Nonlocal Nonlinear Schrodinger Research Articles

New solutions of coupled nonlocal NLS and coupled nonlocal mKdV equations

We provide several novel solutions of the coupled Ablowitz–Musslimani (AM) version of the nonlocal nonlinear Schrödinger (NLS) equation and the coupled nonlocal modified Korteweg–de Vries (mKdV) equation. In each case we compare and contrast the corresponding solutions of the relevant coupled local equations. Further, we provide new solutions of the coupled local NLS and coupled local mKdV equations which are not the solutions of the corresponding nonlocal equations. We also show that the nonlocal coupled (as well as uncoupled) mKdV equations have hidden Galilean invariance and admit novel solutions.

Ring-like two-breather structures of a partially nonlocal NLS system with different two-directional diffractions under a parabolic potential

Via a relation of the hypothetic solution, a variable-coefficient partially nonlocal nonlinear Schrödinger(NLS) system with different diffractions under a parabolic potential is associated with the constant-coefficient system. Ring-like two-breather solution of the variable-coefficient partially nonlocal NLS system is constructed by mapping the related solution of the constant-coefficient system. The characteristics of two-breather in the constant-coefficient system influence the properties of ring-like two-breather in the variable-coefficient partially nonlocal NLS system in xyt plane. In the xyz plane, the layer number of the ring-like two-breather along the z-direction augments with the add of the Hermite parameter.

Integrable reductions of the multi-component Kaup–Newell equations

We investigate the local and nonlocal integrable reductions of the multi-component Kaup–Newell equations and derive a range of novel integrable derivative nonlinear Schrödinger (NLS) equations. Firstly, we demonstrate that the multi-component Kaup–Newell equations can undergo one local reduction and three nonlocal reductions, specifically space-reversal, time-reversal, and space–time-reversal reductions. These reductions result in four multi-component derivative NLS equations. Secondly, focusing on the four-component Kaup–Newell equations, we introduce six joint reductions, which are combinations of the aforementioned local and nonlocal reductions. Through these joint reductions between the local reduction and nonlocal reductions, we discover three scalar nonlocal derivative NLS equations, including a T-reversal derivative NLS equation. Furthermore, through the joint reductions among the nonlocal reductions, we identify three scalar nonlocal derivative NLS equations with four variables, namely (x,t), (-x,t), (x,−t), and (−x,−t). All these reduced systems are integrable systems, meaning they possess infinitely many conserved integrals.

New solutions of nonlocal NLS, mKdV and Hirota equations

In this paper, we provide several novel solutions of the Ablowitz–Musslimani and Yang’s versions of the nonlocal nonlinear Schrödinger (NLS) equation, nonlocal modified Korteweg–de Vries (mKdV) as well as nonlocal Hirota equations. In each case we compare and contrast with the corresponding solutions of the relevant local equation. In addition, we provide new solutions of the local NLS, local mKdV and local Hirota equations which are not the solutions of the corresponding nonlocal equations.

A nonlocal finite-dimensional integrable system related to the nonlocal nonlinear Schrödinger equation hierarchy

Based on the Lenard gradient sequence, a hierarchy of the nonlocal nonlinear Schrödinger (NNLS) equations is obtained. Using the Lax representation, the nonlocal finite-dimensional integrable system with Lie–Poisson structure is presented. Then, under coordinate transformation, the nonlocal finite-dimensional integrable system with Lie–Poisson structure can be expressed as the canonical Hamiltonian system of the standard symplectic structures. Moreover, the parametric representation of the NNLS equation and the nonlocal complex modified Korteweg–de Vries (NcmKdV) equation are constructed. Finally, according to the Hamilton–Jacobi theory, the action–angle variables are built and the inversion problem related to the Lie–Poisson Hamiltonian systems is discussed.

Gauge equivalent structures of the integrable (2+1)-dimensional nonlocal nonlinear Schrödinger equations and their applications

In this paper, we are concerned with the gauge equivalent structures for the integrable (2+1)-dimensional nonlocal nonlinear Schrödinger (NLS) equations. Through constructing the gauge transformation, we prove that these (2+1)-dimensional nonlocal equations, both focusing and defocusing, are gauge equivalent to two types of coupled (2+1)-dimensional Heisenberg ferromagnet equations and two types of coupled (2+1)-dimensional modified Heisenberg ferromagnet equations. As an appropriate extension, we further illustrate that the nonlocal NLS equation is gauge equivalent to two types of coupled Heisenberg ferromagnet equations and two types of coupled modified Heisenberg ferromagnet equations, while its discrete version is gauge equivalent to two types of coupled discrete Heisenberg ferromagnet equations and two types of coupled discrete modified Heisenberg ferromagnet equations, respectively. From its invariance with the combined parity-reflection and time-reversal operators, we can observe that there exist significant differences and intimate connections between standard and nonlocal equations. On the other hand, by using the Darboux transformation and some limit techniques, two types of deformed soliton solutions, namely, the deformed exponential solitons and the deformed rational solitons for the (2+1)-dimensional nonlocal defocusing NLS equation are given explicitly. With no loss of generality, two deformed soliton interactions and their various degenerate cases are discussed and illustrated through some figures.

Exact solutions and Darboux transformation for the reverse space–time non-local fifth-order non-linear Schrödinger equation

In this paper, the non-local reverse space−time fifth-order non-linear Schrödinger(NLS) equation has been investigated, which is proposed by the non-local reduction of Ablowitz–Kaup–Newell–Segur (AKNS) scattering problems. The determinant representation of the Nth Darboux transformation for the non-local reverse space−time fifth-order NLS equation is obtained. Some interesting non-linear wave solutions, including soliton, complexiton, and rogue wave solutions, are derived by the Darboux transformation. Moreover, the dynamics of non-linear wave solutions are illustrated with the corresponding evolution plots, and the results show that the non-local fifth-order NLS equation has new different properties from the local case.

Some general bright soliton solutions and interactions for a (2＋1)-dimensional nonlocal nonlinear Schrödinger equation

In this paper, a novel application of Darboux transformation is presented for (2＋1)-dimensional nonlocal nonlinear Schrödinger equation. The Darboux transformation is a power method to solve some (1＋1)-dimensional classical nonlinear Schrödinger (NLS) equations, however, there is less work of (2＋1)-dimensional ((2＋1)-D) nonlocal nonlinear Schrödinger(NNLS) equation with Darboux transformation. With the development of science, the NNLS equation gradually appears, where the nonlocality is the reverse spatial field or reverse time field. We focus on how to solve the (2＋1)-D NNLS equation with reverse time field q(x,y,−t). Using the Darboux transformation, some novel (2＋1)-D nonlocal soliton solutions are derived on a background of kink waves, including 1-soliton solution, 2-soliton solution, bright soliton and soliton interaction on kink wave backgrounds. This method is a novel application of Darboux transformation, which can be extend to some other nonlocal nonlinear or higher-dimensional soliton equations on kink wave backgrounds.

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.

Interference of Gaussian and/or Airy beams in coupled PT-symmetric nonlocal system

We investigate the interference of Gaussian and/or Airy beams based on coupled parity-time (PT) symmetric nonlocal nonlinear Schrödinger (NNLS) equation, which can be decoupled into two paraxial diffraction equations through a general linear transformation under the balance among nonlocal nonlinear effects. Various superposed solutions of the coupled NNLS equation can be obtained with the help of linear transformation and the solutions of linear diffraction equation. Based on these solutions, we analytically explore the abundant interference phenomena of Gaussian beam train, two Gaussian and/or Airy beams, which can be modulated by the linear transformation, the transverse displacement and relative phase difference between adjacent beams. The results presented here may be helpful to design complex interferometric patterns that can be applied to photoetching or optical printing etc.

Novel superposed kinklike and pulselike solutions for several nonlocal nonlinear equations

We show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the nonlocal mKdV equation also admits the superposed (hyperbolic) kink–antikink solution. In addition, we show that while the nonlocal Ablowitz–Musslimani variant of the NLS admits complex parity-time reversal-invariant kink and pulse solutions, neither the local NLS nor the Yang variant of the nonlocal NLS admits such solutions. Finally, except for the Yang variant of the nonlocal NLS, we show that the other three nonlocal equations admit both the kink and pulse solutions in the same model.

Binary Darboux transformation and new soliton solutions of the focusing nonlocal nonlinear Schrödinger equation

For the focusing nonlocal nonlinear Schrödinger (NNLS) equation, we establish the N-fold binary Darboux transformation (BDT) and give a complete proof on the form invariance of Lax pair. Then, by choosing the tanh-function solution as a seed, we implement the one-fold BDT and obtain two new types of soliton solutions including the exponential and exponential-and-rational types. Both two types of solutions are nonsingular with a wide range of parameter regimes, and they can describe the elastic soliton interactions over the nonzero background with the asymptotic phase difference π as x→±∞. Also, we derive the expressions of all asymptotic solitons and discuss the parametric conditions for different soliton profiles. It turns out that each asymptotic soliton can display both the dark and antidark soliton profiles or exhibit no spatial localization, which confirms that the focusing NNLS equation admits a rich variety of soliton interactions like the defocusing NNLS case.

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.

Rational soliton solutions of nonlocal multicomponent nonlinear Schrödinger equations

In this paper, we present a generalized Darboux transformation for nonlocal multicomponent nonlinear Schrödinger (NLS) equations. Also, we provide a unified formula for Nth-order rational soliton solution of discussed nonlocal multicomponent NLS equations. In particular, the first- and second-order rational soliton solutions of nonlocal three-component NLS equations are obtained. Moreover, we discuss the wave structures of these rational solitons of nonlocal three-component NLS equations with different parameters.

Background-enhanced collapse instability of optical speckle beams in nonlocal nonlinear media

We study the focusing dynamics of incoherent wave-packets (speckle beams) in the presence of a nonlocal nonlinear response in the framework of the nonlocal nonlinear Schrödinger (NLS) equation. In the highly nonlocal regime, we show that the speckle beam develops a collective incoherent collapse instability — at variance with a conventional collapse that is inherently a coherent object, here it is the incoherent beam as a whole that develops a collapse. More specifically, we study the impact of a homogeneous incoherent background on the development of the incoherent collapse instability. Despite the fact that the homogeneous background is modulationally stable, we show that it significantly strengthens the formation of the incoherent collapse instability. Our theoretical analysis is based on a wave turbulence formulation of the Vlasov equation, which allows us to introduce an effective hydrodynamic model that is subsequently solved by the method of characteristics. A quantitative agreement is obtained between the simulations of the NLS equation, the Vlasov equation and the hydrodynamic model, without using adjustable parameters. The interaction between the background and the collapsing structure is described by means of a coupled system for the singular and smooth components of the solution of the Vlasov equation. The theory reveals that the mechanism underlying the background-induced collapse enhancement is due to a transfer of momentum from the background to the collapsing structure, while there is no ‘mass’ exchange among them. Furthermore, we show that a strong background level significantly boosts the collapse dynamics. Our work should stimulate the development of nonlinear experiments aimed at observing the incoherent collapse phenomenon in nonlinear thermal media. From a broader perspective, these investigations pave the way for the study of novel forms of global incoherent collective behaviors in wave turbulence, such as the formation of incoherent rogue waves.

Three types of Darboux transformation and general soliton solutions for the space-shifted nonlocal [formula omitted] symmetric nonlinear Schrödinger equation

Under investigation is the space-shifted nonlocal PT symmetric nonlinear Schrödinger (NLS) equation, which is a novel nonlocal reduction of the classical AKNS system proposed by Ablowitz and Musslimani (2021). We construct three types of Darboux transformation with the help of the symmetry conditions of the linear matrix spectral problem. Several kinds of analytical solutions such as the periodic, breather-like and bounded soliton solutions under the zero background are derived from three kinds of spectral configurations on the complex plane. Dynamics of these solutions to the space-shifted nonlocal PT symmetric NLS equation are shown.

Some reverse space(RS) rational solutions for the nonlocal coupled nonlinear Schrödinger equations on the plane wave backgrounds

A number of integrable nonlocal nonlinear Schrödinger (NNLS) type systems have been recently proposed. In this letter, a Darboux transformation (DT) of the nonlocal coupled nonlinear Schrödinger (NCNLS) equations is presented with the aid of the loop group method. The associated N-fold Darboux transformation is given on plane wave backgrounds. Starting from a special reverse space (RS) Lax pairs, the NCNLS equations are constructed, then we obtain one-, two- and N-soliton solution formulas of the NCNLS equations with N-fold Darboux transformation. As a consequence, two kinds of RS solutions are derived from plane wave backgrounds. Some novel soliton solutions and rational solutions are derived with the nonzero seed solutions through complex computations, including the bright soliton, kink soliton and breather wave soliton. The results may be useful to explain the corresponding wave phenomena in nonlocal wave modes.

Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation

We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation iqt(x,t)+qxx(x,t)+2q2(x,t)q̄(−x,t)=0with symmetric boundary conditions: q(x,t)→Ae2iA2t as x→±∞, where A>0 is an arbitrary constant. We describe the asymptotic stage of modulation instability for the NNLS equation by computing the large-time asymptotics of the solution q(x,t) of this initial value problem. We show that it exhibits a non-universal, in a sense, behavior: the asymptotics of |q(x,t)| depends on details of the initial data q(x,0). This is in a sharp contrast with the local classical NLS equation, where the long-time asymptotics of the solution depends on the initial value through the phase parameters only. The main tool used in this work is the inverse scattering transform method applied in the form of the matrix Riemann–Hilbert problem. The Riemann–Hilbert problem associated with the original initial value problem is analyzed asymptotically by the nonlinear steepest decent method.

Dbar‐approach to coupled nonlocal NLS equation and general nonlocal reduction

AbstractThe coupled nonlocal nonlinear Schrödinger (NLS) equation is studied by virtue of the Dbar‐problem. Two spectral transform matrices are introduced to define two associated Dbar‐problems. The relations between the coupled nonlocal NLS potential and the solution of the Dbar‐problem are constructed. The spatial transform method is extended to obtain the coupled nonlocal NLS equation and its conservation laws. The general nonlocal reduction of the coupled nonlocal NLS equation to the nonlocal NLS equation is discussed in detail. The explicit solutions are derived.

A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials

Abstract We study the derivation of the Gibbs measure for the nonlinear Schrödinger (NLS) equation from many-body quantum thermal states in the mean-field limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded $L^p$ interaction potentials on $\mathbb{T}^d$ for $d=1,2,3$. This extends the author’s earlier joint work with Fröhlich et al. [ 45], where the regime of defocusing and bounded interaction potentials was considered. When $d=1$, we give an alternative proof of a result previously obtained by Lewin et al. [ 69]. Our proof is based on a perturbative expansion in the interaction. When $d=1$, the thermal state is the grand canonical ensemble. As in [ 45], when $d=2,3$, the thermal state is a modified grand canonical ensemble, which allows us to estimate the remainder term in the expansion. The terms in the expansion are analysed using a graphical representation and are resummed by using Borel summation. By this method, we are able to prove the result for the optimal range of $p$ and obtain the full range of defocusing interaction potentials, which were studied in the classical setting when $d=2,3$ in the work of Bourgain [ 15].