Stability in \(H^1\) of the sum of \(K\) solitary waves for some nonlinear Schrödinger equations. (English) Zbl 1099.35134

Summary: We consider nonlinear Schrödinger (NLS) equations in \(\mathbb R^d\) for \(d=1, 2\), and 3. We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let \(R_k(t,x)\) be \(K\) solitary wave solutions of the equation with different speeds \(v_1,v_2,\dots,v_K\). Provided that the relative speeds of the solitary waves \(v_k-v_{k-1}\) are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the \(R_k(t)\) is stable for \(t\geq 0\) in some suitable sense in \(H^1\).
To prove this result, we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear Schrödinger equation. This property is similar to the \(L^2\) monotonicity property that has been proved by Y. Martel and F. Merle [ J. Math. Pures Appl. (9) 79, No. 4, 339–425 (2000; Zbl 0963.37058)] for the generalized Korteweg-de Vries (gKdV) equations and that was used to prove the stability of the sum of \(K\) solitons of the gKdV equations by the authors of the present article [Commun. Math. Phys. 231, 347–373 (2002; Zbl 1017.35098)].


35Q55 NLS equations (nonlinear Schrödinger equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51 Soliton equations
35B35 Stability in context of PDEs
Full Text: DOI Euclid


[1] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I: Existence of a ground state , Arch. Rational Mech. Anal. 82 (1983), 313–345. · Zbl 0533.35029
[2] V. S. Buslaev and G. S. Perelman, “On the stability of solitary waves for nonlinear Schrödinger equations” in Nonlinear Evolution Equations , Amer. Math. Soc. Transl. Ser. 2 164 , Amer. Math. Soc., Providence, 1995, 75–98. · Zbl 0841.35108
[3] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations , Comm. Math. Phys. 85 (1982), 549–561. · Zbl 0513.35007
[4] S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (2001), 1110–1145. · Zbl 1031.35129
[5] K. El Dika, Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation , Discrete Contin. Dyn. Syst. 13 (2005), 583–622. · Zbl 1083.35019
[6] K. El Dika and Y. Martel, Stability of \(N\)-solitary waves for the generalized BBM equations , Dyn. Partial Differ. Equ. 1 (2004), 401–437. · Zbl 1080.35116
[7] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, I: The Cauchy problem, general case , J. Funct. Anal. 32 (1979), 1–32. · Zbl 0396.35028
[8] M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry , I , J. Funct. Anal. 74 (1987), 160–197. · Zbl 0656.35122
[9] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145. · Zbl 0541.49009
[10] J. H. Maddocks and R. L. Sachs, On the stability of KdV multi-solitons , Comm. Pure Appl. Math. 46 (1993), 867–901. · Zbl 0795.35107
[11] Y. Martel, Asymptotic \(N\)-soliton-like solutions of the subcritical and critical generalized Korteweg –.de Vries equations , Amer. J. Math. 127 (2005), 1103–1140. · Zbl 1090.35158
[12] Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg –.de Vries equation , J. Math. Pures Appl. (9) 79 (2000), 339–425. · Zbl 0963.37058
[13] -, Asymptotic stability of solitons for subcritical generalized KdV equations , Arch. Ration. Mech. Anal. 157 (2001), 219–254. · Zbl 0981.35073
[14] -, Instability of solitons for the critical generalized Korteweg –.de Vries equation , Geom. Funct. Anal. 11 (2001), 74–123. · Zbl 0985.35071
[15] Y. Martel, F. Merle, and T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum of \(N\) solitons for subcritical gKdV equations , Comm. Math. Phys. 231 (2002), 347–373. · Zbl 1017.35098
[16] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation , J. Amer. Math. Soc. 14 (2001), 555–578. JSTOR: · Zbl 0970.35128
[17] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves , Comm. Math. Phys. 164 (1994), 305–349. · Zbl 0805.35117
[18] G. Perelman, “Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations” in Spectral Theory, Microlocal Analysis, Singular Manifolds , Math. Top. 14 , Akademie, Berlin, 1997, 78–137. · Zbl 0931.35164
[19] -, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations , Comm. Partial Differential Equations 29 (2004), 1051–1095. · Zbl 1067.35113
[20] I. Rodnianski, W. Schlag, and A. D. Soffer, Asymptotic stability of \(N\)-soliton states of NLS , to appear in Comm. Pure Appl. Math. · Zbl 1130.81053
[21] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations , SIAM J. Math. Anal. 16 (1985), 472–491. · Zbl 0583.35028
[22] -, Lyapunov stability of ground states of nonlinear dispersive evolution equations , Comm. Pure Appl. Math. 39 (1986), 51–67. · Zbl 0594.35005
[23] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media , Soviet Physics JETP 34 (1972), 62–69.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.