Modulational instability in the nonlocal -model

Wyller, John; Królikowski, Wiesław; Bang, Ole; Petersen, Dan; Rasmussen, Jens Juul

doi:10.1016/j.physd.2007.01.002

“…The nonlocality of nonlinearity may be caused by either transport processes, such as heat conduction [10] or ballistic atomic transport [11] and diffusion [12], charge separation [13], or long-range particle interaction, as in dipolar Bose-Einstein condensates [14][15][16] or nematic liquid crystals [17][18][19]. It has also been demonstrated that a parametric nonlinear wave interaction, such as second-harmonic generation, is in fact well described by a nonlocal nonlinearity, which has enabled accurate descriptions of quadratic solitons [20], modulational instability [21], and soliton pulse compression [22][23][24] in quadratic nonlinear materials.…”

Section: Introductionmentioning

confidence: 99%

Anomalous interaction of nonlocal solitons in media with competing nonlinearities

Esbensen

¹

,

Bache

²

,

Bang

³

et al. 2012

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“…They may cause an appearance of non-physical absolute instability of the problem solution if the spectral invariants are not taken into account. It should be mentioned that a presence of such singularities was also discussed in [32] at analysis of the modulation instability occurring in χ (2) -medium. We showed that the energy conservation law is valid at certain conditions on the incident pulse spectra: the spectra must not contain non-zero spectral amplitudes at two specific frequencies.…”

Section: Discussionmentioning

confidence: 95%

Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion

Trofimov

¹

,

Stepanenko

²

,

Razgulin

³

2019

PLoS ONE

4

0

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An interaction of laser pulse, containing a few cycles, with substance is a modern problem, attracting attention of many researches. The frequency conversion is a key problem for a generation of such pulses in various ranges of frequencies. Adequate description of such pulse interaction with a medium is based on a slowly evolving wave approximation (SEWA), which has been proposed earlier for a description of propagation of the laser pulse, containing a few cycles, in a medium with cubic nonlinear response. Despite widely applicability of the frequency conversion for various nonlinear optics problems solutions, SEWA has not been applied and developed for a theoretical investigation of the frequency doubling process until present time. In this study the set of generalized nonlinear Schrödinger equations describing a second harmonic generation of the super-short femtosecond pulse is derived. The equations set contains terms, describing the pulses self-steepening, and the second order dispersion (SOD) of the pulse, a diffraction of the beam as well as mixed derivatives. We propose the transform of the equations set to a type, which does not contain both the mixed derivatives and time derivatives of the nonlinear terms. This transform allows us to derive the integrals of motion of the problem: energy, spectral invariants and Hamiltonian. We show the existence of two specific frequencies (singularities in the Fourier space) inherent to the problem. They may cause an appearance of non-physical absolute instability of the problem solution if the spectral invariants are not taken into account. Moreover, we claim that the energy preservation at the laser pulses propagation may not occur if these invariants do not preserve. Developed conservation laws, in particular, have to be used for developing of the conservative finite-difference schemes, preserving the conservation laws difference analogues, and for developing of adequate theory of the modulation instability of the laser pulses, containing a few cycles.

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“…where ℏ > 0 is the adimensionalized Planck constant, γ ∈ ℝ is a parameter which is relevant in several applications in Physics for which we refer to [21,32], and which we assume here to be positive, V and K are given potentials, for the moment real continuous functions, and the nonlinearity is in the range 2N N−2 ⩽ p < 4N N−2 . The restriction on the Euclidean dimension is motivated by the fact that critical limit equations, related to (1.1) as ℏ → 0, possess explicit solutions which fail to have finite L 2 -energy in low dimension.…”

Section: Introductionmentioning

confidence: 99%

Blow-Up Phenomena and Asymptotic Profiles Passing from H ¹-Critical to Super-Critical Quasilinear Schrödinger Equations

Cassani

¹

,

Wang

²

2021

Advanced Nonlinear Studies

2

0

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We study the asymptotic profile, as ℏ → 0 {\hbar\rightarrow 0} , of positive solutions to - ℏ 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u - ℏ 2 + γ ⁢ u ⁢ Δ ⁢ u 2 = K ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , x ∈ ℝ N , -\hbar^{2}\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^{2}=K(x)\lvert u\rvert^{p-2% }u,\quad x\in\mathbb{R}^{N}, where γ ⩾ 0 {\gamma\geqslant 0} is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L 2 {L^{2}} -energy solutions. We investigate the concentrating behavior of solutions when γ > 0 {\gamma>0} and, differently from the case γ = 0 {\gamma=0} where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for γ > 0 {\gamma>0} we find a different concentration behavior of solutions in the case p = 2 ⁢ N N - 2 {p=\frac{2N}{N-2}} and when 2 ⁢ N N - 2 < p < 4 ⁢ N N - 2 {\frac{2N}{N-2}<p<\frac{4N}{N-2}} . This phenomenon does not occur when γ = 0 {\gamma=0} .

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Modulational instability in the nonlocal -model

Cited by 21 publications

References 30 publications

Anomalous interaction of nonlocal solitons in media with competing nonlinearities

Anomalous interaction of nonlocal solitons in media with competing nonlinearities

Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion

Blow-Up Phenomena and Asymptotic Profiles Passing from H ¹-Critical to Super-Critical Quasilinear Schrödinger Equations

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Modulational instability in the nonlocal -model

Cited by 21 publications

References 30 publications

Anomalous interaction of nonlocal solitons in media with competing nonlinearities

Anomalous interaction of nonlocal solitons in media with competing nonlinearities

Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion

Blow-Up Phenomena and Asymptotic Profiles Passing from H 1-Critical to Super-Critical Quasilinear Schrödinger Equations

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Product

Resources

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Blow-Up Phenomena and Asymptotic Profiles Passing from H ¹-Critical to Super-Critical Quasilinear Schrödinger Equations