New exact solutions for higher order nonlinear Schrödinger equation in optical fibers

Nonlinear Schrodinger equation (NLSE) is now one of the prominent of modern physics, mathematics and chemistry. Over these fields, the NLSE is also applied in new emerging fields such as quantum information and econophysics. In this paper we investigate for new exact solutions of higher order nonlinear Schrodinger’s equation. This method allows to carry out the solution process of nonlinear wave equations more thoroughly and conveniently by computer algebra systems such as the Maple and Mathematica. In addition to providing a different way of solving the Schrodinger equation for such systems, the simplicity of the algorithm renders it a great pedagogical value.     

Optical solitons for non-Kerr law nonlinear Schrödinger equation with third and fourth order dispersions

In this paper, we obtain optical soliton solutions for non-Kerr law nonlinear Schrödinger equation (NLSE) with third order (3OD) and fourth order dispersions (4OD). We will use two integration schemes, namely sin-cosine method and Bernoulli’s equation approach with five laws of nonlinearities. Sine-cosine method is applicable to Kerr, power and anti-cubic laws, this method provides bright soliton solutions. The second method is applicable to parabolic and cubic quintic laws, this method generates dark soliton. The results may be used in discussing the propagation of optical solitons in highly dispersive media with Kerr, power, anti-cubic, parabolic and cubic quintic law nonlinearities.     

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.     

Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.     

Stability and oscillations of two-dimensional solitons described by the perturbed nonlinear Schrödinger equation

A perturbation theory for determining the stability characteristics of spatial optical solitons with a 2D transverse profile in a transparent medium with a weak saturation of nonlinear refractive index is developed. For Kerr nonlinearity, a new solution of linearized equations for weak soliton perturbations is found. Using this solution, an expression for the stability characteristic is deduced, which, in the case of unstable solitons, determines their decay length and, in the case of stable solitons, shows the presence of perturbations with anomalously weak damping (internal modes) and determines their oscillation period.     

Limit behavior of saturated approximations of nonlinear Schrödinger equation

We consider the solutionu _?(t) of the saturated nonlinear Schrödinger equation (1) $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} u + \varepsilon \left| u \right|^{q - 1} uandu(0,.) = \varphi (.)$$ where \(N \geqslant 2,\varepsilon > 0,1 + 4/N< q< (N + 2)/(N - 2),u:\mathbb{R} \times \mathbb{R}^N \to \mathbb{C},\varphi \) , ? is a radially symmetric function inH ¹(R ^N ). We assume that the solution of the limit equation is not globally defined in time. There is aT>0 such that \(\mathop {\lim }\limits_{t \to T} \left\| {u(t)} \right\|_{H^1 } = + \infty \) , whereu(t) is solution of (1) $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} uandu(0,.) = \varphi (.)$$ For ?>0 fixed,u _?(t) is defined for all time. We are interested in the limit behavior as ?→0 ofu _?(t) fort≥T. In the case where there is no loss of mass inu _? at infinity in a sense to be made precise, we describe the behavior ofu _? as ? goes to zero and we derive an existence result for a solution of (1) after the blow-up timeT in a certain sense. Nonlinear Schrödinger equation with supercritical exponents are also considered.     

Rapidly decaying solutions of the nonlinear Schrödinger equation

We consider global solutions of the nonlinear Schrödinger equation

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.     

Effect of phase shift in shape changing collision of solitons in coupled nonlinear Schrödinger equations

Soliton interactions in systems modelled by coupled nonlinear Schr?dinger (CNLS) equations and encountered in phenomena such as wave propagation in optical fibers and photorefractive media possess unusual features: shape changing intensity redistributions, amplitude dependent phase shifts and relative separation distances. We demonstrate these properties in the case of integrable 2-CNLS equations. As a simple example, we consider the stationary two-soliton solution which is equivalent to the so-called partially coherent soliton (PCS) solution discussed much in the recent literature. Received 1st October 2001 / Received in final form 4 February 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: lakshman@bdu.ernet.in     

Discrete nonlinear Schrödinger equation and polygonal solitons with applications to collapsed proteins

We introduce a novel generalization of the discrete nonlinear Schr?dinger equation. It supports solitons that we utilize to model chiral polymers in the collapsed phase and, in particular, proteins in their native state. As an example we consider the villin headpiece HP35, an archetypal protein for testing both experimental and theoretical approaches to protein folding. We use its backbone as a template to explicitly construct a two-soliton configuration. Each of the two solitons describe well over 7.000 supersecondary structures of folded proteins in the Protein Data Bank with sub-angstrom accuracy suggesting that these solitons are common in nature.     

Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator

The Schrödinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, including asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space H ¹.     

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.     

Periodic wavetrains for systems of coupled nonlinear Schrödinger equations

Exact, periodic wavetrains for systems of coupled nonlinear Schrödinger equations are obtained by the Hirota bilinear method and theta functions identities. Both the bright and dark soliton regimes are treated, and the solutions involve products of elliptic functions. The validity of these solutions is verified independently by a computer algebra software. The long wave limit is studied. Physical implications will be assessed.     

Optical solitons and complexitons of the Schrödinger–Hirota equation

The Schrödinger–Hirota equation governs the propagation of optical solitons in a dispersive optical fiber. In this paper, this equation will be solved by the ansatz method for bright and dark 1-soliton solution. The power law nonlinearity will be assumed. By using the tanh method, some additional solutions will be derived. Finally, the numerical simulations will be given.     

Evolution of solitons described by the higher-order nonlinear Schrödinger equation. II. Numerical investigation

The resonant radiation and evolution of a soliton described by the higher-order nonlinear Schrödinger equation is estigated numerically. The results are in good agreement with the analytical theory [V.I. Karpman, Phys. Lett. A 244 698) 397].     

Focusing and multi-focusing solutions of the nonlinear Schrödinger equation

A method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic Schrödinger equations in two and three dimensions and describe their universal properties. The same method is applied to simulate the multi-focusing phenomena produced by simple models of saturating nonlinearities.     

Existence of solitary waves for the discrete Schrödinger equation coupled to a nonlinear oscillator

The discrete Schrödinger equation with a nonlinearity concentrated at a single point is an interesting and important model to study the long-time behavior of solutions, including the asymptotic stability of solitary waves and properties of global attractors. In this note, the global well-posedness of this equation and the existence of solitary waves is proved and the properties of these waves are studied.     

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.