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Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces. (English) Zbl 1497.35418

Summary: The long-time behaviour of solutions to the defocussing modified Korteweg-de Vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of P. A. Deift and X. Zhou [Commun. Pure Appl. Math. 48, No. 3, 277–337 (1995; Zbl 0869.34047); Commun. Pure Appl. Math. 56, No. 8, 1029–1077 (2003; Zbl 1038.35113)] and its reformulation by M. Dieng et al. [Fields Inst. Commun. 83, 253–291 (2019; Zbl 1441.35047)] through \(\overline{\partial}\)-derivatives. To extend the asymptotics to solutions with initial data in lower-regularity spaces, we apply a global approximation via PDE techniques.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q15 Riemann-Hilbert problems in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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