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Weak asymptotic method for the study of propagation and interaction of infinitely narrow \(\delta\)-solitons. (English) Zbl 1054.35079

The authors present a new method for studying the interaction of solitons for nonintegrable Korteweg-de Vries (KdV) type equations with small dispersion. The equation under consideration is \[ u_t+(u^m)_x+\varepsilon^2u_{xxx}=0\tag{1} \] where \(m\) is integer \((m > 2)\), and \(\varepsilon\to 0\) is a small parameter. The main goal of the paper is to propose some general procedure (the weak asymptotic method) for describing the interaction of nonlinear waves. The authors encounter a problem of constructing a suitable definition of the weak asymptotic solution in the case in which the dispersion tends to zero (the so-called zero limit dispersion problem, see [P. D. Lax and C. D. Levermore, Commun. Pure Appl. Math. 36, 253-290 (1983; Zbl 0532.35067) and ibid. 36, 571–593 (1983; Zbl 0527.35073)]). However, in contrast to the papers mentioned above, the authors do not deal with oscillating solutions of the KdV equation or even with more general solutions: they consider in fact a special class of solutions, namely solitons. So, in the case considered by the authors, the zero dispersion limit leads to a system of differential equations, instead of integro-differential equations obtained in the papers cited above. In Section 1 auxiliary formulas of the weak asymptotic method are presented. In Sections 2 and 3, \(\varepsilon - \delta\) soliton interaction in the KdV models is discussed.
This article could be useful to people interested in physical applications of nonlinear wave theory.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35C20 Asymptotic expansions of solutions to PDEs
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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