Collapse in the nonlinear Schrödinger equation

A new class of exact solutions with a singularity at finite time (collapse) is obtained for the nonlinear Schrödinger equation.     

The fractional discrete nonlinear Schrödinger equation

We examine a fractional version of the discrete nonlinear Schrödinger (dnls) equation, where the usual discrete laplacian is replaced by a fractional discrete laplacian. This leads to the replacement of the usual nearest-neighbor interaction to a long-range intersite coupling that decreases asymptotically as a power-law. For the linear case, we compute both, the spectrum of plane waves and the mean square displacement of an initially localized excitation in closed form, in terms of regularized hypergeometric functions, as a function of the fractional exponent. In the nonlinear case, we compute numerically the low-lying nonlinear modes of the system and their stability, as a function of the fractional exponent of the discrete laplacian. The selftrapping transition threshold of an initially localized excitation shifts to lower values as the exponent is decreased and, for a fixed exponent and zero nonlinearity, the trapped fraction remains greater than zero.     

Statistical mechanics of the nonlinear Schrödinger equation

We investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian $$H(\phi ) = \int_\Omega {[\frac{1}{2}|\nabla \phi |^2 - (1/p) |\phi |^p ] dx}$$ is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, whenΩ is the circle and theL ² norm of the field (which is conserved by the dynamics) is bounded byN, the Gibbs measureυ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only ifp andN are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, asN and the temperature are varied.     

Some nonlinear extensions for the schrödinger equation

We investigate extensions for the Schrodinger equation, by considering different non-linear dependencies on the kinetic term, which admit the standard probabilistic interpretation for the wave-function. For one of the cases, we show that the Heisenberg equation is invariant in form and a modified Hamilton-Jacobi equation emerges from the semi-classical approximation for the wave-function. Furthermore, we find solutions assuming time scaling functions different from the standard case and observe nonstandard relaxation processes for the wave-function.     

Conservation laws of the generalized nonlocal nonlinear Schrodinger equation

The derivations of several conservation laws of the generalized nonlocal nonlinear Schr?dinger equation are presented. These invariants are the number of particles, the momentum, the angular momentum and the Hamiltonian in the quantum mechanical analogy. The Lagrangian is also presented.     

Periodic nonlinear Schrödinger equation and invariant measures

In this paper we continue some investigations on the periodic NLSEiu _u +iu _xx +u|u| ^p-2 (p6) started in [LRS]. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measrue (after suitableL ²-truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [B1] on periodic NLS and KdV type equations.     

Solitons for the cubic-quintic nonlinear Schrödinger equation with Raman effect in nonlinear optics

Considering the ultrashort optical soliton propagation in the non-Kerr media, the cubic-quintic nonlinear Schrödinger equation with Raman effect is studied through the dependent variable transformation and Hirota method. Based on symbolic computation, the bilinear form, the explicit one- and two-soliton solutions for the equation are presented. The constraint parametric condition for the existence of soliton solutions is also derived. Propagation characteristics and interaction behaviors of the solitons are graphically shown and discussed: (1) Overtaking elastic interactions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) propagation in parallel of the two solitons.     

Rapidly decaying solutions of the nonlinear Schrödinger equation

We consider global solutions of the nonlinear Schrödinger equation

Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives KleinGordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.     

Energy minimization related to the nonlinear Schrödinger equation

In this the window of the Sobolev gradient technique to the problem of minimizing a Schrödinger functional associated with a nonlinear Schrödinger equation. We show that gradients act in a suitably chosen Sobolev space (Sobolev gradients) can be used in finite-difference and finite-element settings in a computationally efficient way to find minimum energy states of Schrödinger functionals.     

A certain critical two-soliton solution of the nonlinear Schroedinger equation

An exact two-soliton solution of the nonlinear Schroedinger equation is derived by using the Hirota direct approach. This solution describes such a critical process that two still solitons separated infinitely approach and then pass through each other and keep straight on infinitely.     

Long-time relaxation processes in the nonlinear Schrödinger equation

The nonlinear Schrödinger equation, known in low-temperature physics as the Gross-Pitaevskii equation, has a large family of excitations of different kinds. They include sound excitations, vortices, and solitons. The dynamics of vortices strictly depends on the separation between them. For large separations, some kind of adiabatic approximation can be used. We consider the case where an adiabatic approximation can be used (large separation between vortices) and the opposite case of a decay of the initial state, which is close to the double vortex solution. In the last problem, no adiabatic parameter exists (the interaction is strong). Nevertheless, a small numerical parameter arises in the problem of the decay rate, connected with an existence of a large centrifugal potential, which leads to a small value of the increment. The properties of the nonlinear wave equation are briefly considered in the Appendix A.     

Approximate symmetry reduction for perturbed nonlinear Schrdinger equation

We investigate the one-dimensional nonlinear Schrödinger equation with a perturbation of polynomial type. The approximate symmetries and approximate symmetry reduction equations are obtained with the approximate symmetry perturbation theory.     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger equation (NLS) with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions,one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results,some previous one-and two-soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one-and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.     

Parametric instability in the pure-quartic nonlinear Schr?dinger equation

We study the nonlinear stage of modulation instability(MI) in the non-intergrable pure-quartic nonlinear Schr?dinger equation where the fourth-order dispersion is modulated periodically. Using the three-mode truncation, we reveal the complex recurrence of parametric resonance(PR) breathers, where each recurrence is associated with two oscillation periods(PR period and internal oscillation period). The nonlinear stage of parametric instability admits the maximum energy exchange between the spectr...     

A solution spectrum of the nonlinear schrödinger equation

The soliton solutions of the form=A/coshkx and=B tanhkx of the nonlinear Schrödinger equation have been considered with respect to many problems. In this paper, it is shown that the nonlinear Schrödinger equation also possesses a solution manifold that generalizes the above soliton functions and provides a discrete spectrum of eigenfunctions and eigenvalues. With the help of a slight modification of these eigenfunctions, it is possible to extend them to the relativistic case, where they become solutions of a nonlinear Klein-Gordon equation associated with a discrete mass spectrum.