Partially Coherent Solitons on a Finite Background

Akhmediev, Nail; Ankiewicz, Adrian

doi:10.1103/physrevlett.82.2661

Cited by 252 publications

(248 citation statements)

References 31 publications

Supporting

Mentioning

238

Contrasting

Unclassified

Order By: Relevance

“…Physically, the solution (Φ, ψ) denotes the two-component beam in Kerr-like photorefractive media(cf. [1]). The positive constant µ j is for self-focusing in the j-th component of the beam.…”

Section: Introductionmentioning

confidence: 99%

Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations

Lin

Wei

2006

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

Conventionally, to learn wave collapse and optical turbulence, one must study finite-time blow-up solutions of one-component self-focusing nonlinear Schrödinger equations (NLSE). Here we consider simultaneous blow-up solutions of two-component system of self-focusing NLSE. By studying the associated self-similar solutions, we prove two components of solutions blow up at the same time. These self-similar solutions may come from solitary wave solutions with multi-bumps forming abundant geometric patterns which cannot be found in one-component self-focusing NLSE. Our results may provide the first step to investigate optical turbulence in two-component system of NLSE.

show abstract

Section: Introductionmentioning

confidence: 99%

Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations

Lin

Wei

2006

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…It turns out that one can obtain from Liouville-type theorems a variety of results on qualitative properties of solutions such as: universal, pointwise, a priori estimates of local solutions; universal and singularity estimates; decay estimates; universal bound of global solutions, initial and final blow-up rates, etc..., see [24,25] and references therein. In addition, it was shown in [34] that the parabolic system (1) can be used in the study of solutions of the corresponding elliptic problems, provided one can show suitable a priori bound of the global solutions of (1). This a priori bound property is a consequence of the Liouville-type theorems.…”

Section: Introductionmentioning

confidence: 99%

“…Now, for any R > 0, we rescale Finally, to treat the general case, we recall that the Liouville-type property of Theorem 2.1 for bounded solutions is sufficient for the proof of Proposition 3.1 (see the paragraph after Proposition 3.1). But after a time shift, formula (13) in Proposition 3.1 guarantees that any solution of (1) in R N × (−T, T ) has to satisfy u(x, t) ≤ CT −1/(p−1) in R N × (−T /2, T /2). The conclusion then follows by letting T → ∞.…”

Section: Using (23) Again and Integrating By Parts In T We Obtain (18)mentioning

confidence: 99%

“…In this paper, we shall use a different approach to establish a Liouville-type theorem for problem (1) in the whole space in a larger range of p and for any m. We shall then treat Liouville-type theorems in the half-space by reduction to the whole space case.…”

Section: Introductionmentioning

confidence: 99%

“…In nonlinear optics, the solution u i stands for the i-th component of the beam in Kerr-like photorefractive media (see e.g. [1]). The positive constant β ii is for self-focusing in the i-th component of the beam.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

Phan

Souplet²

2016

Math. Ann.

View full text Add to dashboard Cite

Abstract. We prove a Liouville-type theorem for semilinear parabolic systems of the formin the whole space R N × R. Very recently, Quittner [Math. Ann., DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for m = 2 in dimension N ≤ 2, and partial results in higher dimensions in the range p < N/(N − 2). By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions N ≥ 3. In particular, our results solve the important case of the parabolic Gross-Pitaevskii system -i.e. the cubic case r = 1 -in space dimension N = 3, for any symmetric (m, m)-matrix (β ij ) with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouvilletype theorem in the half-space R N + × R. As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.

show abstract

Energy‐critical scattering for focusing inhomogeneous coupled Schrödinger systems

Ghanmi,

Saanouni

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

This work investigates the time asymptotics of the inhomogeneous coupled Schrödinger equations , where , and . Here, one treats the energy‐critical regime and . This is the index of the invariant Sobolev norm under the dilatation . To the authors knowledge, the technique used in order to prove the scattering of an energy global solution to the above problem in the sub‐critical regime is no more applicable for . In order to overcome this difficulty, one uses the Kenig–Merle road map. In order to avoid a singularity of the source term, one considers the case , which restricts the space dimensions to . Moreover, in order to use the Sobolev injection , one restricts the space dimensions to . Compared with the previous work for the first author (Inhomogeneous coupled non‐linear Schrödinger systems. J. Math. Phys. 62, 101508 (2021)), the method consisting on dividing the integrals in the unit ball and its complementary seems not sufficient to conclude in the present study because of the energy‐critical exponent. Instead, one uses some Caffarelli–Kohn–Nirenberg weighted type inequalities.

show abstract

Partially Coherent Solitons on a Finite Background

Cited by 252 publications

References 31 publications

Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations

Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations

A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

Energy‐critical scattering for focusing inhomogeneous coupled Schrödinger systems

Contact Info

Product

Resources

About