NLS-type Equations Research Articles

Novel high-order explicit energy-preserving schemes for NLS-type equations based on the Lie-group method

In this paper, by taking the NLS-type equations as examples, we propose a novel high-order explicit energy-preserving Lie-group method for Hamiltonian PDEs. The main idea is to combine recently developed generalized SAV (G-SAV) approach (Cheng et al., 2021) and explicit Lie group method, in which the fourth-order or higher-order Lie group method only composes two exponentials in each stage, and this is the key point that the proposed methods can preserve energy of Hamiltonian PDEs. This is our first attempt to construct high-order explicit energy-preserving methods without using projection technique (Hairer et al., 2006 [14]), and some stability conclusions of our proposed methods for the NLS-type equations are also presented. Finally, we present 2D and 3D numerical simulations to demonstrate the stability and accuracy.

Numerical simulations and analytical approach for three-component coupled NLS-type equations in fiber optics

Numerical simulations and analytical approach for three-component coupled NLS-type equations in fiber optics

Sensitivity analysis and analytical study of the three-component coupled NLS-type equations in fiber optics

Sensitivity analysis and analytical study of the three-component coupled NLS-type equations in fiber optics

The large time asymptotic solutions of nonlinear Schrödinger type equations

We give a short description of the proof of asymptotic-completeness for NLS-type equations with radial data in three dimensions. We also show some aspects of the method by giving a new proof of Asymptotic Completeness for the two-body Quantum Scattering case.

NLS-type equations from quadratic pencil of Lax operators: Negative flows

We formulate and study an integrable model of Nonlinear Schrödinger (NLS)-type through its Lax representation, where one of the Lax operators is quadratic and the other has a rational dependence on the spectral parameter. We discuss the associated spectral problem, the Riemann-Hilbert problem formulation, the conserved quantities, as well as a generalisation for symmetric spaces. In addition we explore the possibilities for modelling with higher order NLS (HNLS) integrable equations and in particular, the relevance of the proposed system.

Optical solitons of NLS-type differential equations by extended direct algebraic method

In this paper, we examine the optical soliton solutions of nonlinear partial differential equations belonging to the nonlinear Schrödinger (NLS) class which includes cubic focusing NLS equation and paraxial NLS equation in a Kerr media. We construct the optical soliton of these models using a new extended direct algebraic method (NEDAM). The resulting solutions carry a variety of new families including singular solutions of Types 1 and 2, dark, darkly bright and dark singular soliton solutions. Sufficient conditions are provided for these structures to exist. NEDAM is successfully applied to supposed nonlinear model and provides us some new and interesting solitary wave solutions. These solutions contain the trigonometric-, hyperbolic- and exponential-type functions. Moreover, to present the physical importance of the considered model, some 3D and 2D diagrams of obtained solutions are plotted by using Mathematica under the suitable choice of involving parameters values.

Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.

Superregular solutions for a coupled nonlinear Schrödinger system in a two-mode nonlinear fiber

For the increase of the transmission capacity in optical communication systems, the so-called few-mode fibers are used for people to design the mode division multiplexing transmission. In this paper, we analytically obtain and graphically display the superregular solutions for a coupled nonlinear Schrödinger (NLS) system which describes the wave evolution in a two-mode nonlinear fiber, where the superregular solutions are the analogue of superregular breathers for certain scalar NLS-type equations. On the nonzero-zero (or proportional nonzero-nonzero) background, regular solutions describe the regular nonlinear waves which are located in a finite t domain but do not perturb the background with t being big enough, and superregular solutions are a subset of regular solutions which describe the nonlinear superposition of breathers and dark-bright (or breather-like) solitons developing from the perturbations on the dark-bright (or breather-like) solitons at a certain z, where z and t denote the evolution dimension and temporal distribution dimension, respectively. On the nonzero-zero background, superregular solutions are constructed in three cases: trivial case, a pair of breathers case and single breather case, and then other superregular solutions could be constructed according to the analyses for such three cases. Superregular solutions on the proportional nonzero-nonzero background are derived via the superregular solutions on the nonzero-zero background and an orthogonal transformation.

Riemann–Hilbert approach and nonlinear dynamics in the nonlocal defocusing nonlinear Schrödinger equation

The nonlocal defocusing nonlinear Schrodinger (ND-NLS) equation is comparatively studied via the Riemann–Hilbert approach. Firstly, via spectral analysis, the spectral structure of the ND-NLS equation is investigated, which is different to those of the other three NLS-type equations, i.e., the local focusing nonlinear Schrodinger (LF-NLS) equation, the local defocusing nonlinear Schrodinger (LD-NLS) equation and the nonlocal focusing nonlinear Schrodinger (NF-NLS) equation. Secondly, by solving the Riemann–Hilbert problem corresponding to the reflectionless cases, multi-soliton solutions are obtained for the ND-NLS equation. Thirdly, we prove that, if parameters are suitably chosen, the multi-soliton solutions of the ND-NLS equation can be reduced to those of the LF-NLS equation and the LD-NLS equation, respectively. Fourthly, the multi-soliton solutions of the ND-NLS equation are demonstrated to possess repeated singularities generally, but they can also remain analytic for appropriate soliton parameters. Moreover, the multi-soliton dynamics are graphically illustrated using Mathematica symbolic computations. These results show that the solution structure and the nonlinear dynamics in the ND-NLS equation are rather different from those of the LF-NLS equation, the LD-NLS equation and the NF-NLS equation.

On symmetry of traveling solitary waves for dispersion generalized NLS

We consider dispersion generalized nonlinear Schrödinger equations (NLS) of the form where denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers . The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein–Gordon and half-wave equations.

Two completely explicit and unconditionally convergent Fourier pseudo-spectral methods for solving the nonlinear Schrödinger equation

This paper aims to construct and analyze two new Fourier pseudo-spectral (FPS) methods for the general nonlinear Schrödinger (NLS) equation. The two FPS methods have two merits: unconditional convergence and complete explicitness in the practical computation. Further more, by introducing a modified mass functional and a modified energy functional, the two FPS methods are proved to preserve the total mass and energy in the discrete sense. Besides the standard energy method, the key techniques used in our numerical analysis are a mathematical induction argument and a lifting technique. Without any restriction on the grid ratio and initial value, we establish the optimal error estimate of the two FPS methods for solving the general NLS equation, while previous work just is valid for the cubic NLS equation and requires small initial value for the focusing case. These two FPS methods are proved to be spectrally accurate in space and second-order accurate in time, respectively. The analysis framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the NLS-type equations. We investigate the effect of the nonlinear term on the progression simulation of the plane wave, the conservation of the invariants and the effect of initial data on the blow-up solution via different parameters. Numerical results are reported to show the accuracy and efficiency of the proposed methods.

Entwining Yang–Baxter maps related to NLS type equations

We construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense.

Determination of the blow up point for complex nonautonomous ODE with cubic nonlinearity

We present a method for determination of the blow up point for a complex nonautonomous ordinary differential equation with a cubic nonlinearity. This equation describes stationary nonlinear modes for two important NLS-based models: (i) the Gross–Pitaevskii equation with complex potential and repulsive nonlinearity and (ii) the Lugiato–Lefever equation in the case of normal dispersion. We derive and justify an asymptotic expansion in the vicinity of the blow up point. This expansion is employed for the construction of a numeric procedure for the computation of the coordinate of the blow up point. We illustrate applications of the proposed procedure by two examples, where exact analytical solutions are available. It is shown that it allows one to find the blow up point with high accuracy. The method may be efficiently used for the search of soliton solutions for vector and scalar NLS-type equations within the strategy of ’filtering out’ of blow up solutions (Alfimov et al., Physica D, 394, 39 (2019)).

Riemann–Hilbert method and multi-soliton solutions for three-component coupled nonlinear Schrödinger equations

An integrable three-component coupled nonlinear Schrödinger (NLS) equation is considered in this work. We present the scattering and inverse scattering problems of the three-component coupled NLS equation by using the Riemann–Hilbert formulation. Furthermore, according to the Riemann–Hilbert method, the multi-soliton solutions of this equation are derived. We also analyze the collision dynamic behaviors of these solitons. Moreover, a new phenomenon for two-soliton collision is displayed, which is unique and not common in integrable systems. It is hoped that our results can help enrich the nonlinear dynamics of the NLS-type equations.

High-order soliton matrices for Sasa–Satsuma equation via local Riemann–Hilbert problem

A study of high-order soliton matrices for Sasa–Satsuma equation in the framework of the Riemann–Hilbert problem approach is presented. Through a standard dressing procedure, soliton matrices for simple zeros and elementary high-order zeros in the Riemann–Hilbert problem for Sasa–Satsuma equation are constructed, respectively. It is noted that pairs of zeros are simultaneously tackled in the situation of the high-order zeros, which is different from other NLS-type equation. Furthermore, the generalized Darboux transformation for Sasa–Satsuma equation is also presented. Moreover, collision dynamics along with the asymptotic behavior for the two-solitons are analyzed, and long time asymptotic estimations for the high-order one-soliton are concretely calculated. In this case, two double-humped solitons with nearly equal velocities and amplitudes can be observed.

A coupled nonlinear Schrödinger-type equation and its Darboux transformation

A new coupled nonlinear Schrödinger (NLS)-type equation is proposed by means of the negative power flow of a spectral problem. Resorting to the gauge transformation of the spectral problem and the reduction technique, Darboux transformations for the coupled NLS-type equation and its reduction are constructed, by which explicit solutions of the two coupled NLS-type equations can be engendered from their known solutions. This process can be done continually and will usually yield a series of solutions including multi-soliton solutions. As an illustrate example, one- and two-soliton solutions of the latter coupled NLS-type equation are obtained from a trivial solution.

Modelling of the spatial evolution of extreme laboratory wave crest and trough heights with the NLS-type equations

The statistical properties of long-crested nonlinear wave time series measured in an offshore basin have been analyzed in different aspects such as the distributions of surface elevation, wave crest, wave trough, and wave period. Comparison with linear, second-order and third-order theoretical models indicates that although bound wave effects also contribute to the deviation from a Gaussian process, it is the modulational instability that primarily determines the discrepancy in the evolution process in the presence of strong nonlinearity. Interestingly enough, wave crest is more sensitive to the quasi-resonant four-wave interaction effect than wave trough and the scaled maximal wave crest presents a linear regression model with the coefficient of kurtosis. Meanwhile, the estimation of the observed statistical properties is reconstructed on the basis of an ensemble of 100 wave series simulated by the NLS-type equations and compared favourably with the experimental results in most cases. Moreover, with the increased third-order nonlinear effect the difference between NLS and Dysthe simulations is enlarged and mainly reflected on the distribution of wave crest.

Direct dynamical energy cascade in the modified KdV equation

In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg–de Vries (mKdV) equation. The particularity of this study consists in considering the problem essentially in the Fourier space. A dynamical energy cascade model of this process originally proposed for the focusing NLS-type equations is transposed to the mKdV setting using the existing connections between the KdV-type and NLS-type equations. The main predictions of the D-cascade model are outlined and validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.

Coding of nonlinear states for the Gross–Pitaevskii equation with periodic potential

We study nonlinear states for the NLS-type equation with additional periodic potential U(x), also called the Gross–Pitaevskii equation, GPE, in theory of Bose–Einstein Condensate, BEC. We prove that if the nonlinearity is defocusing (repulsive, in the BEC context) then under some conditions there exists a homeomorphism between the set of all nonlinear states for GPE (i.e. real bounded solutions of some nonlinear ODE) and the set of bi-infinite sequences of numbers from 1 to N for some integer N. These sequences can be viewed as codes of the nonlinear states. We present numerical arguments that for GPE with cosine potential these conditions hold in certain areas of the plane of the external parameters. This implies that for these values of parameters all the nonlinear states can be described in terms of the coding sequences.

Localized Nonlinear Waves in Nonlinear Schr¨odinger Equation with Nonlinearities Modulated in Space and Time

In this paper, the generalized sub-equation method is extended to investigate localized nonlinear waves of the one-dimensional nonlinear Schrödinger equation (NLSE) with potentials and nonlinearities depending on time and on spatial coordinates. With the help of symbolic computation, three families of analytical solutions of this NLS-type equation are presented. Based on these solutions, periodically and quasiperiodically oscillating solitons (dark and bright) and moving solitons are observed. Some implications to Bose-Einstein condensates are also discussed