Wave Structures and Nonlinear Balances in a Family of 1+1 Evolutionary PDEs

We study the following family of evolutionary 1+1 PDEs that describe the balance between convection and stretching for small viscosity in the dynamics of 1D nonlinear waves in fluids: m_t + \underbrace{um_x \} _{(-2mm)\hbox{convection}(-2mm)} + \underbrace{b u_xm \} _{(-2mm)\hbox{stretching}(-2mm)} = \underbrace{\nu m_{xx}\ }_{(-2mm)\hbox{viscosity}}, \quad\hbox{with}\quad u=g*m . Here $u=g*m$ denotes $u(x)=\int_{-\infty}^\infty g(x-y)m(y) dy .$ We study exchanges of stability in the dynamics of solitons, peakons, ramps/cliffs, leftons, stationary solutions and other solitary wave solutions associated with this equation under changes in the nonlinear balance parameter $b$.Comment: 69 pages, 26 figure

Discrete peakons

We demonstrate for the first time the possibility for explicit construction in a discrete Hamiltonian model of an exact solution of the form $\exp(-|n|)$, i.e., a discrete peakon. These discrete analogs of the well-known, continuum peakons of the Camassa-Holm equation [Phys. Rev. Lett. {\bf 71}, 1661 (1993)] are found in a model different from their continuum siblings. Namely, we observe discrete peakons in Klein-Gordon-type and nonlinear Schr\"odinger-type chains with long-range interactions. The interesting linear stability differences between these two chains are examined numerically and illustrated analytically. Additionally, inter-site centered peakons are also obtained in explicit form and their stability is studied. We also prove the global well-posedness for the discrete Klein-Gordon equation, show the instability of the peakon solution, and the possibility of a formation of a breathing peakon.Comment: Physica D, in pres