Summary: We investigate the following family of evolutionary PDEs that describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids:
Here denotes . This convolution (or filtering) relates velocity to momentum density by integration against the kernel . We shall choose to be an even function so that and have the same parity under spatial reflection. When , this equation is both reversible in time and parity invariant. We shall study the effects of the balance parameter and the kernel on the solitary wave structures and investigate their interactions analytically for and numerically for small or zero viscosity.
This family of equations admits the classic Burgers ”ramps and cliffs” solutions, which are stable for with small viscosity.
For , the Burgers ramps and cliffs are unstable. The stable solution for moves leftward instead of rightward and tends to a stationary profile. When and , this profile is given by for and by for .
For , the Burgers ramps and cliffs are again unstable. The stable solitary traveling wave for and is the “pulson” , which restricts to the “peakon” in the special case when . Nonlinear interactions among these pulsons or peakons are governed by the superposition of solutions for and ,
These pulson solutions obey a finite-dimensional dynamical system for the time-dependent speeds and positions . We study the pulson and peakon interactions analytically, and we determine their fate numerically under adding viscosity.
Finally, as outlook, we propose an -dimensional vector version of this evolutionary equation with convection and stretching, namely,
for a defining relation, . These solutions show quasi-one-dimensional behavior for that we find numerically to be stable for . The corresponding superposed solutions of the vector -equation in dimensions exist, with coordinates , , , and parameters ,
These are momentum surfaces (or filaments for ), defined on surfaces (or curves) , . For , the satisfy canonical Hamiltonian equations for geodesic motion on the space of -vector valued -surfaces with cometric .