Nonlocal NLS Research Articles

PT-symmetric PINN for integrable nonlocal equations: Forward and inverse problems.

Since the PT-symmetric nonlocal equations contain the physical information of the PT-symmetric, it is very appropriate to embed the physical information of the PT-symmetric into the loss function of PINN, named PTS-PINN. For general PT-symmetric nonlocal equations, especially those equations involving the derivation of nonlocal terms due to the existence of nonlocal terms, directly using the original PINN method to solve such nonlocal equations will face certain challenges. This problem can be solved by the PTS-PINN method, which can be illustrated in two aspects. First, we treat the nonlocal term of the equation as a new local component so that the equation is coupled at this time. In this way, we successfully avoid differentiating nonlocal terms in neural networks. On the other hand, in order to improve the accuracy, we make a second improvement, which is to embed the physical information of the PT-symmetric into the loss function. Through a series of independent numerical experiments, we evaluate the efficacy of PTS-PINN in tackling the forward and inverse problems for the nonlocal NLS equation, the nonlocal derivative NLS equation, the nonlocal (2+1)-dimensional NLS equation, and the nonlocal three-wave interaction systems. The numerical experiments demonstrate that PTS-PINN has good performance. In particular, PTS-PINN has also demonstrated an extraordinary ability to learn large space-time scale rogue waves for nonlocal equations.

Ring-like partially nonlocal extreme wave of a (3+1)-dimensional NLS system with partially nonlocal nonlinearity and external potential

This paper aims to study extreme waves localized in (3+1)-dimensional space based on a (3+1)-dimensional nonautonomous partially nonlocal NLS system under the linear and parabolic potentials, which is simplified into a (2+1)-dimensional autonomous equation via a converting formula. Based on solutions of the autonomous NLS system, an approximate solution of ring-like extreme wave is found. Characteristics and evolution of ring-like extreme wave are explored in an exponential system. The influence of the magnitude and exponential parameters of diffraction on ring-like extreme wave is studied. Moreover, the impact of the Hermite polynomial parameter q, the radius parameter R and the thickness parameter w on ring-like extreme wave is also discussed.

Spectral structure and even-order soliton solutions of a defocusing shifted nonlocal NLS equation via Riemann-Hilbert approach

Spectral structure and even-order soliton solutions of a defocusing shifted nonlocal NLS equation via Riemann-Hilbert approach

Spectral structures and soliton dynamical behaviors of two shifted nonlocal NLS equations via a novel Riemann–Hilbert approach: A reverse-time NLS equation and a reverse-spacetime NLS equation

By extending the traditional Riemann–Hilbert (RH) approach of soliton equations to a novel version, this paper is devoted to studying two shifted nonlocal NLS equations: a reverse-time NLS equation with a temporal parameter t0, and a reverse-spacetime NLS equation with the spatial and temporal parameters x0,t0. It is shown that the defocusing cases of the two shifted nonlocal NLS equations are equivalent to their focusing cases under linear transformations, only the focusing cases of the two shifted nonlocal NLS equations need to be treated in this work which differs from the classical local NLS equation whose focusing and defocusing cases should be treated separately. Firstly, the spectral analysis of the Lax pair of the two shifted nonlocal NLS equations are performed, from which spectral structures of the two shifted equations are revealed in detail, respectively. Secondly, based on the obtained symmetry relations of the scattering data, the soliton solutions of the two shifted nonlocal NLS equations are rigorously obtained by introducing a novel reduction-proof technique which is different but more direct than the traditional RH approach. It is demonstrated that, due to the existence of the shifted parameters, the spectral analysis for deriving the symmetry relations of the scattering data and thus the calculations of the soliton solutions for the two shifted nonlocal NLS equations involve more tedious and ingenious computations than those unshifted ones without shifted parameters. Finally, the dynamical behaviors underlying the obtained soliton solutions are theoretically explored and graphically illustrated by highlighting the roles that the shifted parameters play, which manifest the peculiar soliton characteristics that may have potential applications of the shifted nonlocal reverse-time and reverse-spacetime NLS equations.

Vector ring-like freak wave triplets of 3D partially nonlocal NLS system with a linear potential

Vector ring-like freak wave triplets of 3D partially nonlocal NLS system with a linear potential

Generalized -dressing method for coupled nonlocal NLS equation

In this paper, we investigate a general integrable coupled nonlocal nonlinear Schrödinger (cnNLS) equation with reverse space and reverse time. First, we introduce a 4-component nonlocal nonlinear Schrödinger (nNLS) equation. Based on two 3 × 3 matrices -problems, we obtain the N-soliton solutions of the 4-component nNLS equation by constructing two spectral transformation matrices R and with some specific scattering data and . We have the symmetry condition for R and . We express the determinant in the N-soliton solution in the form of sums with the help of Cauchy matrix properties to facilitate follow-up reduction. The general nonlocal reduction of the 4-component nNLS equation to the cnNLS equation is discussed in detail. After obtaining the 1-soliton solution of cnNLS equation, we analyse the nonsingular region of the single soliton solution, with spectral parameters k 1 and λ 1 are conjugate. We also draw some typical images of 1-soliton solution. The 2-soliton solutions for cnNLS equation are derived and their asymptotic behaviours are discussed. After that we discuss the dynamic behaviour of the two waves in 2-soliton solution under the condition of different spectral parameters values.

Existence of global solutions to the nonlocal Schrödinger equation on the line

AbstractWe address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (nonlocal NLS) equation under the initial data with the small‐norm. The nonlocal NLS equation was first introduced by Ablowitz and Musslimani as a new nonlocal reduction of the well‐known Ablowitz–Kaup–Newell–Segur system. The main technical difficulty for proving its global well‐posedness on the line in is due to the fact that mass and energy conservation laws, being nonlocal, do not preserve any reasonable norm and may be negative. In this paper, we use the inverse scattering transform approach to prove the existence of global solutions in based on the representation of a Riemann–Hilbert (RH) problem associated with the Cauchy problem of the nonlocal NLS equation. A key of this approach is, by applying the Volttera integral operator and Cauchy integral operator, to establish a Lipschitz bijective map between the solution of the nonlocal NLS equation and reflection coefficients associated with the RH problem. By using the reconstruction formula and estimates on the solution of the time‐dependent RH problem, we further affirm the existence of a unique global solution to the Cauchy problem for the nonlocal NLS equation.

On multi-hump solutions of reverse space-time nonlocal nonlinear Schrödinger equation

In this article multi-soliton solutions of reverse space-time nonlocal nonlinear Schr ödinger (NLS) equation have been constructed. Darboux transformation is applied to the associated linear eigenvalue problem for the generalized NLS equation and we obtain a determinant formula for multi-soliton solutions. Under suitable reduction conditions and appropriate choice of spectral parameters, the generalized expression of first-order nontrivial solution gives some novel solutions such as double-hump and flat-top soliton solutions for reverse space-time nonlocal NLS equation. The dynamics and interaction of double-hump soliton solutions are studied in detail and it is indicated that these solutions undergo collisions without any energy redistribution. For higher-order double-hump solutions, the relative velocities of solitons play a crucial role to have humps and also induce nonlinear interference in the collision zone. The dynamics of individual decaying and growing unstable and stable double-humps as well as their interactions are explained and illustrated.

The solutions of classical and nonlocal nonlinear Schr\"{o}dinger equations with nonzero backgrounds: Bilinearisation and reduction approach

In this paper we develop a bilinearisation-reduction approach to derive solutions to the classical and nonlocal nonlinear Schr\"{o}dinger (NLS) equations with nonzero backgrounds. We start from the second order Ablowitz-Kaup-Newell-Segur coupled equations as an unreduced system. With a pair of solutions $(q_0,r_0)$ we bilinearize the unreduced system and obtain solutions in terms of quasi double Wronskians. Then we implement reductions by introducing constraints on the column vectors of the Wronskians and finally obtain solutions to the reduced equations, including the classical NLS equation and the nonlocal NLS equations with reverse-space, reverse-time and reverse-space-time, respectively. With a set of plane wave solution $(q_0,r_0)$ as a background solution, we present explicit formulae for these column vectors. As examples, we analyze and illustrate solutions to the focusing NLS equation and the reverse-space nonlocal NLS equation. In particular, we present formulae for the rouge waves of arbitrary order for the focusing NLS equation.

Global conservative solutions of the nonlocal NLS equation beyond blow-up

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \mathrm{i}\partial_t q(x, t)+\partial_{x}^2q(x, t)+2\sigma q^{2}(x, t)\overline{q(-x, t)} = 0 $ with initial data $ q(x, 0)\in H^{1, 1}(\mathbb{R}) $. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $ H^{1, 1} $ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

Global Cauchy problems for the nonlocal (derivative) NLS in [formula omitted

We consider the Cauchy problem for the nonlocal (derivative) NLS in super-critical function spaces Eσs for which the norms are defined by‖f‖Eσs=‖〈ξ〉σ2s|ξ|fˆ(ξ)‖L2,s<0,σ∈R. Any Sobolev space Hr is a subspace of Eσs, i.e., Hr⊂Eσs for any r,σ∈R and s<0. Let s<0 and σ>−1/2 (σ>0) for the nonlocal NLS (for the nonlocal derivative NLS). We show the global existence and uniqueness of the solutions if the initial data belong to Eσs and their Fourier transforms are supported in (0,∞), the smallness conditions on the initial data in Eσs are not required for the global solutions.

Reduced Non-Local Integrable NLS Hierarchies by Pairs of Local and Non-Local Constraints

Reduced Non-Local Integrable NLS Hierarchies by Pairs of Local and Non-Local Constraints

Statistical approach of modulational instability in the class of nonlocal NLS equation involving nonlinear Kerr-like responses with non-locality: Exact and approximated solutions

The purpose of this work is the investigation of the behavior of modulated signal light in the class of optical waveguide whose the dynamics is governed by the nonlocal nonlinear Schrödinger (NNLS) equation involving nonlinear Kerr-like responses with nonlocality. Firstly, from the NNLS equation, the modulational instability (MI) is investigated for both the coherent and partially-coherent cases. The coherent MI is investigated by using the deterministic approach, i.e. the well-known Benjamin–Feir method, while for the partially coherent case (The propagating signal lights are not always monochromatic in real applications), the statistical approach is used starting from the Wigner Moyal transform, also known as Klimontovich’s statistical average method, and all these studies are checked by numerical investigations. The modulated bright soliton as well as periodic signal as solutions of the waveguide equation are next found. Then we find the implicit exact solutions, that the profiles are just obtained numerically which are not new in the literature. In addition, we use the perturbation method to find the approximated explicit bright and periodic solutions, where profiles are compared to the implicit ones.

Lagrangian nonlocal nonlinear Schrödinger equations

In 2013 Ablowitz and Musslimani (AM) obtained a nonlocal generalization of the NLS equation which is integrable, but does not have a standard Lagrangian structure. The present paper shows that there exist two new nonlocal NLS equations, similar to the AM model, which do possess Lagrangian structures. We show that these two models (LN1 and LN2) possess solitary wave solutions which remain trapped in the neighborhood of the origin (x=0), and solitary waves which are able to escape from the origin. These two types of solutions are obtained by direct numerical solutions, and also by a variational method. In the case of LN2 model, the variational approach explains the existence of these two types of solutions. Collisions of breathers which obey the equation LN2 are numerically studied, and the results show that these breathers are robust solutions.

Soliton solutions of the shifted nonlocal NLS and MKdV equations

We find one- and two-soliton solutions of shifted nonlocal NLS and MKdV equations. We discuss the singular structures of these soliton solutions and present some of the graphs of them.

Riemann-Hilbert problems and soliton solutions of nonlocal reverse-time NLS hierarchies

Riemann-Hilbert problems and soliton solutions of nonlocal reverse-time NLS hierarchies

Quasi binormal Schrodinger evolution of wave polarizatıon field of light wıth repulsive type

In this paper, we study the evolution of the wave polarization vector in the tangent direction of the curved path. This path is assumed to be the trajectory of the propagated light beam. The polarization state of the wave is described by the unit complex transverse field component by eliminating the longitudinal field component. We obtain new relationship between the geometric phase and the parallel transportation law of the wave polarization vector of the evolving light beam in the tangent direction of the curved path. Moreover, we present a new geometric interpretation of the quasi binormal evolution of the wave polarization vector via the nonlinear Schrodinger equation of repulsive type in the tangent direction. Finally, we find a space-time nonlocal NLS reduction for equation system.

Rogue waves and periodic solutions of a nonlocal nonlinear Schrödinger model

In the present work, a nonlocal nonlinear Schr\"odinger (NLS) model is studied by means of a recent technique that identifies solutions of partial differential equations, by considering them as fixed points in {\it space-time}. This methodology allows to perform a continuation of well-known solutions of the local NLS model to the nonlocal case. Four different examples of this type are presented, namely (a) the rogue wave in the form of the Peregrine soliton, (b) the generalization thereof in the form of the Kuznetsov-Ma breather, as well as two spatio-temporally periodic solutions in the form of elliptic functions. Importantly, all four waveforms can be continued in intervals of the parameter controlling the nonlocality of the model. The first two can be continued in a narrower interval, while the periodic ones can be extended to arbitrary nonlocalities and, in fact, present an intriguing bifurcation whereby they merge with (only) spatially periodic structures. The results suggest the generic relevance of rogue waves and related structures, as well as periodic solutions, in nonlocal NLS models.

A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation

We give a microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation (NLS) from many-body quantum theory. The starting point of our proof is [11] on the time-independent problem and [16] on the corresponding problem on a finite lattice. An important new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaussian structure of the free quantum field and prevents the use of the Wick theorem. We overcome it by means of complex analytic methods. Our methods apply to the nonlocal NLS with bounded convolution potential. In the periodic setting, we also consider the local NLS, arising from short-range interactions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in Xs,b spaces.

GBDT and algebro-geometric approaches to explicit solutions and wave functions for nonlocal NLS

We apply the GBDT version of the Bäcklund–Darboux transformation to the nonlocal NLS (focusing and defocusing cases). The matrix case is included and solutions in the form of rectangular matrix functions are dealt with. In the case of the trivial initial solution, wide classes of the wave functions and multipole solutions are constructed explicitly. Families of explicit examples are considered in detail. Some initial results and representations for the more complicated algebro-geometric solutions are obtained as well.