Nonlocal Davey Stewartson Research Articles

Coherent structures of lump-soliton for the PT-symmetric nonlocal Davey-Stewartson III equation on a background of constant or periodic line waves

Coherent structures of lump-soliton for the PT-symmetric nonlocal Davey-Stewartson III equation on a background of constant or periodic line waves

Bright solitons on periodic background in the nonlocal Davey–Stewartson I equation with fully space-shifted $$\mathcal{P}\mathcal{T}$$-symmetry

Bright solitons on periodic background in the nonlocal Davey–Stewartson I equation with fully space-shifted $$\mathcal{P}\mathcal{T}$$-symmetry

The multiple bright soliton pairs of the fully PT-symmetric nonlocal Davey-Stewartson I equation

This article investigates the dynamics of multiple bright soliton pair interactions in the fully PT -symmetric nonlocal Davey–Stewartson I equation. The bright soliton pair solutions are derived by employing the bilinear KP-hierarchy reduction method, and are expressed in terms of determinants. To study the interactions of the multiple soliton pairs, the long-time asymptotic analysis for these soliton solutions is performed by using the analysis of determinants, and the asymptotic expressions of the N individual soliton pair solutions are given as the sum of expressions for the 2N single soliton solutions. The asymptotics shows that the soliton pairs only exhibit elastic collisions and the two solitons in each soliton pair share equal amplitudes

Rational and semi-rational solutions to the nonlocal Davey–Stewartson III equation

We utilize the bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique to obtain solutions for the nonlocal DS III equation. These solutions are elegantly expressed in terms of an N×N determinant, incorporating various parameter reduction constraints. It is important to note that these solutions are nontrivial and do not necessarily satisfy the local DS III equation. In the case of an even value of N, we derive solutions on constant backgrounds, while for odd values of N, solutions on periodic backgrounds are generated. To gain a deeper understanding of the dynamics involved, we conduct a thorough analysis of the derived solutions, accompanied by informative plots. Overall, our findings showcase the power of the bilinear method and KP hierarchy reduction technique in obtaining meaningful solutions for the nonlocal DS III equation, shedding light on its complex behavior and enriching our understanding of the system.

Dark soliton solutions for a [formula omitted]-dimensional nonlocal Davey–Stewartson system

A (2+1)-dimensional nonlocal Davey–Stewartson system has been considered in this paper. Dark-soliton solutions in Gram determinant for such nonlocal Davey–Stewartson system have been demonstrated with the aid of the Kadomtsev–Petviashvili hierarchy reduction. According to the dark-soliton solutions, interactions between the dark–dark, antidark–antidark and dark–antidark solitons have been graphically presented. The interactions between two solitons (two dark, one dark and one anti-dark, two anti-dark) are all parallel. Furthermore, various types of the interactions among dark four solitons are also demonstrated: interactions between double dark two solitons, between double antidark two solitons, between dark two solitons and antidark two solitons, and between there mixed types. The interactions are all elastic.

The line rogue wave solutions of the nonlocal Davey-Stewartson I equation with PT symmetry based on the improved physics-informed neural network.

In the paper, we employ an improved physics-informed neural network (PINN) algorithm to investigate the data-driven nonlinear wave solutions to the nonlocal Davey-Stewartson (DS) I equation with parity-time (PT) symmetry, including the line breather, kink-shaped and W-shaped line rogue wave solutions. Both the PT symmetry and model are introduced into the loss function to strengthen the physical constraint. In addition, since the nonlocal DS I equation is a high-dimensional coupled system, this leads to an increase in the number of output results. The PT symmetry also needs to be learned that is not given in advance, which increases challenges in computing for multi-output neural networks. To address these problems, our objective is to assign various levels of weight to different items in the loss function. The experimental results show that the improved algorithm has better prediction accuracy to a certain extent compared with the original PINN algorithm. This approach is feasible to investigate complex nonlinear waves in a high-dimensional model with PT symmetry.

Dromion solutions of PT-symmetric [formula omitted]-nonlocal Davey-Stewartson I equation

A PT-symmetric nonlocal Davey-Stewartson I equation is considered, in which u¯(x,y,t) in the classical equation is replaced by u¯(−x,−y,t). Using the nonlinear constraint from 2+1 dimensions to 1+1 dimensions and Darboux transformation in 1+1 dimensions, 2m×2n dromion solutions are obtained. It is proved that under certain conditions, the derived solutions are always globally defined and decay exponentially at spacial infinity. Moreover, each asymptotic solution as t→±∞ has exactly 4mn peaks. The local behavior of each peak is also given.

Soliton solutions to the reverse-time nonlocal Davey–Stewartson III equation

This study derives solitons solutions of the reverse-time nonlocal Davey–Stewartson III equation by the Kadomtsev–Petviashvili hierarchy reduction method and Hirota’s bilinear method. The solutions are expressed as N×N Gram-type determinants with different parametric reduction conditions. N-soliton and line breather solutions on both constant and periodic backgrounds are derived. The dynamics of these solutions are discussed. All possible configurations of these solutions are illustrated for 1≤N≤4. Both intersecting and parallel solitons are presented. In particular, the elastic and inelastic collisions of the two parallel-soliton solutions are determined. In the inelastic case, the amplitudes of the solitons change after collision. Moreover, the parametric conditions for determining the inelastic collisions are derived, and all possible types of inelastic behaviors are obtained and displayed.

Soliton solutions to the nonlocal Davey–Stewartson III equation

The soliton solutions on both constant and periodic backgrounds of the nonlocal Davey–Stewartson III equation are derived by using the bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. These solutions are presented as [Formula: see text] Gram-type determinants, with [Formula: see text] a positive integer. Typical dynamics of these soliton solutions are investigated in analytical and graphical aspects. Two types of soliton solutions are generated with different [Formula: see text]. When [Formula: see text] is even, solitons on the constant background can be constructed, whereas solitons appear on the periodic background for odd [Formula: see text]. Under suitable parameter restrictions, we show the regularity of solutions and display all patterns of two- and four-soliton solutions.

Formula omitted]-symmetric nonlocal Davey–Stewartson I equation: General lump-soliton solutions on a background of periodic line waves

We study the general lump-type soliton solutions on a background of periodic line waves in the PT-symmetric nonlocal Davey–Stewartson I (DS I) equation. By using the Kadomtsev–Petviashvili hierarchy reduction method, new families of semi-rational solutions termed as lump-soliton solutions to the nonlocal DS I equation are constructed. Under particular parameters restrictions we obtain: (a) different types of lumps sitting on a background of periodic line waves and (b) different types of lumps interacting with line solitons on a background of periodic line waves. The interactions of lumps and line solitons are inelastic, generating three generic dynamical scenarios: (i) lumps fusing into line solitons, (ii) lumps fissioning from line solitons, and (iii) a combined process consisting of lumps fusing into and fissioning from line solitons.

Nonintegrable spatial discrete nonlocal nonlinear schrödinger equation.

Integrable and nonintegrable discrete nonlinear Schrödinger equations (NLS) are significant models to describe many phenomena in physics. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time, and reverse space-time nonlocal integrable equations, including the nonlocal NLS equation, nonlocal sine-Gordon equation, nonlocal Davey-Stewartson equation, etc. Moreover, the integrable nonlocal discrete NLS has been exactly solved by inverse scattering transform. In this paper, we study a nonintegrable discrete nonlocal NLS, which is a direct discrete version of the reverse space nonlocal NLS. By applying discrete Fourier transform and modified Neumann iteration, we present its stationary solutions numerically. The linear stability of the stationary solutions is examined. Finally, we study the Cauchy problem for the nonlocal NLS equation numerically and find some different and new properties on the numerical solutions comparing with the numerical solutions of the Cauchy problem for the NLS equation.

Formula omitted]-symmetric nonlocal Davey–Stewartson I equation: Soliton solutions with nonzero background

Various solutions to the PT-symmetric nonlocal Davey–Stewartson (DS) I equation with nonzero boundary condition are derived by constraining different tau functions of the Kadomtsev–Petviashvili hierarchy combined with the Hirota bilinear method. From the first type of tau functions of the nonlocal DS I equation, we construct: (a) general line soliton solutions sitting on either a constant background or on a background of periodic line waves and (b) general lump-type soliton solutions. We find two generic types of line solitons that we call usual line solitons and new-found ones. The usual line solitons exhibit elastic collisions, whereas the new-found ones, in the evolution process, change their waveforms from an antidark (dark) shape to a dark (antidark) one. The general lump-type soliton solutions describe the interaction between 2N-line solitons and 2N-lumps, which give rise to different dynamical scenarios: (i) fusion of line solitons and lumps into line solitons, (ii) fission of line solitons into lumps and line solitons, and (iii) a combination of fusion and fission processes. By constraining another type of tau functions combined with the long wave limit method, periodic line waves, rogue waves, and semi-rational solutions to the nonlocal DS I equation are obtained in terms of determinants whose matrix elements have simple algebraic expressions. Finally, different types of general solutions of the nonlocal nonlinear Schrödinger equation, namely general higher-order breathers and mixed solutions consisting of higher-order breathers and rogue waves sitting on either a constant background or on a background of periodic line waves are obtained as reductions of the corresponding solutions of the nonlocal DS I equation.

Breather and lump solutions for nonlocal Davey–Stewartson II equation

A binary Darboux transformation for the nonlocal Davey–Stewartson II equation is constructed by the Darboux matrix method. As applications, N-soliton, breather and lump solutions of the considered equation have been obtained. In particular, the dynamics of the single soliton, two-soliton, breather and lump solutions are discussed and illustrated through some figures.

Dynamics of rogue waves in the partially [formula omitted]-symmetric nonlocal Davey–Stewartson systems

In this work, we study the dynamics of rogue waves in the partially PT-symmetric nonlocal Davey–Stewartson(DS) systems. Using the Darboux transformation method, general rogue waves in the partially PT-symmetric nonlocal DS equations are derived. For the partially PT-symmetric nonlocal DS-I equation, the solutions are obtained and expressed in term of determinants. For the partially PT-symmetric DS-II equation, the solutions are represented as quasi-Gram determinants. It is shown that the fundamental rogue waves in these two systems are rational solutions which arises from a constant background at t→−∞, and develops finite-time singularity on an entire hyperbola in the spatial plane at the critical time. It is also shown that the interaction of several fundamental rogue waves is described by the multi rogue waves. And the interaction of fundamental rogue waves with dark and anti-dark rational travelling waves generates the novel hybrid-pattern waves. However, no high-order rogue waves are found in this partially PT-symmetric nonlocal DS systems. Instead, it can produce some high-order travelling waves from the high-order rational solutions.

Rogue waves of the nonlocal Davey–Stewartson I equation

Recently, Fokas presented a nonlocal Davey–Stewartson I (DSI) equation (Fokas 2016 Nonlinearity 29 319–24), which is a two-spatial dimensional analogue of the nonlocal nonlinear Schrödinger (NLS) equation (Ablowitz and Musslimani 2013 Phys. Rev. Lett. 110 064105), involving a self-induced parity-time-symmetric potential. For this equation, high-order periodic line waves and line breathers are derived by employing the bilinear method. The long wave limit of these periodic solutions yields two kinds of fundamental rogue waves, namely, kink-shaped and W-shaped line rogue waves. The interaction of fundamental line rogue waves generate higher-order rogue waves, which have several interesting patterns with different curvy profiles. Furthermore, semi-rational solutions are constructed, which are line rogue waves on a background of periodic line waves. Finally, two particular solutions of the nonlocal NLS equation, namely, a first-order rogue wave and a semi-rational solution are obtained as reductions of the corresponding solutions of the nonlocal DSI equation.

Darboux Transformations and Global Explicit Solutions for Nonlocal Davey–Stewartson I Equation

AbstractFor the nonlocal Davey–Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained. Like other nonlocal equations, many solutions of this equation may have singularities. However, by suitable choice of parameters in the solutions of the Lax pair, it is proved that the solutions obtained from seed solutions which are zero and an exponential function of t, respectively, by a Darboux transformation of degree n are global solutions of the nonlocal Davey–Stewartson I equation. The derived solutions are soliton solutions when the seed solution is zero, in the sense that they are bounded and have n peaks, and “dark cross soliton” solutions when the seed solution is an exponential function of t, in the sense that they are bounded and their norms change fast along some crossing straight lines.

Several reverse-time integrable nonlocal nonlinear equations: Rogue-wave solutions.

A study of rogue-wave solutions in the reverse-time nonlocal nonlinear Schrödinger (NLS) and nonlocal Davey-Stewartson (DS) equations is presented. By using Darboux transformation (DT) method, several types of rogue-wave solutions are constructed. Dynamics of these rogue-wave solutions are further explored. It is shown that the (1 + 1)-dimensional fundamental rogue-wave solutions in the reverse-time NLS equation can be globally bounded or have finite-time blowing-ups. It is also shown that the (2 + 1)-dimensional line rogue waves in the reverse-time nonlocal DS equations can be bounded for all space and time or develop singularities in critical time. In addition, the multi- and higher-order rogue waves exhibit richer structures, most of which have no counterparts in the corresponding local nonlinear equations.

Reductions of Darboux transformations for the PT-symmetric nonlocal Davey–Stewartson equations

In this letter, a study of the reductions of the Darboux transformations (DTs) for the PT -symmetric nonlocal Davey–Stewartson (DS) equations is presented. Firstly, a binary DT is constructed in integral form for the PT -symmetric nonlocal DS-I equation. Secondly, an elementary DT is constructed in differential form for the PT -symmetric nonlocal DS-II equation. Afterwards, a new binary DT in integral form is also found for the nonlocal DS-II equation. Moreover, it is shown that the symmetry properties in the corresponding Lax-pairs of the equations are well preserved through these DTs. Thirdly, based on above DTs, the fundamental rogue waves and rational travelling waves are obtained.

Rational and semi-rational solutions of the [formula omitted]-nonlocal Davey–Stewartson I equation

The nonlocal Davey–Stewartson (DS) I equation with a parity-time-symmetric potential with respect to the y-direction, which is called the y-nonlocal DS I equation, is a two-dimensional analogue of the nonlocal nonlinear Schrödinger (NLS) equation. The multi-breather solutions for the y-nonlocal DS I equation are derived by using the Hirota bilinear method. Lump-type solutions and hybrid solutions consisting of lumps sitting on periodic line waves are generated by long wave limits of the obtained soliton solutions. Also, various types of analytical solutions for the nonlocal NLS equation with negative nonlinearity, including both the Akhmediev breathers and the Peregrine rogue waves sitting on periodic line waves, can be generated with appropriate constraints on the parameters of the obtained exact solutions of the y-nonlocal DS I equation. Particularly, we show that a family of hybrid solitons describing the Peregrine rogue wave that coexists with the Akhmediev breather, both of them sitting on a spatially-periodic background can be thus obtained.

Nonsingular localized wave solutions for the nonlocal Davey–Stewartson I equation with zero background

In this paper, via the elementary Darboux transformation, we study the nonsingular localized wave solutions of the partially parity-time [Formula: see text] symmetric nonlocal Davey–Stewartson I equation with zero background. In addition to the common dromion and line-soliton solutions, we obtain some new localized wave solutions including the periodical-soliton, quasi-line-soliton and defected-line-soliton solutions. Meanwhile, we give the exact nonsingular parametric conditions for the derived solutions to display different localized wave structures. In addition, we discuss the dynamical behavior of the obtained nonlinear localized wave solutions with graphical illustration.