Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary pde

Abstract

We study exchange of stability in the dynamics of solitary wave solutions under changes in the nonlinear balance in a 1+1 evolutionary partial differential equation related both to shallow water waves and to turbulence. We find that solutions of the equation $ m_t + um_x +b u_xm = νm_{xx} $ with $m = u - α^2 u_{xx}$ for fluid velocity $u(x,t)$ change their behavior at the special values $b=0,\pm1,\pm2,\pm3$.

15 pages, 8 figures. For this replacement of the original submission, we: (1) Introduced key explanations that clarify the differences between Figs 1 and 2, versus 3 and 4. (2) Expanded the introduction to provide added motivation and precise definitions. (3) Added section and subsection headings to make the plan of the paper more evident. (4) Added a brief summary