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Wave interactions and the analysis of the perturbed Burgers’ equation. (English) Zbl 1084.35078
Summary: In multiple-front solutions of the Burgers’ equation, all the fronts, except for two, are generated through the inelastic interaction of exponential wave solutions of the Lax pair associated with the equation. The inelastically generated fronts are the source of two interrelated difficulties encountered in the standard normal form expansion of the approximate solution of the perturbed Burgers’ equation, when the zero-order term is a multiple-front solution: (i) the higher-order terms in the expansion are not bounded; (ii) the normal form (equation obeyed by the zero-order approximation) is not asymptotically integrable; its solutions lose the simple wave structure of the solutions of the unperturbed equation. The freedom inherent in the normal form method allows a simple modification of the expansion procedure, making it possible to overcome both problems in more than one way. The loss of asymptotic integrability is shifted from the normal form to the higher-order terms (part of which has to be computed numerically) in the expansion of the solution. The front-velocity update is different from the one obtained in the standard analysis.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
37K55Perturbations, KAM for infinite-dimensional systems
35C20Asymptotic expansions of solutions of PDE