Nonlinear self-modulation: An exactly solvable model (original) (raw)
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Nonlinear Dynamics, 2021
We investigate the modulation instability (MI) analysis of a nonautonomous (3+1)-dimensional coupled nonlinear Schrödinger (NLS) equation with timedependent dispersion and phase modulation coefficients. By employing standard linear stability analysis, we obtain an explicit expression for the MI gain as a function of dispersion, phase modulation, perturbation wave numbers and an initial incidence power. The nonautonomous coupled NLS equation is found to be modulationally unstable for the same sign of dispersion and phase modulation coefficients. This equation is modulationally stable for zero-dispersion and or phase modulation coefficients. But non-zero dispersion coefficient, it is modulationally stable/unstable on distinct bandwidth of wave numbers. The trigonometric, exponential, algebraic function of time and constant have been chosen as test functions for dispersion and phase modulation to find the effect on the MI analysis. The effect of focusing and defocusing medium on the MI analysis has also been investigated. The MI bandwidth in the focusing medium is found to be larger than defocusing medium. It is found that the modulation instability of the equation can be managed by proper choice of the dispersion and phase modulation parameters.
Modulation Theory for Self-Focusing in the Nonlinear Schrödinger–Helmholtz Equation
Numerical Functional Analysis and Optimization, 2009
The nonlinear Schrödinger-Helmholtz (SH) equation in N space dimensions with 2σ nonlinear power was proposed as a regularization of the classical nonlinear Schrödinger (NLS) equation. It was shown that the SH equation has a larger regime (1 ≤ σ < 4 N ) of global existence and uniqueness of solutions compared to that of the classical NLS (0 < σ < 2 N ). In the limiting case where the Schrödinger-Helmholtz equation is viewed as a perturbed system of the classical NLS equation, we apply modulation theory to the classical critical case (σ = 1, N = 2) and show that the regularization prevents the formation of singularities of the NLS equation. Our theoretical results are supported by numerical simulations.
Modulational Instability of Double-Periodic Waves in the Nonlinear Schrödinger Equation
2020
It is shown how to compute the modulational instability rates for the doubleperiodic solutions to the cubic NLS (nonlinear Schrödinger) equation by using the Lax linear equations. The wave function modulus of the double-periodic solutions is periodic both in space and time coordinates; such solutions generalize the standing waves which have the time-independent and space-periodic wave function modulus. Similar to other waves in the NLS equation, the double-periodic solutions are modulationally unstable and this instability is related to the bands of the Lax spectrum outside the imaginary axis. A simple numerical method is used to compute the modulational instability bands and to compare the instability rates of the double-periodic solutions with those of the standing periodic waves.
The Integrable Nature of Modulational Instability
Siam Journal on Applied Mathematics, 2015
We investigate the nonlinear stage of the modulational (or Benjamin-Feir) instability by characterizing the initial value problem for the focusing nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBC) at infinity. We do so using the recently formulated inverse scattering transform (IST) for this problem. While the linearization of the NLS equation ceases to be valid when the perturbations have grown sufficiently large compared to the background, the results of the IST remain valid for all times and therefore provide a convenient way to study the nonlinear stage of the modulational instability. We begin by studying the spectral problem for the Dirac operator (i.e., the first half of the Lax pair for the NLS equation) with piecewise constant initial conditions which are a generalization to NZBC of a potential well and a potential barrier. Since the scattering data uniquely determine the time evolution of the initial condition via the inverse problem, the study of these kinds of potentials provides a simple means of investigating the growth of small perturbations of a constant background via IST. We obtain several results. First, we prove that there are arbitrarily small perturbations of the constant background for which there are discrete eigenvalues, which shows that no area theorem is possible for the NLS equation with NZBC. Second, we prove that there is a class of perturbations for which no discrete eigenvalues are present. In particular, this latter result shows that solitons cannot be the primary vehicle for the manifestation of the instability, contrary to a recent conjecture. We supplement these results with a numerical study about the existence, number, and location of discrete eigenvalues in other situations. Finally, we compute the small-deviation limit of the IST, and we compare it with the direct linearization of the NLS equation around a constant background, which allows us to precisely identify the nonlinear analogue of the unstable Fourier modes within the IST. These are the Jost eigenfunctions for values of the scattering parameter belonging to a finite interval of the imaginary axis around the origin. Importantly, this last result shows that the IST contains an automatic mechanism for the saturation of the modulational instability.
A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation
Physics Letters A, 1998
In this Letter we i n troduce a systematic perturbation method for analyzing the e ect of small perturbations on critical self focusing by reducing the perturbed critical nonlinear Schr odinger equation PNLS to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simpli ed, depending on whether PNLS is power conserving or not. An importantand somewhat surprising result is that various small defocusingperturbations lead to a universal form for the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations.
Induced modulation instability and recurrence in nonlocal nonlinear media
Journal of Physics B: Atomic, Molecular and Optical Physics, 2008
The long-term behaviour of spatial modulation instability in nonlocal nonlinear Kerr media is studied theoretically and numerically. Seeding the modulation instability with a periodic modulation leads to an energy transfer to higher-order modes and the long-term behaviour of the system shows the Fermi-Pasta-Ulam recurrence. For large spatial frequencies, close to the cut-off frequency, the stable solution can be found analytically. The propagation with an initial state different from the stable solution results in a propagation circling around the stable point. For general frequencies, the calculation of the stable point is performed numerically. Comparison of the calculations with earlier reported experimental results in nematic liquid crystals shows a satisfactory agreement.
Results in Physics, 2017
The nonlinear Schrödinger equations (NLSEs) describe the promulgation of ultra-short pluse in optical fibers. The modify unstable nonlinear Schrödinger equation (mUNLSE) is a universal equation of the class of nonlinear integrable systems in NLSEs, which governs certain instabilities of modulated wave-trains. This equation also describes the time evolution of disturbances in marginally stable or unstable media. In the current work, the aim is to investigate the mUNLSE analytically by utilizing proposed modified extended mapping method. New exact solutions are constructed in the different form such as exact dark soliton, exact bright soliton, bright-dark soliton, solitary wave, elliptic function in different form and periodic solutions of mUNLSE. Furthermore, we also present the formation conditions of the bright soliton and dark soliton of this equation. The modulation instability analysis is implemented to discuss the stability analysis of the attained solutions and the movement role of the waves is examined, which confirms that all constructed solutions are exact and stable.
Modulational instability of a wave scattered by a nonlinear center
Physical review, 1993
We consider scattering of a quantum particle by a potential which includes a 5 function whose amplitude is nonlinear in the wave function. Solution of the scattering problem in this model is nonunique in a certain interval of amplitudes of the incident wave. We demonstrate that the nonlinearity gives rise to an oscillatory instability of the scattering solutions, which is a localized version of the well-known modulational instability of the nonlinear Schrodinger equation. We also consider a nonlinear regime slightly above the instability threshold. The results obtained can be applied to the problem of single-particle tunneling through an ultrashort junction in the presence of multiparticle interaction. Our prediction is that the instability gives rise to an ac component in the transmitted current.
Computational Methods for Differential Equations, 2021
The current study utilizes the extended sinh-Gordon equation expansion and ($frac{G^{prime}}{G^2}$)-expansion function methods in constructing various optical soliton and other solutions to the (2+1)-dimensional hyperbolic nonlinear Schr${ddot o}$dinger's equation which describes the elevation of water wave surface for slowly modulated wave trains in deep water in hydrodynamics. We secure different kinds of solutions like optical dark, bright, singular, combo solitons as well as hyperbolic and trigonometric functions solutions. Moreover, singular periodic wave solutions are recovered and the constraint conditions which provide the guarantee to the soliton solutions are also reported. In order to shed more light on these novel solutions, graphical features 3D, 2D and contour with some suitable choice of parameter values have been depicted. We also discuss the stability analysis of the studied nonlinear model with aid of modulation instability analysis.