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 $m_t+u{m}_x+b{u}_{x}m=\nu m_{xx}$ with $m=u-{\alpha }^2{u}_{xx}$ for fluid velocity $u(x,t)$ change their behavior at the special values $b=0,\pm 1,\pm 2,\pm 3$.

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

References in corpus (4)

• An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion
• The Navier-Stokes-alpha model of fluid turbulence
• Wave Structures and Nonlinear Balances in a Family of 1+1 Evolutionary PDEs
• A New Integrable Equation with Peakon Solutions

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