Darboux transformation,infinite conservation laws,and exact solutions for the nonlocal Hirota equation with variable coefficients

This article presents the construction of a nonlocal Hirota equation with variable coefficients and its Darboux transformation.Using zero-seed solutions,1-soliton and 2-soliton solutions of the equation are constructed through the Darboux transformation,along with the expression for N-soliton solutions.Influence of coefficients that are taken as a function of time instead of a constant,i.e.,coefficient functionδ(t),on the solutions is investigated by choosing the coefficient functionδ(t),and the dynamics of the solutions are analyzed.This article utilizes the Lax pair to construct infinite conservation laws and extends it to nonlocal equations.The study of infinite conservation laws for nonlocal equations holds significant implications for the integrability of nonlocal equations.

Global attractor for Hirota equation

The long time behavior of solution for Hirota equation with zero order dissipation is studied. The global weak attractor for this system in Hper^k is constructed. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in Hper^k.

Rogue-wave pair and dark-bright-rogue wave solutions of the coupled Hirota equations

Novel explicit rogue wave solutions of the coupled Hirota equations are obtained by using the Darboux transformation. In contrast to the fundamental Peregrine solitons and dark rogue waves, we present an interesting rogue-wave pair that involves four zero-amplitude holes for the coupled Hirota equations. It is significant that the corresponding expressions of the rogue-wave pair solutions contain polynomials of the fourth order rather than the second order. Moreover, dark-bright- rogue wave solutions of the coupled Hirota equations are given, and interactions between Peregrine solitons and dark-bright solitons are analyzed. The results further reveal the dynamical properties of rogue waves for the coupled Hirota equations.

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations,we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem.The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair.By Lax pair,an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found.The solutions with specific properties are distinct from those of the local Hirota equation.In order to further describe the properties and the dynamic features of the solutions explicitly,several kinds of graphs are depicted.

Long-time Asymptotics for the Reverse Space-time Nonlocal Hirota Equation with Decaying Initial Value Problem:without Solitons

In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation.Furthermore,using the approach of Deift-Zhou nonlinear steepest descent,the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived.For the reverse space-time nonlocal Hirota equation,since the symmetries of its scattering matrix are different with the local Hirota equation,the v(λ_(i))(i=0,1)would like to be imaginary,which results in theδ_(λi)^(0)contains an increasing t(±Imv(λ_(i)))/2,and then the asymptotic behavior for nonlocal Hirota equation becomes differently.

Asymptotic analysis of multi-valley dark soliton solutions in defocusing coupled Hirota equations

We construct uniform expressions of such dark soliton solutions encompassing both single-valley and double-valley dark solitons for the defocusing coupled Hirota equation with high-order nonlinear effects utilizing the uniform Darboux transformation,in addition to proposing a sufficient condition for the existence of the above dark soliton solutions.Furthermore,the asymptotic analysis we perform reveals that collisions for single-valley dark solitons typically exhibit elastic behavior;however,collisions for double-valley dark solitons are generally inelastic.In light of this,we further propose a sufficient condition for the elastic collisions of double-valley dark soliton solutions.Our results offer valuable insights into the dynamics of dark soliton solutions in the defocusing coupled Hirota equation and can contribute to the advancement of studies in nonlinear optics.

Inverse Scattering Transform and Soliton Solutions for the Hirota Equation with N Distinct Arbitrary Order Poles

We employ the Riemann-Hilbert(RH)method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions.Through the spectral analysis,the asymptoticity,symmetry,and analysis of the Jost functions are obtained,which play a key role in constructing the RH problem.Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem.Choosing some appropriate parameters of the resulting solutions,we further derive the soliton solutions with different order poles,including four cases of a fourthorder pole,two second-order poles,a third-order pole and a first-order pole,and four first-order points.Finally,the dynamical behavior of these solutions are analyzed via graphic analysis.

Multi-soliton solutions for the three types of nonlocal Hirota equations via Riemann-Hilbert approach

The purpose of the paper is to formulate multi-soliton solutions for the nonlocal Hirota equations via the Riemann-Hilbert(RH)approach.The RH problems are constructed and the zero structures are studied via performing spectral analysis of the Lax pair.Then we consider three types of nonlocal Hirota equations by discussing different symmetry reductions of the potential matrix.On the basis of the resulting matrix RH problem under the restriction of the reflectionless case,we successfully obtain the multi-soliton solutions of the nonlocal Hirota equations.

Long-time asymptotics for the initial-boundary value problem of coupled Hirota equation on the half-line

The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line.We show that the solution of the coupled Hirota equation can be expressed in terms of the solution of a 3×3 matrix Riemann-Hilbert problem formulated in the complex k-plane.The relevant jump matrices are explicitly given in terms of the matrix-valued spectral functions s(k)and S(k)that depend on the initial data and boundary values,respectively.Then,applying nonlinear steepest descent techniques to the associated 3×3 matrix-valued Riemann-Hilbert problem,we can give the precise leading-order asymptotic formulas and uniform error estimates for the solution of the coupled Hirota equation.

Nonlinear waves of the Hirota and the Maxwell Bloch equations in nonlinear optics

In this paper, considering the Hirota and the Maxwell–Bloch （H-MB） equations which are governed by femtosecond pulse propagation through a two-level doped fiber system, we construct the Darboux transformation of this system through a linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions （both bright and dark breathers） of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.

Generation and propagation of subpicosecond pulse train

Higher-order nonlinear Schrodinger equation with the Hirota constraint conditions is considered, and an analytic solution, which can describe the modulational instability process, is presented. Based on the solution, a new pulse train without continuous wave （CW） background is generated in quadratures and the propagation of the pulse train is discussed in detail by simulating numerically. The results show that, unlike the propagation of the picosecond pulse train, under the effects of the higher-order terms, the pulse train cannot propagate along the fibre when the energy is very high; however, for some medium energy the pulse train can stably propagate. We also investigate the stability of the pulse train against violation of the Hirota conditions, and the results show that the pulse train can still propagate stably when the Hirota conditions are broken.

Data-driven parametric soliton-rogon state transitions for nonlinear wave equations using deep learning with Fourier neural operator

In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equation,Hirota equation,and NLS equation with the generalized PT-symmetric Scarf-II potentials.Specifically,we analyze the state transitions of different types of solitons(e.g.bright solitons,breathers,peakons,rogons,and periodic waves)appearing in these complex nonlinear wave equations.By checking the absolute errors between the predicted solutions and exact solutions,we can find that the FNO with the Ge Lu activation function can perform well in all cases even though these solution parameters have strong influences on the wave structures.Moreover,we find that the approximation errors via the physics-informed neural networks(PINNs)are similar in magnitude to those of the FNO.However,the FNO can learn the entire family of solutions under a given distribution every time,while the PINNs can only learn some specific solution each time.The results obtained in this paper will be useful for exploring physical mechanisms of soliton excitations in nonlinear wave equations and applying the FNO in other nonlinear wave equations.

Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy

In this paper,the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations.After this,two kinds of Darboux transforma-tions in different directions and their mixed Darboux transformations of this hierarchy are constructed.These symmetries and Darboux transformations might be useful in Gromov-Witten theory of CP1.