Existence and breaking property of real loop-solutions of two nonlinear wave equations
Dynamical analysis has revealed that, for some nonlinear wave equations, loop- and inverted loop-solutions are actually visual artifacts. The so-called loop-soliton solution consists of three solutions, and is not a real solution. The paper answers the question as to whether or not nonlinear wave equations exist for which a ``real'' loop-solution exists, and if so, what are the precise parametric representations of these loop traveling wave solutions. The main result of the paper is the following theorem. Theorem. For any of the equations \[ u_{xxx}+\frac{A[A^3-2(3-4A^2)u^2+3(A-2)u^4]}{c(A-u^2)^4}u_t=0 \] and \[ u_{xxx}+\frac{4[B+(2B-3)u]}{c(1-u)^4}u_t=0, \] its corresponding traveling wave system has at least one family of open orbits which gives rise to a family of breaking loop-solutions of the underlying equation. In addition, these open orbits lie in an area bounded by a singular straight line and the stable and unstable manifolds of a saddle point of the system.
scientific article; zbMATH DE number 5573912
scientific article