Focusing nonlinear schrödinger equation and wave-packet collapse

Sulem, Catherine; Sulem, P. L.

doi:10.1016/s0362-546x(96)00168-x

Cited by 836 publications

(1,523 citation statements)

References 32 publications

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“…This transformation is analogous to the so-called lens transformation [18] which has been also used in other BEC problems [19,20] and differs (mainly because of the extra freedom provided by the definition of the new time τ ) from the transformations which are frequent in the context of dispersion-managed solitons [15]. For simplicity and without loss of generality we choose t 0 = 0, ℓ(0) = 1, L(0) = 1 and τ (0) = 0.…”

Section: Model Statement and Equations For The Transformation Parametersmentioning

confidence: 97%

“…(4) An obvious application is to start with stationary solutions of Eq. (3) which are known to exist for σ = −1 [18] and from them construct breathing and blowing up solutions of Eq. (1).…”

Section: Model Statement and Equations For The Transformation Parametersmentioning

confidence: 99%

“…(3) (i.e. the ground state [18]) when σ = −1. Then we have self-similar collapsing solutions with decaying nonlinear coefficient given by Eq.…”

Section: Systems Without Dispersion Management α(T) ≡mentioning

confidence: 99%

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Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

Pérez-Garcı́a

Torres

Konotop

2006

Physica D: Nonlinear Phenomena

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In this paper we use a similarity transformation connecting some families of Nonlinear Schrödinger equations with time-varying coefficients with the autonomous cubic nonlinear Schrödinger equation. This transformation allows one to apply all known results for that equation to the original non-autonomous case with the additional dynamics introduced by the transformation itself. In particular, using stationary solutions of the autonomous nonlinear Schrödinger equation we can construct exact breathing solutions to multidimensional non-autonomous nonlinear Schrödinger equations. An application is given in which we explicitly construct time dependent coefficients leading to solutions displaying weak collapse in three-dimensional scenarios. Our results can find physical applicability in mean field models of Bose-Einstein condensates and in the field of dispersion-managed optical systems.

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Section: Model Statement and Equations For The Transformation Parametersmentioning

confidence: 97%

Section: Model Statement and Equations For The Transformation Parametersmentioning

confidence: 99%

See 1 more Smart Citation

Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

Pérez-Garcı́a

Torres

Konotop

2006

Physica D: Nonlinear Phenomena

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“…The nonlinear Schrödinger equation also describes various phenomena arising in: self-channelling of a highpower ultra-short laser in matter, in the theory of Heisenberg ferromagnets and magnons, in dissipative quantum mechanics, in condensed matter theory, in plasma physics (e.g., the Kurihara superfluid film equation). We refer to Ablowitz, Prinari and Trubatch [1], Grosse and Martin [19] and Sulem [38] for a modern overview, including applications.…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

Dinu¹

2006

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. We establish the existence of an entire solution for a class of stationary Schrödinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows-up at infinity. The abstract framework is related to Lebesgue-Sobolev spaces with variable exponent. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the Mountain Pass Lemma without the PalaisSmale condition for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz [35] on the existence of ground-state solutions of the nonlinear Schrödinger equation.

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“…In this paper we study systems ruled by the two-dimensional nonlinear Schrö-dinger equation (NLSE) [1,2] with a cubic time-dependent nonlinearity. More precisely we look for solutions of the Cauchy problem i ∂u ∂t = − 1 2 △ + g(t)|u| 2 u (1a)…”

Section: Introductionmentioning

confidence: 99%

Numerical studies of stabilized Townes solitons

Montesinos

Pérez-Garcı́a

2005

Mathematics and Computers in Simulation

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We study numerically stabilized solutions of the two-dimensional Schrödinger equation with a cubic nonlinearity. We discuss in detail the numerical scheme used and explain why choosing the right numerical strategy is very important to avoid misleading results. We show that stabilized solutions are Townes solitons, a fact which had only been conjectured previously. Also we make a systematic study of the parameter regions in which these structures exist.

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Focusing nonlinear schrödinger equation and wave-packet collapse

Cited by 836 publications

References 32 publications

Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

Numerical studies of stabilized Townes solitons

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