Gordon equation

Abstract

We analyze the existence and uniqueness of monotone traveling wavefront for a generalized nonlinear Klein–Gordon model ∂2ϕ∂t2−p+∂ϕ∂x2∂2ϕ∂x2+V′(ϕ)=0, using classical arguments of ordinary differential equations, with V(x) a potentials family containing the ϕ‐four potential Vfalse(xfalse)=M0false(1−x2false)2 and the sine‐Gordon‐type potential Vfalse(xfalse)=false(1false/2false)false(1+cosfalse(πxfalse)false). Also for these specific potentials, we give estimations of their monotone kink and anti‐kink solutions.