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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)].

MSC:

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
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