Dynamics of solitons in the fourth-order nonlocal nonlinear Schrödinger equation (original) (raw)

Nonlocal nonlinear Schrödinger equations and their soliton solutions

Journal of Mathematical Physics

We study standard and nonlocal nonlinear Schrödinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the Hirota bilinear method we first find soliton solutions of the coupled NLS system of equations then using the reduction formulas we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function |q(t, x)| 2 for the standard and nonlocal NLS equations.

Dark and singular optical solitons with competing nonlocal nonlinearities

Optica Applicata, 2016

In this work, we study the dynamics of optical solitons in a synthetic nonlocal nonlinear media. The nonlinear dynamical model which describes the propagation of optical solitons in the weakly nonlocal nonlinear media with parabolic law nonlinearity is investigated analytically. The tool of integration that is the Riccati equation mapping approach is introduced to extract exact traveling wave solutions. As a result, an explicit dark soliton, singular soliton and periodic solutions are derived.

Bright and dark solitons of a weakly nonlocal Schrödinger equation involving the parabolic law nonlinearity

Optik, 2021

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Wide localized solitons in systems with time- and space-modulated nonlinearities

Physical Review E, 2012

In this work we apply point canonical transformations to solve some classes of nonautonomous nonlinear Schrödinger equation namely, those which possess specific cubic and quintic-time and space dependent-nonlinearities. In this way we generalize some procedures recently published which resort to an ansatz to the wavefunction and recover a time and space independent nonlinear equation which can be solved explicitly. The method applied here allow us to find wide localized (in space) soliton solutions to the nonautonomous nonlinear Schrödinger equation, which were not presented before. We also generalize the external potential which traps the system and the nonlinearities terms.

A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons

Results in Physics, 2021

The present paper explores a high-order nonlinear Schrödinger equation in a non-Kerr law media with the weak non-local nonlinearity describing solitons' propagation through nonlinear optical fibers. To this end, the real and imaginary parts of the model are firstly extracted using a wave variable transformation. The modified Kudryashov method and symbolic computations are then adopted to successfully retrieve optical solitons of the model. The results presented in the current study demonstrate the great performance of the modified Kudryashov method in handling high-order nonlinear Schrödinger equations.

Singular soliton molecules of the nonlinear Schrödinger equation

Physical Review E, 2020

We derive an exact solution to the local nonlinear Schrödinger equation (NLSE) using the Darboux transformation method. The new solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically-decaying seed solution, we obtain a two-soliton solution with diverging peaks, which we denote as singular soliton molecule. We find that the new solution has a finite binding energy. We calculate the force and potential of interaction between the two solitons, which turn out to be of molecular-type. The robustness of the bond between the two solitons is also verified. Furthermore, we obtain a new solution to the nonlocal NLSE using the same method and seed solution. The new solution in this case corresponds to an elastic collision of a soliton, a breather soliton on flat background, and a breather soliton on a background with linear ramp. Finally, we consider an NLSE which is nonlocal in time rather than space. Although we did not find a Lax pair to this equation, we derive three exact solutions.

The nonlocal nonlinear Schrödinger and Maxwell – Bloch equation

International Journal of Mathematics and Physics, 2017

In this paper, the nonlocal nonlinear Schrödinger and Maxwell-Bloch equations are is introduced. A particular case of this system, namely the Schrödinger equation, is integrable by the inverse scattering method as shown in the work of M. Ablowitz and Z. Musslimani. Following their idea, we prove the integrability of the nonlocal nonlinear Schrödinger and Maxwell-Bloch equation using its Lax pairs. Also the Darboux transformations are constructed, and soliton solutions are obtained from different "seed" solutions using them. One-fold, twofold and N-fold determinant representations are obtained by this transformation. Moreover, soliton and solitons-like solutions, such as dynamic and topological soliton, periodic, domain walls, kink, lamp, bright and dark solitons, bright and dark rogue waves, bright and dark positons, etc., of this equation are built. In future papers, we will investigate the conservation laws of the nonlocal nonlinear Schrödinger and Maxwell-Bloch equation using the Lax pair.

Finite-length Soliton Solutions of the Local Homogeneous Nonlinear Schrödinger Equation

Physica Scripta, 1998

We found a new kind of soliton solutions for the 5-parameter family of the potential-free Stenflo-Sabatier-Doebner-Goldin nonlinear modifications of the Schr\"odinger equation. In contradistinction to the "usual'' solitons like {\cosh[b(x-kt)]}^{-a}\exp[i(kx-ft)], the new {\em Finite-Length Solitons} (FLS) are nonanalytical functions with continuous first derivatives, which are different from zero only inside some finite regions of space. The simplest one-dimensional example is the function which is equal to {\cos[g(x-kt)]}^{1+d}\exp[i(kx-ft)] (with d>0) for |x-kt|<\pi/(2g), being identically equal to zero for |x-kt|>\pi/(2g). The FLS exist even in the case of a weak nonlinearity, whereas the ``usual'' solitons exist provided the nonlinearity parameters surpass some critical values.

Quadratic solitons as nonlocal solitons

Physical Review E, 2003

We show that quadratic solitons are equivalent to solitons of a nonlocal Kerr medium. This provides new physical insight into the properties of quadratic solitons, often believed to be equivalent to solitons of an effective saturable Kerr medium. The nonlocal analogy also allows for novel analytical solutions and the prediction of novel bound states of quadratic solitons.

Peregrine Solitons of the Higher-Order, Inhomogeneous, Coupled, Discrete, and Nonlocal Nonlinear Schrödinger Equations

Frontiers in Physics, 2020

This study reviews the Peregrine solitons appearing under the framework of a class of nonlinear Schrödinger equations describing the diverse nonlinear systems. The historical perspectives include the various analytical techniques developed for constructing the Peregrine soliton solutions, followed by the derivation of the general breather solution of the fundamental nonlinear Schrödinger equation through Darboux transformation. Subsequently, we collect all forms of nonlinear Schrödinger equations, involving systematically the effects of higher-order nonlinearity, inhomogeneity, external potentials, coupling, discontinuity, nonlocality, higher dimensionality, and nonlinear saturation in which Peregrine soliton solutions have been reported.