Soliton dynamics and scattering

Soffer, Avy

doi:10.4171/022-3/24

Cited by 58 publications

(50 citation statements)

References 45 publications

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“…Such an assertion would be consistent with the (somewhat imprecise) soliton resolution conjecture; see [11], [15], [17] for further discussion.…”

Section: Introductionsupporting

confidence: 68%

A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential

Tao

2008

Dynamics of Partial Differential Equations

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Abstract. We study the asymptotic behavior of large data solutions in the energy spaceis a real potential (which could contain bound states), and 1 +is an exponent which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as t → +∞, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which converges in H to a compact attractor K, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in H. The main novelty of this result is that K is a global attractor, being independent of the initial energy of the initial data; in particular, no matter how large the initial data is, all but a bounded amount of energy is radiated away in the limit.

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“…Such an assertion would be consistent with the (somewhat imprecise) soliton resolution conjecture; see [11], [15], [17] for further discussion.…”

Section: Introductionsupporting

confidence: 68%

A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential

Tao

2008

Dynamics of Partial Differential Equations

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“…Thus we can try to weaken the conjecture in this case by allowing the pseudo-soliton component to merely be almost periodic in time, rather than be an actual soliton. As we shall see, this weakened statement is related to the petite conjecture in [21].…”

Section: Assumptions On the Solutionmentioning

confidence: 66%

A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations

Tao

2007

Dynamics of Partial Differential Equations

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Abstract. We study the asymptotic behavior of large data solutions to Schrödinger equations iut + ∆u = F (u) in R d , assuming globally bounded H 1 x (R d ) norm (i.e. no blowup in the energy space), in high dimensions d ≥ 5 and with nonlinearity which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as t → +∞, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which converges in H 1 x (R d ) to a compact attractor, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in H 1 x (R d ). This is despite the total lack of any dissipation in the equation. This statement can be viewed as weak form of the "soliton resolution conjecture". We also obtain a more complicated analogue of this result for the non-spherically-symmetric case. As a corollary we obtain the "petite conjecture" of Soffer in the high dimensional non-critical case.

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“…Indeed, soliton solutions represent a perfect balance between the focusing forces of the nonlinearity and the dispersive forces of the linear component. The basic line of thought in the subject, motivated by heuristics (Soffer [24]), rigorous partial results (Tao [27,28]), numerical simulation (Sulem-Sulem [26]), and analogy with the completely integrable one-dimensional case, is that a solution of (1.1) either completely disperses as t → ∞ (linear effects dominate), blows-up in finite time (nonlinear effects dominate) or the solution resolves into a sum of solitons 1 In view of the connection between solutions Q to (1.2) and solutions u(t) = e it Q to (1.1), and the fact that u(t) L 2 ∇u(t) L 2 is a scale invariant quantity for solutions u(t) to (1.1), it might be more natural to classify the family of solutions Q to (1.2) in terms of the quantity Q L 2 ∇Q L 2 rather than the mass. However, any solution Q to (1.2) must satisfy the Pohozhaev identity Q L 2 ∇Q L 2 = √ 3 Q 2 L 2 , and thus the two classifications are equivalent.…”

Section: Introductionmentioning

confidence: 99%

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

Holmer

Roudenko

2008

Commun. Math. Phys.

287

349

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Abstract. We consider the problem of identifying sharp criteria under which radial H 1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂ t u + ∆u + |u| 2 u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the, where u 0 denotes the initial data, and M [u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution e it Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if, then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution e it Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if, then the solution blows-up in finite time. The technique employed is parallel to that employed by in their study of the energy-critical NLS.

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Soliton dynamics and scattering

Cited by 58 publications

References 45 publications

A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential

A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential

A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

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