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Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case. (English) Zbl 1465.35360

In this paper, the author studies the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations(NLS) \(i\partial_tu+\partial_x^2 u=F(|u|^2)u\) in the subcritical case with symmetry and spectrum assumptions.
The asymptotic stability of solitons for (NLS) started from V. S. Buslaev and G. S. Perel’man [St. Petersbg. Math. J. 4, No. 6, 63–102 (1992; Zbl 0853.35112); translation from Algebra Anal. 4, No. 6, 63–102 (1992)] where the one dimension (NLS) case was considered and further refined in [V. S. Buslaev and G. S. Perel’man Transl., Ser. 2, Am. Math. Soc. 164, 75–98 (1995; Zbl 0841.35108)] and [V. S. Buslaev and C. Sulem, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, No. 3, 419–475 (2003; Zbl 1028.35139)]. Their work was also extended to high dimensions and multi-solitons by many authors.
The aim of this paper is to prove asymptotic stability of solitons for (NLS) by using the vector field method to obtain more decay of the solution to the linearized equation with respect to time. Meanwhile, the author applied the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J. 86, No. 1, 109–142 (1997; Zbl 0874.35114)] to control the high moments of the solutions emerging from the vector fields method. The vector field method appeared originally in [S. Klainerman, Commun. Pure Appl. Math. 33, 43–101 (1980; Zbl 0405.35056)] where the small data global well-posedness of quasilinear wave equations was solved.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
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References:

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