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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)].

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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\textit{Y. Martel} et al., Duke Math. J. 133, No. 3, 405--466 (2006; Zbl 1099.35134)

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