Dynamics of Lump Solutions in a 2 + 1 NLS Equation

Villarroel, Javier; Prada, J.; Estévez, P.G.

doi:10.1111/j.1467-9590.2009.00440.x

Cited by 57 publications

(29 citation statements)

References 27 publications

(36 reference statements)

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“…/2)), while the minimal value is − 12, which corresponds to the point (z � 0, t � − 1). Similarly, Figure 2 reveals the structure of the lump solution (14) in the case of z � t � 0. e maximal value 49/72 corresponds to the points of (…”

Section: Complexitymentioning

confidence: 95%

“…Various solutions of bilinear equations, including solitons, positons, and complexitons, can be obtained via Wronskian formulation [13], while lump solutions can be obtained by taking long wave limit of solitons [3]. Lump is stable as soliton, the main difference between them lies in the fact that soliton decay exponentially in certain directions while a lump is a localized wave that decay rationally in all directions in space and moves with a uniform velocity [14]. In addition, soliton have a relation between amplitude and width but lump waves have no such relation.…”

Section: Introductionmentioning

confidence: 99%

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A Study on Lump and Interaction Solutions to a (3 + 1)‐Dimensional Soliton Equation

2019

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Based on bilinear formulation of a (3 + 1)-dimensional soliton equation, lump solution and related interaction solutions are investigated. e lump solutions of the soliton equation are classi ed into three cases with nonsingularity conditions being given. e interaction solutions between lump and a stripe soliton are obtained in eight cases, which have interesting fusing and ssion behaviors with changing time. e interaction solutions of the soliton equation between a lump and a resonant pair of stripe solitons are also given, and we nd that the lump just exist for a nite period during the interaction process.

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Section: Complexitymentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

A Study on Lump and Interaction Solutions to a (3 + 1)‐Dimensional Soliton Equation

2019

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“…This suggests that rogue waves can appear as a reduction of variables in the lump solutions. As it is well known, lumps are solutions whose meromorphic structure guarantees their stability [18]. This is the main motivation to propose a modified NLSE in 2 + 1 dimensions similarly to the generalization considered in Ref.…”

Section: Introductionmentioning

confidence: 99%

Lump solitons in a higher-order nonlinear equation in2+1dimensions

et al. 2016

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We propose and examine an integrable system of nonlinear equations that generalizes the nonlinear Schrödinger equation to 2 + 1 dimensions. This integrable system of equations is a promising starting point to elaborate more accurate models in nonlinear optics and molecular systems within the continuum limit. The Lax pair for the system is derived after applying the singular manifold method. We also present an iterative procedure to construct the solutions from a seed solution. Solutions with one-, two-, and three-lump solitons are thoroughly discussed.

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“…which generalizes the defocusing nonlinear Schrödinger equations (NLS) to two dimensions ( [11][12][13]). Note that integrable equations that generalize KdV and NLS to higher dimensions, while of the utmost interest, are extremely rare.…”

Section: Introductionmentioning

confidence: 99%

Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

Villarroel

Montero

2017

Stud Appl Math

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We study here the initial value problem for a two-dimensional Korteweg-de Vries (KdV) equation, first derived by Calogero and Bogoyavlenskii, by means of the inverse scattering transform. The dynamics of the discrete spectrum of an associated Schrödinger operator is far richer than that of KdV equation. Even for optimal eigenvalues, generic smooth solutions may develop shocks with multiple branches and/or cusp singularities in finite time. However, evolution may move poles of the transmission coefficient off the imaginary axis, destroy or even create them. We characterize conditions to prevent these pathologies before explosion time and describe ample classes of solutions, corresponding to both continuous and discrete spectrum. We also find that in certain conditions new eigenvalues might be created; in these cases a minimal set of initial spectral data must incorporate additionally the transmission coefficient T 0 (y, k), k ∈ C on the entire plane. The previous results are applied to describe the Cauchy problem corresponding to initial data combinations of delta terms and derivatives and show that for long time the delta singularity may persist or be smoothed to a cusp-discontinuity. Finally, we give conditions under which the evolution u(x, y, t) is reduced to the classical KdV.

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Dynamics of Lump Solutions in a 2 + 1 NLS Equation

Cited by 57 publications

References 27 publications

A Study on Lump and Interaction Solutions to a (3 + 1)‐Dimensional Soliton Equation

A Study on Lump and Interaction Solutions to a (3 + 1)‐Dimensional Soliton Equation

Lump solitons in a higher-order nonlinear equation in2+1dimensions

Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

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