2021
DOI: 10.1093/imamat/hxab040
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Behaviour of solutions to the 1D focusing stochastic L2-critical and supercritical nonlinear Schrödinger equation with space-time white noise

Abstract: We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions.… Show more

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Cited by 14 publications
(13 citation statements)
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“…While the probability is very low, it is positive; this is consistent with results proved in [9,Thm 5.1]. We ran a similar experiment in the L 2 -critical case and did not observe any blow-up trajectories in 2000 runs for a variety of values of β; this is consistent with [31,Thm 2.7]. We conclude this section with mentioning that a similar positive probability of blow-up in finite time we observed in the case of space-time white noise in [32, end of Section 5].…”
Section: Probability Of Blow-upsupporting
confidence: 90%
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“…While the probability is very low, it is positive; this is consistent with results proved in [9,Thm 5.1]. We ran a similar experiment in the L 2 -critical case and did not observe any blow-up trajectories in 2000 runs for a variety of values of β; this is consistent with [31,Thm 2.7]. We conclude this section with mentioning that a similar positive probability of blow-up in finite time we observed in the case of space-time white noise in [32, end of Section 5].…”
Section: Probability Of Blow-upsupporting
confidence: 90%
“…One major difference (and difficulty) compared to the deterministic setting is that energy is not necessarily conserved in the stochastic perturbations. In the SNLS equation (1.1) with multiplicative noise (defined via the Stratonovich integral) the mass is conserved a.s., see [8], which allows to prove global existence of solutions in the L 2 -critical setting with M (u 0 ) < M (Q); see [31]. To further understand global behavior in the L 2 -supercritical setting one needs to control energy (as can be seen from Theorem 1).…”
Section: Introductionmentioning
confidence: 99%
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“…When the noise is of non-conservative type, the explosion, however, can be prevented with high probability as long as the strengthen of noise is large enough, which reflects the damped effect of non-conservative noise ( [7]). We also refer to [50] for the global well-posedness below the threshold in the mass-(super)critical case.…”
Section: Introductionmentioning
confidence: 99%
“…Such a trace formula for the energy will thus unfortunately not be as simple as the one for the mass. Very recent studies have been carried on for (mainly) the Crank-Nicolson scheme in the preprint [41]. In particular, it is observed that this numerical scheme does not verify an exact trace formula for the mass, see also the numerical experiments below.…”
Section: 1mentioning
confidence: 99%