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Long-time dynamics of variable coefficient modified Korteweg-de Vries solitary waves. (English) Zbl 1112.35136

Summary: We study the long-time behavior of solutions to the Korteweg-de Vries-type equation \(\partial_tu= -\partial_x(\partial_x^2u+ f(u)-b(t,x)u)\), with initial conditions close to a stable, \(b=0\) solitary wave. The coefficient \(b\) is a bounded and slowly varying function, and \(f\) is a nonlinearity. For a restricted class of nonlinearities, we prove that for long time intervals, such solutions have the form of the solitary wave, whose center and scale evolve according to a certain dynamical law involving the function \(b(t,x)\), plus an \(H^1(\mathbb R)\)-small fluctuation. The result is stronger than those previously obtained for general nonlinearities \(f\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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