Collapse for the higher-order nonlinear Schrödinger equation

We consider the asymptotic behavior of the solutions of a nonlinear Schrödinger (NLS) model incorporating linear and nonlinear gain/loss. First, we describe analytically the dynamical regimes (depending on the gain/loss strengths), for finite-time collapse, decay, and global existence of solutions in the dynamics. Then, for all the above parametric regimes, we use direct numerical simulations to study the dynamics corresponding to algebraically decaying initial data. We identify crucial differences between the dynamics of vanishing initial conditions, and those converging to a finite constant background: in the former (latter) case we find strong (weak) collapse or decay, when the gain/loss parameters are selected from the relevant regimes. One of our main results, is that in all the above regimes, non-vanishing initial data transition through spatiotemporal, algebraically decaying waveforms. While the system is nonintegrable, the evolution of these waveforms is reminiscent to the evolution of the Peregrine rogue wave of the integrable NLS limit. The parametric range of gain and loss for which this phenomenology persists is also touched upon.

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“…Then, applying Theorem II.1 for k ≥ 2, and employing the arguments of Refs. [47,48], we can prove the following Theorem:…”

Section: Analytical Considerationsmentioning

confidence: 98%

Spatiotemporal algebraically localized waveforms for a nonlinear Schrödinger model with gain and loss

Anastassi

Fotopoulos

Frantzeskakis

et al. 2017

Physica D: Nonlinear Phenomena

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“…We may show that the analytical upper estimate (2.12)-(2.13) on the collapse time T max is sharp for the initial data (3.1), as it is also in its continuous counterparts [24,26]. In fact, it can be verified that the evolution of either discrete or continuous plane-waves is governed by the same ODE dynamics, as we may effectively transfer the arguments of [24,26], from the continuous, to the discrete set-up: substituting in (1.1) the ansatz ψ n = W (t)e i Kxn , we find that W satisfies the equation…”

Section: Numerical Resultsmentioning

confidence: 78%

“…We may observe the similarity of the balance law (2.10) to the corresponding balance laws satisfied by the solutions of the continuous NLS counterparts incorporating gain and loss [24,26]. Indeed, Eq.…”

Section: Upper Bound For the Collapse Time For Periodic Boundary Condmentioning

confidence: 66%

“…Remark II.1. We may observe the identicality of the functional form of the analytical upper bounds for the problem (P), to the corresponding bounds derived for the continuous NLS counterparts incorporating gain and loss, when the latter are supplemented with periodic boundary conditions, [24,26]. This is due to the similarity of the balance law (2.10), and the analogy of the Cauchy-Schwarz inequality (A.5) with its continuous counterpart for integrals on finite intervals.…”

Section: Upper Bound For the Collapse Time For Periodic Boundary Condmentioning

confidence: 73%

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Collapse dynamics for the discrete nonlinear Schrödinger equation with gain and loss

Fotopoulos

Karachalios

Koukouloyannis

2019

Communications in Nonlinear Science and Numerical Simulation

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We discuss the finite-time collapse, also referred as blow-up, of the solutions of a discrete nonlinear Schrödinger (DNLS) equation incorporating linear and nonlinear gain and loss. This DNLS system appears in many inherently discrete physical contexts as a more realistic generalization of the Hamiltonian DNLS lattice. By using energy arguments in finite and infinite dimensional phase spaces (as guided by the boundary conditions imposed), we prove analytical upper and lower bounds for the collapse time, valid for both the defocusing and focusing cases of the model. In addition, the existence of a critical value in the linear loss parameter is underlined, separating finite timecollapse from energy decay. The numerical simulations, performed for a wide class of initial data, not only verified the validity of our bounds, but also revealed that the analytical bounds can be useful in identifying two distinct types of collapse dynamics, namely, extended or localized. Pending on the discreteness /amplitude regime, the system exhibits either type of collapse and the actual blow-up times approach, and in many cases are in excellent agreement, with the upper or the lower bound respectively. When these times lie between the analytical bounds, they are associated with a nontrivial mixing of the above major types of collapse dynamics, due to the corroboration of defocusing/focusing effects and energy gain/loss, in the presence of discreteness and nonlinearity. arXiv:1809.08025v2 [nlin.PS]

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“…[32], all possible regimes except γ > 0, δ < 0, are associated with finite-time collapse or decay. Furthermore, a critical value γ * can be identified in the regime γ < 0, δ > 0, which separates finite-time collapse from the decay of solutions.…”

Section: Motivation and Presentation Of The Modelmentioning

confidence: 96%

Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

et al. 2016

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The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects, gives rise to rich dynamics: this extends from Poincaré-Bendixson-type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections, or space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that the third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (i.e., even in the absence of the other higher-order effects) for the existence of the periodic, quasi-periodic and chaotic spatiotemporal structures. Suitable lowdimensional phase space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

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Collapse for the higher-order nonlinear Schrödinger equation

Cited by 6 publications

References 29 publications

Spatiotemporal algebraically localized waveforms for a nonlinear Schrödinger model with gain and loss

Spatiotemporal algebraically localized waveforms for a nonlinear Schrödinger model with gain and loss

Collapse dynamics for the discrete nonlinear Schrödinger equation with gain and loss

Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

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