Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation

Xu, Hang

doi:10.1007/s00209-016-1768-9

Cited by 31 publications

(36 citation statements)

References 16 publications

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“…a cascade of energy from low to high frequencies. In fact two main issues have been extensively studied in the literature: the first one concerns a priori bounds on how fast higher order Sobolev norms can grow along the flow associated with Hamiltonian PDEs (see [2,3,4,5,10,11,12,23,24,25,26,28,31] and all the references therein); the second one concerns the existence of global solutions whose higher order Sobolev norms are unbounded (see [9,13,14,15,16,30] and all the references therein).…”

Section: Introductionmentioning

confidence: 99%

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On the growth of Sobolev norms for NLS on 2- and 3-dimensional manifolds

2017

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Using suitable modified energies, we study higher-order Sobolev norms' growth in time for the nonlinear SchrÃ¶dinger equation (NLS) on a generic 2- or 3-dimensional compact manifold. In two dimensions, we extend earlier results that dealt only with cubic nonlinearities, and get polynomial-in-time bounds for any higher-order nonlinearities. In three dimensions, we prove that solutions to the cubic NLS grow at most exponentially, while for the subcubic NLS we get polynomial bounds on the growth of the H2 norm

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Section: Introductionmentioning

confidence: 99%

“…only the second term on the r.h.s. of (30) and all the other terms give a smaller power of u L ∞ H 2k ). Summarizing we get…”

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confidence: 98%

On the growth of Sobolev norms for NLS on 2- and 3-dimensional manifolds

2017

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“…This question, raised by Bourgain in [1], [2] for the defocusing nonlinear Schrödinger equation on the torus, led to several contributions constructing solutions with a small initial Sobolev norm of high regularity and a big Sobolev norm at some later time, see [4], [5], [8], [13], [16], [19], [15], [14], [12]. The actual existence of unbounded trajectories was proved in [21], [17], [18], [7], [23], [24], [3], [22], [9]. In this paper, we intend to study a case where a weak damping can promote unbounded trajectories in Sobolev spaces with high regularity.…”

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confidence: 99%

On a Damped Szegö Equation (With an Appendix in Collaboration With Christian Klein)

Gérard¹,

Grellier²

2020

SIAM J. Math. Anal.

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We investigate how damping the lowest Fourier mode modifies the dynamics of the cubic Szegő equation. We show that there is a nonempty open subset of initial data generating trajectories with high Sobolev norms tending to infinity. In addition, we give a complete picture of this phenomenon on a reduced phase space of dimension 6. An appendix is devoted to numerical simulations supporting the generalisation of this picture to more general initial data. Contents 14 4.2. The stable manifold of the periodic orbit 16 Appendix A. Numerical simulationsin collaboration with C. Klein(Université de Bourgogn References 33

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“…Using the method presented in [5], similar results have been obtained by adding a potential (see [4]), a harmonic trapping (see [6]), or by considering different derivatives along the Euclidean direction and the periodic one (see [12]). In the last mentioned article, Xu exhibits a scattering between the Schrödinger equation i∂ t U + ∆ R U − |∇ T |U = |U | 2 U in the spatial domain R × T and the cubic Szegő equation.…”

Section: Introductionmentioning

confidence: 67%