Dynamics of fundamental and double-pole breathers and solitons for a nonlinear Schr¨odinger equation with sextic operator under non-zero boundary conditions

We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinearSchr¨odinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons.

From breather solutions to lump solutions:A construction method for the Zakharov equation

Periodic solutions of the Zakharov equation are investigated.By performing the limit operationλ_(2l-1)→λ_(1)on the eigenvalues of the Lax pair obtained from the n-fold Darboux transformation,an order-n breather-positon solution is first obtained from a plane wave seed.It is then proven that an order-n lump solution can be further constructed by taking the limitλ_(1)→λ_(0)on the breather-positon solution,because the unique eigenvalueλ_(0)associated with the Lax pair eigenfunctionΨ(λ_(0))=0 corresponds to the limit of the infinite-periodic solutions.A convenient procedure of generating higher-order lump solutions of the Zakharov equation is also investigated based on the idea of the degeneration of double eigenvalues in multi-breather solutions.

Breather and its interaction with rogue wave of the coupled modified nonlinear Schrodinger equation

We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.

Nonlinear perturbation of a high-order exceptional point:Skin discrete breathers and the hierarchical power-law scaling

We study the nonlinear perturbation of a high-order exceptional point(EP)of the order equal to the system site number L in a Hatano-Nelson model with unidirectional hopping and Kerr nonlinearity.Notably,we find a class of discrete breathers that aggregate to one boundary,here named as skin discrete breathers(SDBs).The nonlinear spectrum of these SDBs shows a hierarchical power-law scaling near the EP.Specifically,the response of nonlinear energy to the perturbation is given by E_(m)∝Γ~(α_(m)),whereα_(m)=3^(m-1)is the power with m=1,...,L labeling the nonlinear energy bands.This is in sharp contrast to the L-th root of a linear perturbation in general.These SDBs decay in a double-exponential manner,unlike the edge states or skin modes in linear systems,which decay exponentially.Furthermore,these SDBs can survive over the full range of nonlinearity strength and are continuously connected to the self-trapped states in the limit of large nonlinearity.They are also stable,as confirmed by a defined nonlinear fidelity of an adiabatic evolution from the stability analysis.As nonreciprocal nonlinear models may be experimentally realized in various platforms,such as the classical platform of optical waveguides,where Kerr nonlinearity is naturally present,and the quantum platform of optical lattices with Bose-Einstein condensates,our analytical results may inspire further exploration of the interplay between nonlinearity and non-Hermiticity,particularly on high-order EPs,and benchmark the relevant simulations.

Soliton molecules,T-breather molecules and some interaction solutions in the(2+1)-dimensional generalized KDKK equation

By employing the complexification method and velocity resonant principle to N-solitons of the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(KDKK)equation,we obtain the soliton molecules,T-breather molecules,T-breather–L-soliton molecules and some interaction solutions when N≤6.Dynamical behaviors of these solutions are discussed analytically and graphically.The method adopted can be effectively used to construct soliton molecules and T-breather molecules of other nonlinear evolution equations.The results obtained may be helpful for experts to study the related phenomenon in oceanography and atmospheric science.

Periodic,quasiperiodic and chaotic discrete breathers in a parametrical driven two-dimensional discrete diatomic Klein-Gordon lattice

We study a two-dimensional （2D） diatomic lattice of anhaxmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers （DBs） can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the 2D discrete diatomic Klein-Gordon lattice with hard and soft on-site potentials. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers （QDBs） and chaotic discrete breathers （CDBs） by changing the amplitude of the driver. But the DBs and QDBs with symmetric and anti-symmetric profiles that are centered at a heavy atom are more stable than at a light atom, because the frequencies of the DBs and QDBs centered at a heavy atom are lower than those centered at a light atom.

Periodic, quasiperiodic, and chaotic breathers in two-dimensional discrete β-Fermi-Pasta-Ulam lattice

Using numerical method, we investigate whether periodic, quasiperiodic, and chaotic breathers are supported by the two-dimensional discrete Fermi-Pasta-Ulam （FPU） lattice with linear dispersion term. The spatial profile and time evolution of the two-dimensional discrete fl-FPU lattice are segregated by the method of separation of variables, and the numerical simulations suggest that the discrete breathers （DBs） are supported by the system. By introducing a periodic interaction into the linear interaction between the atoms, we achieve the coupling of two incommensurate frequencies for a single DB, and the numerical simulations suggest that the quasiperiodic and chaotic breathers are supported by the system, too.

Different Kinds of Discrete Breathers in Three Types of One-Dimensional Models

The dynamics of different kinds of discrete breathers in three types of one-dimensionai monatomic chains with on-site and inter-site potentiais are investigated. The existence and evolution of symmetric breather, antisymmetric breather, and multibreather in one-dimensionai models are proved by using rotating wave approximation, local anharmonic approximation, and the numerical method. The linear stability of these breathers is investigated by using Lyapunov stable anaiysis. The localization and stability of breathers in three types of models correlate closely to the system nonlinear parameter β.

Existences and stabilities of bright and dark breathers in a general one-dimensional discrete monatomic Chain

A general one-dimensional discrete monatomic model is investigated by using the multiple-method. It is proven that the discrete bright breathers （DBBs） and discrete dark breathers （DDBs） exist in this model at the anti-continuous limit, and then the concrete models of the DBBs and DDBs are also presented by the multiple-scale approach （MSA） and the quasi-discreteness approach （QDA）. When the results are applied to some particular models, the same conclusions as those presented in corresponding references are achieved. In addition, we use the method of the linearization analysis to investigate this system without the high order terms of ε. It is found that the DBBs and DDBs are linearly stable only when coupling parameter χ is small, of which the limited value is obtained by using an analytical method.

Modulational instability, quantum breathers and two-breathers in a frustrated ferromagnetic spin lattice under an external magnetic field

The modulational instability, quantum breathers and two-breathers in a frustrated easy-axis ferromagnetic zig-zag chain under an external magnetic field are investigated within the Hartree approximation. By means of a linear stability analysis, we analytically study the discrete modulational instability and analyze the effect of the frustration strength on the discrete modulational instability region. Using the results from the discrete modulational instability analysis, the presence conditions of those stationary bright type localized solutions are presented. On the other hand, we obtain the analytical expressions for the stationary bright localized solutions and analyze the effect of the frustration on their emergence conditions. By taking advantage of these bright type single-magnon bound wave functions obtained, quantum breather states in the present frustrated ferromagnetic zig-zag lattice are constructed. What is more, the analytical forms for quantum two-breather states are also obtained. In particular, the energy level formulas of quantum breathers and two-breathers are derived.

Two-Dimensional Breather Lattice Solutions and Compact-Like Discrete Breathers and Their Stability in Discrete Two-Dimensional Monatomic β-FPU Lattice

We restrict our attention to the discrete two-dimensional monatomic β-FPU lattice. We look for two- dimensional breather lattice solutions and two-dimensional compact-like discrete breathers by using trying method and analyze their stability by using Aubry＇s linearly stable theory. We obtain the conditions of existence and stability of two-dimensional breather lattice solutions and two-dimensional compact-like discrete breathers in the discrete two- dimensional monatomic β-FPU lattice.

Formation mechanism of asymmetric breather and rogue waves in pair-transition-coupled nonlinear Schr?dinger equations

Based on the developed Darboux transformation, we investigate the exact asymmetric solutions of breather and rogue waves in pair-transition-coupled nonlinear Schr?dinger equations. As an example, some types of exact breather solutions are given analytically by adjusting the parameters. Moreover, the interesting fundamental problem is to clarify the formation mechanism of asymmetry breather solutions and how the particle number and energy exchange between the background and soliton ultimately form the breather solutions. Our results also show that the formation mechanism from breather to rogue wave arises from the transformation from the periodic total exchange into the temporal local property.

Soliton, breather, and rogue wave solutions for solving the nonlinear Schrodinger equation using a deep learning method with physical constraints

The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.

Elegant Ince-Gaussian breathers in strongly nonlocal nonlinear media

A novel class of optical breathers, called elegant Ince-Gaussian breathers, are presented in this paper. They are exact analytical solutions to Snyder and Mitchell＇s mode in an elliptic coordinate system, and their transverse structures are described by Ince-polynomials with complex arguments and a Gaussian function. We provide convincing evidence for the correctness of the solutions and the existence of the breathers via comparing the analytical solutions with numerical simulation of the nonlocal nonlinear SchrSdinger equation.

A Breather in Birefringent Fibers

The propagation properties of the breather in birefringent fibers are investigated. The breather can propagate stably in strongly birefringent fibers. The propagation law can be expected. However, random birefringence makes the propagation of the breather more complex. The breather will partly disappear and partly appear, even may split into two smaller breathers. In addition, the varying range of relative time displacement between two components of the breather becomes narrower with the effect of third-order dispersion. If third order dispersion is too strong, the breather behavior will disappear gradually during the transmission. The breather can exist in random birefringent fiber with dispersion management rather than in strongly birefringent fiber.

Discrete gap breathers in a two-dimensional diatomic face-centered square lattice

In this paper we study the existence and stability of two-dimensional discrete gap breathers in a two-dimensional diatomic face-centered square lattice consisting of alternating light and heavy atoms, with on-site potential and coupling potential. This study is focused on two-dimensional breathers with their frequency in the gap that separates the acoustic and optical bands of the phonon spectrum. We demonstrate the possibility of the existence of two-dimensional gap breathers by using a numerical method. Six types of two-dimensional gap breathers are obtained, i.e., symmetric, mirror-symmetric and asymmetric, whether the center of the breather is on a light or a heavy atom. The difference between one-dimensional discrete gap breathers and two-dimensional discrete gap breathers is also discussed. We use Aubry＇s theory to analyze the stability of discrete gap breathers in the two-dimensional diatomic face-centered square lattice.

Breathers and solitons for the coupled nonlinear Schrodinger system in three-spine α-helical protein

We mainly investigate the variable-coefficient 3-coupled nonlinear Schrodinger(NLS)system,which describes soli- ton dynamics in the three-spineα-helical protein with inhomogeneous effect.The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly.The Hirota method is used to solve the constant coefficient NLS equation,and then we get the one-and two-breather solutions of the variable-coefficient NLS equation.The results show that,in the background of plane waves and periodic waves,the breather can be transformed into some forms of combined soliton solutions.The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail.Our results are helpful to study the soliton dynamics inα-helical protein.

Soliton, Breather and Rogue Wave Solutions for the Nonlinear Schrdinger Equation Coupled to a Multiple Self-Induced Transparency System

We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.The N-soliton, N-breather and N th-order rogue wave solutions in the compact determinant representations are derived using the Darboux transformation and limit technique. Dynamics of such solutions from the first-to second-order ones are shown.

Controllable optical superregular breathers in the femtosecond regime

We investigate optical superregular breathers in the femtosecond regime under dispersion management in an inho- mogeneous fiber governed by the nonautonomous higher-order nonlinear Schr6dinger equation （NLSE）. Exact solutions describing the dynamics of superregular breathers are obtained. Furthermore, we discuss the propagation behaviors of controllable superregular breathers, including stabilization and recurrence in an exponential dispersion fiber and a peri- odic distributed fiber system. Particularly, the nonlinear dynamics of superregular modes evolved from an identical initial small-amplitude modulation is analyzed in detail.

Discrete breathers in a model with Morse potentials

Under harmonic approximation, this paper discusses the linear dispersion relation of the one-dimensional chain. The existence and evolution of discrete breathers in a general one-dimensional chain are analysed for two particular examples of soft （Morse） and hard （quartic） on-site potentials. The existence of discrete breathers in one-dimensional and two-dimensional Morse lattices is proved by using rotating wave approximation, local anharmonic approximation and a numerical method. The localization and amplitude of discrete breathers in the two-dimensional Morse lattice with on-site harmonic potentials correlate closely to the Morse parameter a and the on-site parameter к.