Nonlocal Nonlinear Schrodinger Equation Research Articles

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.

Exotic vector freak waves in the nonlocal nonlinear Schrödinger equation

In this paper, general higher-order freak wave solutions of the nonlocal nonlinear Schrödinger equation (NLSE) with parity-time symmetric can be calculated theoretically via a Darboux transformation and a separation of variable technique. This family of solutions are given in separation of variables form. Moreover, in order to understand these obtained solutions better, the main characteristics of two lowest freak wave solutions are discussed clearly and conveniently. They show that the dynamics of these solutions exhibits rich patterns, most of which have no counterparts in the corresponding local equation.

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.

An improved finite integration method for nonlocal nonlinear Schrödinger equations

In this paper, we study the nonlocal nonlinear Schrödinger equation (NNLSE), which is an important extension of the nonlinear Schrödinger equation (NLSE). Since the nonlinearity of NNLSE involves a convolution, numerical approximation of its solution is very expensive. To achieve an efficient numerical method for solving the NNLSE, according to the property of convolution, we firstly present a partial differential equation (PDE) method to transfer the NNLSE from an integro-differential equation to an equivalent or approximate system of PDEs, and the numerical solution of the NNLSE can then be obtained by solving these PDEs. Based on the recently developed finite integration method (FIM), we derive an improved method (IFIM) in this paper. The novelty of this IFIM is that the quadrature method is only used once instead of multiple times for multi-layer integral, so it consumes less running time and provides higher accuracy. In addition, we adapt a second-order operator-splitting method (OS2) at each time-step to ensure a convergent solution for long-time integration. Several numerical experiments are given to verify the efficiency and accuracy of the proposed IFIM for solving the NNLSE.

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.

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.

Propagations of Fresnel diffraction accelerating beam in Schrödinger equation with nonlocal nonlinearity* *Project supported by the National Natural Science Foundation of China (Grant Nos. 61805068, 61875053, and 62074127), China Postdoctoral Science Foundation (Grant No. 2017M620300), and the Fund from the Science and Technology Department of Henan Province, China (Grant No. 202102210111).

Accelerating beams have been the subject of extensive research in the last few decades because of their self-acceleration and diffraction-free propagation over several Rayleigh lengths. Here, we investigate the propagation dynamics of a Fresnel diffraction beam using the nonlocal nonlinear Schrödinger equation (NNLSE). When a nonlocal nonlinearity is introduced into the linear Schrödinger equation without invoking an external potential, the evolution behaviors of incident Fresnel diffraction beams are modulated regularly, and certain novel phenomena are observed. We show through numerical calculations, under varying degrees of nonlocality, that nonlocality significantly affects the evolution of Fresnel diffraction beams. Further, we briefly discuss the two-dimensional case as the equivalent of the product of two one-dimensional cases. At a critical point, the Airy-like intensity profile oscillates between the first and third quadrants, and the process repeats during propagation to yield an unusual oscillation. Our results are expected to contribute to the understanding of NNLSE and nonlinear optics.

On general solitons in the parity-time-symmetric defocusing nonlinear Schrödinger equation

Via the Hirota bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method, two types of solitons, namely general solitons on a background of periodic waves and rational soliton solutions, to the parity-time-symmetric nonlocal nonlinear Schrodinger equation with the defocusing-type nonlinearity for nonzero boundary condition are investigated. These two types of soliton solutions are constructed by constraining the tau functions of single-component KP hierarchy satisfying the dimension reduction and the nonlocal symmetry and the complex conjugated conditions. For the solitons on a background of periodic waves, we first construct the periodic solution to provide the background of periodic waves and then combine the periodic solution with general soliton solution, which generate solitons on a background of periodic waves. For the rational solitons, we mainly investigate the dynamical behaviours of interactions between several individual rational dark–antidark solitons, antidark–antidark solitons, and antidark–dark solitons, which are elastic collisions with no phase shift. The degenerated soliton cases are also discussed, in which only one-antidark soliton survives.

Breather solutions of the nonlocal nonlinear self-focusing Schrödinger equation

The first- and second-order breather solutions of the self-focusing nonlocal nonlinear Schrödinger (NNLS) equation are obtained by employing Hirota's bilinear method. The NNSE also happens to be an example of Schrödinger equation with parity-time (PT) symmetry. With the help of recurrence relations in the Hirota bilinear form, the nth-order breather solutions on the nonzero background of the NNLS equation are obtained, and the collision, superposition and separation of transmission modes is studied respectively. When the parameters describing these breathers are selected as some special values, they display plentiful spatial structures which provide effective methods for controlling the localized optical waves in nonlocal nonlinear media.

Combined soliton solutions of a (1+1)-dimensional weakly nonlocal conformable fractional nonlinear Schrödinger equation in the cubic–quintic nonlinear material

We concentrate on a (1+1)-dimensional weakly nonlocal conformable fractional nonlinear Schrodinger equation with cubic–quintic nonlinearities, and get some exact combined soliton solutions and periodic solutions based on the Jacobian elliptic function method and double function method. We discuss dynamical behaviors of bright soliton, dark soliton, periodic bright soliton train and periodic M-shaped soliton train, and find that shift distances along the x-direction of these solitons all add if the value of the fractional order increases.

TWO-DIMENSIONAL NONLOCAL NONLINEAR SCHRODINGER EQUATION BASED ON THE ABLOWITZ-MUSSLIMANI SYMMETRY CONDITION

The nonlinear Schrodinger equation is a nonlinear partial differential equation and integrable equation that play an essential role in many branches of physics as nonrelativistic quantum mechanics, acoustics, and optics. In this work, motivated by the ideas of Ablowitz and Musslimani, we successfully obtain a two-dimensional nonlocal nonlinear Schrodinger equation where the nonlocality consists of reverse time fields as factors in the nonlinear terms. The nonlocal nonlinear Schrodinger equation admits a great number of good properties that the classical nonlinear Schrodinger equation possesses, e.g. PT-symmetric, admitting Lax-pair, and infinitely many conservation laws. We apply the Darboux transformation method to the two-dimensional nonlinear Schrodinger equation. The idea of this method is having a Lax representation, one can obtain various kinds of solutions of the Nth order with a spectral parameter. The exact solutions and graphical representation of obtained solutions are derived.

Propagation of self-accelerating Hermite complex-variable-function Gaussian wave packets in highly nonlocal nonlinear media

The evolution dynamic properties of self-accelerating Hermite complex-variable-function Gaussian (SHCG) wave packets in highly nonlocal nonlinear media are investigated. Analytical results from a $$(3+1)$$ -dimensional Snyder–Mitchell model show that various SHCG wave packets carrying multi-order vortices rotate smoothly. Increasing the distribution factor will cause the intensity layout to cluster more closely around the center, while the vortices will be farther away. The SHCG wave packets can reverse the positions of their temporal side lobes. The role of the power ratio in determining the rotation period and the angular velocity is also discussed. Furthermore, numerical results of the nonlocal nonlinear Schrodinger equation are simulated to illustrate the effects of different nonlocalities and initial perturbations. The SHCG wave packets show interesting features during propagation, which can provide new ideas for the regulation of the multi-dimensional optical field.