Long time dynamics of highly concentrated solitary waves for the nonlinear Schrödinger equation. (English) Zbl 1304.35632

Summary: In this paper we study the behavior of solutions of a nonlinear Schrödinger equation in presence of an external potential, which is allowed to be singular at one point. We show that the solution behaves like a solitary wave for long time even if we start from an unstable solitary wave, and its dynamics coincides with that of a classical particle evolving according to a natural effective Hamiltonian.


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
35C08 Soliton solutions
Full Text: DOI arXiv


[1] Abou Salem, W.; Fröhlich, J.; Sigal, I. M., Colliding solitons for the nonlinear Schrödinger equation, Comm. Math. Phys., 291, 151-176, (2009) · Zbl 1184.35289
[2] Bellazzini, J.; Benci, V.; Ghimenti, M.; Micheletti, A. M., On the existence of the fundamental eigenvalue of an elliptic problem in \(R^N\), Adv. Nonlinear Stud., 7, 439-458, (2007) · Zbl 1136.47040
[3] Benci, V.; Ghimenti, M.; Micheletti, A. M., The nonlinear Schrödinger equation: soliton dynamics, J. Differential Equations, 249, 3312-3341, (2010) · Zbl 1205.35287
[4] Benci, V.; Ghimenti, M.; Micheletti, A. M., On the dynamics of solitons in the nonlinear Schrödinger equation, Arch. Ration. Mech. Anal., 205, 467-492, (2012) · Zbl 1256.35129
[5] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. I. existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345, (1982) · Zbl 0533.35029
[6] Bonanno, C.; Ghimenti, M.; Squassina, M., Soliton dynamics of NLS with singular potentials, Dyn. Partial Differ. Equ., 10, 177-207, (2013) · Zbl 1300.35130
[7] Bonanno, C.; d’Avenia, P.; Ghimenti, M.; Squassina, M., Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl., 417, 180-199, (2014) · Zbl 1332.35066
[8] Bronski, J. C.; Jerrard, R. L., Soliton dynamics in a potential, Math. Res. Lett., 7, 329-342, (2000) · Zbl 0955.35067
[9] Cazenave, T., Semilinear Schrödinger equations, Courant Lect. Notes Math., vol. 10, (2003), New York University, Courant Institute of Mathematical Sciences New York · Zbl 1055.35003
[10] Fröhlich, J.; Gustafson, S.; Jonsson, B. L.G.; Sigal, I. M., Solitary wave dynamics in an external potential, Comm. Math. Phys., 250, 613-642, (2004) · Zbl 1075.35075
[11] Fröhlich, J.; Gustafson, S.; Jonsson, B. L.G.; Sigal, I. M., Long time motion of NLS solitary waves in a confining potential, Ann. Henri Poincaré, 7, 621-660, (2006) · Zbl 1100.81019
[12] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal., 74, 160-197, (1987) · Zbl 0656.35122
[13] Holmer, J.; Zworski, M., Slow soliton interaction with delta impurities, J. Mod. Dyn., 1, 689-718, (2007) · Zbl 1137.35060
[14] Holmer, J.; Zworski, M., Soliton interaction with slowly varying potentials, Int. Math. Res. Not. IMRN, 2008, (2008), Art. ID rnn026 · Zbl 1147.35084
[15] Keraani, S., Semiclassical limit for nonlinear Schrödinger equations with potential II, Asymptot. Anal., 47, 171-186, (2006) · Zbl 1133.35092
[16] Kwong, M. K., Uniqueness of positive solutions of \(\operatorname{\Delta} u - u + u^p = 0\) in \(\mathbb{R}^n\), Arch. Ration. Mech. Anal., 105, 243-266, (1989) · Zbl 0676.35032
[17] Weinstein, M. I., Modulational stability of ground state of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, 472-491, (1985) · Zbl 0583.35028
[18] Weinstein, M. I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39, 51-67, (1986) · Zbl 0594.35005
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