Interacting localized structures with Galilean invariance

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Abstract
We consider a nonlinear partial differential equation that arises in the study of Hopf bifurcation in extended systems, as in the Kapitza problem. The equation in one space variable and time has dispersion and dissipation, and it is invariant under translation and Galilean boost. This equation contains the Burgers, Korteweg–de Vries, and Kuramoto-Sivashinsky equations as special cases. Numerical studies reveal that the complicated solutions of this equation may be seen as a mixture of elementary, pulselike solutions that, in the course of time, lock in and form stable lattices for a wide range of system parameters. By describing such states as bound states of single pulses, we can calculate the lattice spacings accurately–a simple formula gives these spacings. We also use this multiparticle description to derive equations of motion of unbound, interacting pulses. These equations go to the proper asymptotic states and provide a qualitatively plausible description. However, some quantitative discrepancies with the numerical simulations suggest that further aspects of such problems deserve further exploration.

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