In the article, the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established. Then the new function space is introduced and the stability problem for the finite differ...In the article, the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established. Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space. The approach used is of a simple characteristic in gaining the stability condition of the scheme.展开更多
A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect ...A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.展开更多
A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkin finite element method, which has been proved to be 2nd-order accurate in time and 4th-order in space. The comparison between the exac...A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkin finite element method, which has been proved to be 2nd-order accurate in time and 4th-order in space. The comparison between the exact and numerical solutions of progressive waves shows that this numerical scheme is quite accurate, stable andefflcient. It is also shown that any local disturbance will spread, have a full growth and finally form two progressive waves propagating in both directions. The shape and the speed of the long term progressive waves are determined by the system itself, and do not depend on the details of the initial values.展开更多
In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof ar...In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof are the tails estimate and to demonstrate the Lipschitz continuity and random squeezing property of the solution for the equation defined on R^(N) .展开更多
In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to ...In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.展开更多
A monotone compact implicit finite difference scheme with fourth-order accuracy in space and second-order in time is proposed for solving nonlinear reaction-diffusion equations. An accelerated monotone iterative metho...A monotone compact implicit finite difference scheme with fourth-order accuracy in space and second-order in time is proposed for solving nonlinear reaction-diffusion equations. An accelerated monotone iterative method for the resulting discrete problem is presented. The sequence of iteration converges monotonically to the unique solution of the discrete problem, and the convergence rate is either quadratic or nearly quadratic, depending on the property of the nonlinear reaction. The numerical results illustrate the high accuracy of the proposed scheme and the rapid convergence rate of.the iteration.展开更多
In this paper, by introducing a proper transformation, the （Gr/G）-expansion method is further extended into the nonlinear reaction diffusion equations in mathematical biology whose balancing numbers may be negative ...In this paper, by introducing a proper transformation, the （Gr/G）-expansion method is further extended into the nonlinear reaction diffusion equations in mathematical biology whose balancing numbers may be negative integer. As a result, hyperbolic function solutions and trigonometric function solutions with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions and the periodic wave solutions are also derived from the traveling wave solutions. Moreover, it is observed that the suggested techniques is compatible of such problems.展开更多
In this paper,we prove the existence of martingale solutions of a class of stochastic equations with a monotone drift of polynomial growth of arbitrary order and a continuous diffusion term with superlinear growth.Bot...In this paper,we prove the existence of martingale solutions of a class of stochastic equations with a monotone drift of polynomial growth of arbitrary order and a continuous diffusion term with superlinear growth.Both the nonlinear drift and diffusion terms are not required to be locally Lipschitz continuous.We then apply the abstract result to establish the existence of martingale solutions of the fractional stochastic reaction-diffusion equation with polynomial drift driven by a superlinear noise.The pseudo-monotonicity techniques and the Skorokhod-Jakubowski representation theorem in a topological space are used to pass to the limit of a sequence of approximate solutions defined by the Galerkin method.展开更多
In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple dire...In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method,which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.展开更多
Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-...Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O（△t ＋ h k＋1 ＋ H 2k＋2 d/2 ） （k ≥ 1）, where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.展开更多
In this paper, we discuss the blowing\|up of the solutions of a class of nonlinear reaction\|diffusion equations with the general (or nonlinear) boundary conditions. On some proper assumptions, we conclude that there ...In this paper, we discuss the blowing\|up of the solutions of a class of nonlinear reaction\|diffusion equations with the general (or nonlinear) boundary conditions. On some proper assumptions, we conclude that there is no global smooth solution, i.e., the solutions blow up in the finite time.展开更多
In this paper, the initial value problem of nonlinear reaction-diffusion equation is considered. The Dufort-Frankel finite difference approximation for the long time scheme is given for the d-dimensional reaction-diff...In this paper, the initial value problem of nonlinear reaction-diffusion equation is considered. The Dufort-Frankel finite difference approximation for the long time scheme is given for the d-dimensional reaction-diffusion equation with the two different cases. The global solution and global attractor are discussed for the Dufort-Frankel scheme. Moreover properties of the solution are studied. The error estimate is presented in a finite time region and in the global time region for some special cases. Finally the numerical results for the equation are investigated for Allen-Cahn equation and some other equations and the homoclinic orbit is simulated numerically.展开更多
The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method. The invarianee of some sets under the ...The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method. The invarianee of some sets under the flow of these problems and the vacuum isolation of solutions are obtained by introducing a family of potential wells. Then the threshold result of global existence and nonexistence of solutions are given. Finally, the problem with critical initial conditions are discussed.展开更多
In this paper,we focus on peaked traveling wave solutions of the modified highly nonlinear Novikov equation by dynamical systems approach.We obtain a traveling wave system which is a singular planar dynamical system w...In this paper,we focus on peaked traveling wave solutions of the modified highly nonlinear Novikov equation by dynamical systems approach.We obtain a traveling wave system which is a singular planar dynamical system with three singular straight lines,and derive all possible phase portraits under corresponding parameter conditions.Then we show the existence and dynamics of two types of peaked traveling wave solutions including peakons and periodic cusp wave solutions.The exact explicit expressions of two peakons are given.Besides,we also derive smooth solitary wave solutions,periodic wave solutions,compacton solutions,and kink-like(antikink-like)solutions.Numerical simulations are further performed to verify the correctness of the results.Most importantly,peakons and periodic cusp wave solutions are newly found for the equation,which extends the previous results.展开更多
In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with ...In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods.展开更多
With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixe...With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixed types.Our primary contribution is the establishment of solution existence,illuminating the dynamics of these complex equations.To tackle this challenging problem,we construct an approximate solution sequence and apply the contraction mapping principle to rigorously prove local solution existence.Our results significantly advance the understanding of nonlinear evolution equations of mixed types.Furthermore,they provide a versatile,powerful approach for tackling analogous challenges across physics,engineering,and applied mathematics,making this work a valuable reference for researchers in these fields.展开更多
The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schrodinger and Klein-Gordon equations.These theories encompass both local and global well-posed...The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schrodinger and Klein-Gordon equations.These theories encompass both local and global well-posedness,as well as the existence of blowing-up solutions for large and irregular initial data.The main results presented in this paper can be summarized as follows:(1)Discrete Nonlinear Schrodinger Equation:Global well-posedness in l^(p) spaces for all1≤p≤∞,regardless of whether it is in the defocusing or focusing cases.(2)Discrete Klein-Gordon Equation:Local well-posedness in l^(p) spaces for all 1≤p≤∞.Furthermore,in the defocusing case,we establish global well-posedness in l^(p) spaces for any2≤p≤2σ+2(σ>0).In contrast,in the focusing case,we show that solutions with negative energy blow up within a finite time.These conclusions reveal the distinct dynamic behaviors exhibited by the solutions of the equations in discrete settings compared to their continuous setting.Additionally,they illuminate the significant role that discretization plays in preventing ill-posedness,and collapse for the nonlinear Schrodinger equation.展开更多
In this paper, the nonlinear Schr?dinger equation combining quadratic-cubic nonlinearity is considered, which can be represented by an approximate model of relatively dense quasi-one-dimensional Bose-Einstein condensa...In this paper, the nonlinear Schr?dinger equation combining quadratic-cubic nonlinearity is considered, which can be represented by an approximate model of relatively dense quasi-one-dimensional Bose-Einstein condensate. Based on the bifurcation theory, we proved the existence of solitary and periodic solutions. The methods we take are the trial equation method and the complete discrimination system for polynomial method. Therefore, we obtain the exact chirped solutions, which are more abundant in type and quantity than the existing results, so that the equation has more profound physical significance. These two methods are rigorously mathematical derivation and calculations, rather than based on certain conditional assumptions. In addition, we give some specific parameters to graphing the motion of the solutions, which helps to understand the propagation of nonlinear waves in fiber optic systems.展开更多
The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear...The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear method,we systematically construct single-and double-periodic lump solutions.To provide a detailed insight into the dynamic behavior of the nonlinear waves,we explore diverse mixed solutions,including bright-dark,W-shaped,multi-peak,and bright soliton solutions.Building upon single-periodic lump solutions,we analyze the dynamics of lump waves on both plane-wave and periodic backgrounds using the long-wave limit method.Moreover,we obtain the interaction solutions involving lumps,periodic lumps,and solitons.The interactions among two solitons,multiple lumps,and mixed waves are illustrated and analyzed.Comparative analysis reveals that these multi-lump solutions exhibit richer dynamical properties than conventional single-lump ones.These results contribute to a deeper understanding of nonlinear systems and may facilitate solving nonlinear problems in nature.展开更多
We are concerned with a Camassa-Holm type equation with higher-order nonlinearity including some integrable peakon models such as the Camassa-Holm equation,the Degasperis-Procesi equation,and the Novikov equation.We s...We are concerned with a Camassa-Holm type equation with higher-order nonlinearity including some integrable peakon models such as the Camassa-Holm equation,the Degasperis-Procesi equation,and the Novikov equation.We show that all the horizontal symmetric waves for this equation must be traveling waves.This extends the previous results for the Camassa-Holm and Novikov equations.展开更多
文摘In the article, the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established. Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space. The approach used is of a simple characteristic in gaining the stability condition of the scheme.
基金Supported by NSFC Grant(11401100,10601021)the foundation of Fujian Education Department(JB14021)the innovation foundation of Fujian Normal University(IRTL1206)
文摘A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.
文摘A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkin finite element method, which has been proved to be 2nd-order accurate in time and 4th-order in space. The comparison between the exact and numerical solutions of progressive waves shows that this numerical scheme is quite accurate, stable andefflcient. It is also shown that any local disturbance will spread, have a full growth and finally form two progressive waves propagating in both directions. The shape and the speed of the long term progressive waves are determined by the system itself, and do not depend on the details of the initial values.
基金supported by the NSFC(12271141)supported by the Fundamental Research Funds for the Central Universities(B240205026)the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX24_0821).
文摘In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof are the tails estimate and to demonstrate the Lipschitz continuity and random squeezing property of the solution for the equation defined on R^(N) .
基金Supported in part by Natural Science Foundation of Guangxi(2023GXNSFAA026246)in part by the Central Government's Guide to Local Science and Technology Development Fund(GuikeZY23055044)in part by the National Natural Science Foundation of China(62363003)。
文摘In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.
基金supported in part by NSF of China No.10571059E-Institutes of Shanghai Municipal Education Commission No.E03004+4 种基金Shanghai Priority Academic Discipline,and the Scientific Research Foundation for the Returned Overseas Chinese Scholars of the State Education MinistrySF of Shanghai No.04JC14062the fund of Chinese Education Ministry No.20040270002the Shanghai Leading Academic Discipline Project No.T0401the fund for E-Institutes of Shanghai Municipal Education Commission No.E03004 and the fund No.04DB15 of Shanghai Municipal Education Commission
文摘A monotone compact implicit finite difference scheme with fourth-order accuracy in space and second-order in time is proposed for solving nonlinear reaction-diffusion equations. An accelerated monotone iterative method for the resulting discrete problem is presented. The sequence of iteration converges monotonically to the unique solution of the discrete problem, and the convergence rate is either quadratic or nearly quadratic, depending on the property of the nonlinear reaction. The numerical results illustrate the high accuracy of the proposed scheme and the rapid convergence rate of.the iteration.
文摘In this paper, by introducing a proper transformation, the （Gr/G）-expansion method is further extended into the nonlinear reaction diffusion equations in mathematical biology whose balancing numbers may be negative integer. As a result, hyperbolic function solutions and trigonometric function solutions with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions and the periodic wave solutions are also derived from the traveling wave solutions. Moreover, it is observed that the suggested techniques is compatible of such problems.
文摘In this paper,we prove the existence of martingale solutions of a class of stochastic equations with a monotone drift of polynomial growth of arbitrary order and a continuous diffusion term with superlinear growth.Both the nonlinear drift and diffusion terms are not required to be locally Lipschitz continuous.We then apply the abstract result to establish the existence of martingale solutions of the fractional stochastic reaction-diffusion equation with polynomial drift driven by a superlinear noise.The pseudo-monotonicity techniques and the Skorokhod-Jakubowski representation theorem in a topological space are used to pass to the limit of a sequence of approximate solutions defined by the Galerkin method.
基金Supported by the National Natural Science Foundation of China under Grant No.11275072Research Fund for the Doctoral Program of Higher Education of China under Grant No.20120076110024+3 种基金Innovative Research Team Program of the National Natural Science Foundation of China under Grant No.61321064Shanghai Knowledge Service Platform Project under Grant No.ZF1213Shanghai Minhang District Talents of High Level Scientific Research ProjectTalent Fund and K.C.Wong Magna Fund in Ningbo University
文摘In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method,which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.
基金Supported by the National Natural Science Foundation of China Grant (No. 10771124)the Research Fund for Doctoral Program of High Education by State Education Ministry of China (No. 20060422006)+1 种基金the Program for Innovative Research Team in Ludong Universitythe Discipline Construction Fund of Ludong University
文摘Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O（△t ＋ h k＋1 ＋ H 2k＋2 d/2 ） （k ≥ 1）, where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.
文摘In this paper, we discuss the blowing\|up of the solutions of a class of nonlinear reaction\|diffusion equations with the general (or nonlinear) boundary conditions. On some proper assumptions, we conclude that there is no global smooth solution, i.e., the solutions blow up in the finite time.
基金This project is partly supported by the Natural Science Foundation of China and partly by StateEducation Committee.
文摘In this paper, the initial value problem of nonlinear reaction-diffusion equation is considered. The Dufort-Frankel finite difference approximation for the long time scheme is given for the d-dimensional reaction-diffusion equation with the two different cases. The global solution and global attractor are discussed for the Dufort-Frankel scheme. Moreover properties of the solution are studied. The error estimate is presented in a finite time region and in the global time region for some special cases. Finally the numerical results for the equation are investigated for Allen-Cahn equation and some other equations and the homoclinic orbit is simulated numerically.
基金the National Natural Science Foundation of China(No.10271034)the Basic Research Foundation of Harbin Engineering University(No.HEUF04012)
文摘The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method. The invarianee of some sets under the flow of these problems and the vacuum isolation of solutions are obtained by introducing a family of potential wells. Then the threshold result of global existence and nonexistence of solutions are given. Finally, the problem with critical initial conditions are discussed.
基金Supported by the National Natural Science Foundation of China(12071162)the Natural Science Foundation of Fujian Province(2021J01302)the Fundamental Research Funds for the Central Universities(ZQN-802).
文摘In this paper,we focus on peaked traveling wave solutions of the modified highly nonlinear Novikov equation by dynamical systems approach.We obtain a traveling wave system which is a singular planar dynamical system with three singular straight lines,and derive all possible phase portraits under corresponding parameter conditions.Then we show the existence and dynamics of two types of peaked traveling wave solutions including peakons and periodic cusp wave solutions.The exact explicit expressions of two peakons are given.Besides,we also derive smooth solitary wave solutions,periodic wave solutions,compacton solutions,and kink-like(antikink-like)solutions.Numerical simulations are further performed to verify the correctness of the results.Most importantly,peakons and periodic cusp wave solutions are newly found for the equation,which extends the previous results.
基金Supported by the National Natural Science Foundation of China(11671403,11671236,12101192)Henan Provincial General Natural Science Foundation Project(232300420113)。
文摘In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods.
基金Supported by the National Natural Science Foundation of China(12201368,62376252)Key Project of Natural Science Foundation of Zhejiang Province(LZ22F030003)Zhejiang Province Leading Geese Plan(2024C02G1123882,2024C01SA100795).
文摘With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixed types.Our primary contribution is the establishment of solution existence,illuminating the dynamics of these complex equations.To tackle this challenging problem,we construct an approximate solution sequence and apply the contraction mapping principle to rigorously prove local solution existence.Our results significantly advance the understanding of nonlinear evolution equations of mixed types.Furthermore,they provide a versatile,powerful approach for tackling analogous challenges across physics,engineering,and applied mathematics,making this work a valuable reference for researchers in these fields.
基金in part supported by the NSFC(12171356,12494544)supported by the National Key R&D Program of China(2020 YFA0713300)+1 种基金the NSFC(12531006)the Nankai Zhide Foundation。
文摘The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schrodinger and Klein-Gordon equations.These theories encompass both local and global well-posedness,as well as the existence of blowing-up solutions for large and irregular initial data.The main results presented in this paper can be summarized as follows:(1)Discrete Nonlinear Schrodinger Equation:Global well-posedness in l^(p) spaces for all1≤p≤∞,regardless of whether it is in the defocusing or focusing cases.(2)Discrete Klein-Gordon Equation:Local well-posedness in l^(p) spaces for all 1≤p≤∞.Furthermore,in the defocusing case,we establish global well-posedness in l^(p) spaces for any2≤p≤2σ+2(σ>0).In contrast,in the focusing case,we show that solutions with negative energy blow up within a finite time.These conclusions reveal the distinct dynamic behaviors exhibited by the solutions of the equations in discrete settings compared to their continuous setting.Additionally,they illuminate the significant role that discretization plays in preventing ill-posedness,and collapse for the nonlinear Schrodinger equation.
文摘In this paper, the nonlinear Schr?dinger equation combining quadratic-cubic nonlinearity is considered, which can be represented by an approximate model of relatively dense quasi-one-dimensional Bose-Einstein condensate. Based on the bifurcation theory, we proved the existence of solitary and periodic solutions. The methods we take are the trial equation method and the complete discrimination system for polynomial method. Therefore, we obtain the exact chirped solutions, which are more abundant in type and quantity than the existing results, so that the equation has more profound physical significance. These two methods are rigorously mathematical derivation and calculations, rather than based on certain conditional assumptions. In addition, we give some specific parameters to graphing the motion of the solutions, which helps to understand the propagation of nonlinear waves in fiber optic systems.
基金supported by the Applied Basic Research Program of Shanxi Province,China(Grant Nos.202403021212253 and 202203021221217).
文摘The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear method,we systematically construct single-and double-periodic lump solutions.To provide a detailed insight into the dynamic behavior of the nonlinear waves,we explore diverse mixed solutions,including bright-dark,W-shaped,multi-peak,and bright soliton solutions.Building upon single-periodic lump solutions,we analyze the dynamics of lump waves on both plane-wave and periodic backgrounds using the long-wave limit method.Moreover,we obtain the interaction solutions involving lumps,periodic lumps,and solitons.The interactions among two solitons,multiple lumps,and mixed waves are illustrated and analyzed.Comparative analysis reveals that these multi-lump solutions exhibit richer dynamical properties than conventional single-lump ones.These results contribute to a deeper understanding of nonlinear systems and may facilitate solving nonlinear problems in nature.
基金partially supported by the National Natural Science Foundation of China(Grant No.12201417)the Project funded by the China Postdoctoral Science Foundation(Grant No.2023M733173)partially supported by the National Natural Science Foundation of China(Grant No.12375006)。
文摘We are concerned with a Camassa-Holm type equation with higher-order nonlinearity including some integrable peakon models such as the Camassa-Holm equation,the Degasperis-Procesi equation,and the Novikov equation.We show that all the horizontal symmetric waves for this equation must be traveling waves.This extends the previous results for the Camassa-Holm and Novikov equations.
