2016
DOI: 10.1007/s00209-016-1768-9
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Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation

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Cited by 31 publications
(36 citation statements)
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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%
See 1 more Smart Citation
“…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%
“…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…”
mentioning
confidence: 98%
“…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.…”
mentioning
confidence: 99%
“…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%