This paper studies the coupled Burgers equation and the high-order Boussinesq-Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton)...This paper studies the coupled Burgers equation and the high-order Boussinesq-Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.展开更多
Based on the Hirota bilinear and long wave limit methods,the hybrid solutions of m-lump with n-soliton and nbreather wave for generalized Hirota–Satsuma–Ito(GHSI)equation are constructed.Then,by approximating soluti...Based on the Hirota bilinear and long wave limit methods,the hybrid solutions of m-lump with n-soliton and nbreather wave for generalized Hirota–Satsuma–Ito(GHSI)equation are constructed.Then,by approximating solutions of the GHSI equation along some parallel orbits at infinity,the trajectory equation of a lump wave before and after collisions with n-soliton and n-breather wave are studied,and the expressions of phase shift for lump wave before and after collisions are given.Furthermore,it is revealed that collisions between the lump wave and other waves are elastic,the corresponding collision diagrams are used to further explain.展开更多
This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé...This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.展开更多
In this work,using the Hirota bilinear method,N-soliton solution is obtained for Hirota-Satsuma nonlinear evolution equation:u_t - u_(xxt) - 3u_xu_t + u_x = 0.
The mixed AKNS nonlinear evolution equation in equation, which contains an isospectral term the AKNS system. So searching for its exact and a nonisospectral term, is an important solutions is vital both for the AKNS s...The mixed AKNS nonlinear evolution equation in equation, which contains an isospectral term the AKNS system. So searching for its exact and a nonisospectral term, is an important solutions is vital both for the AKNS system and in mathematical sense. In this paper, the corresponding Lax pair was given, the bilinear forms of the mixed AKNS equation were obtained through introducing the transformation of dependent variables. By using Hirota's bilinear method, the N-soliton solutions were obtained.展开更多
Instead of the usual Hirota ansatz,i.e.,the functions in bilinear equations being chosen as exponentialtypes,a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation.Based on theres...Instead of the usual Hirota ansatz,i.e.,the functions in bilinear equations being chosen as exponentialtypes,a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation.Based on theresulting generalized Hirota ansatz,a family of new explicit solutions for the equation are derived.展开更多
We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive qua...We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive quadratic functions,the exponential and hyperbolic functions.According to the known lump solution in the outset,we obtained the superposition formulas of positive quadratic functions by plausible reasoning.Next,we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory.These two kinds of solutions contained superposition formulas of positive quadratic functions,which were turned into general ternary quadratic functions,the coefficients of which were all rational operation of vector inner product.Then we obtained linear superposition formulas of exponential and hyperbolic function solutions.Finally,for aforementioned various solutions,their dynamic properties were showed by choosing specific values for parameters.From concrete plots,we observed wave characteristics of three kinds of solutions.Especially,we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.展开更多
We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equ...We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equation,we obtain the Manakov model of all-focusing,all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions.Especially,we find that the all-defocusing type Manakov model admits bright-bright soliton solutions.Furthermore,we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions.Therefrom,we go over how the free parameters affect the Manakov model’s allfocusing type’s two-soliton and three-soliton solutions’collision locations,propagation directions,and wave amplitudes.These findings are useful for setting a simulation scene in Rossby waves research.The answers we have found are helpful for studying physical properties of the equation in Rossby waves.展开更多
Based on the Hirota bilinear method,the second extended(3+1)-dimensional Jimbo–Miwa equation is established.By Maple symbolic calculation,lump and lump-kink soliton solutions are obtained.The interaction solutions be...Based on the Hirota bilinear method,the second extended(3+1)-dimensional Jimbo–Miwa equation is established.By Maple symbolic calculation,lump and lump-kink soliton solutions are obtained.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and triangular periodic soliton are derived by combining a multi-exponential function or trigonometric sine and cosine functions with quadratic functions.Furthermore,periodiclump wave solution is derived via the ansatz including hyperbolic and trigonometric functions.Finally,3D plots,2D curves,density plots,and contour plots with particular choices of the suitable parameters are depicted to illustrate the dynamical features of these solutions.展开更多
Based on the direct method of calculating the periodic wave solution proposed by Nakamura,we give an approximate analytical three-periodic solutions of Korteweg-de Vries(KdV)-type equations by perturbation method for ...Based on the direct method of calculating the periodic wave solution proposed by Nakamura,we give an approximate analytical three-periodic solutions of Korteweg-de Vries(KdV)-type equations by perturbation method for the first time.Limit methods have been used to establish the asymptotic relationships between the three-periodic solution separately and another three solutions,the soliton solution,the one-and the two-periodic solutions.Furthermore,it is found that the asymptotic three-soliton solution presents the same repulsive phenomenon as the asymptotic three-soliton solution during the interaction.展开更多
We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota b...We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10871117 and 10571110)
文摘This paper studies the coupled Burgers equation and the high-order Boussinesq-Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12001424 and 12271324)the Natural Science Basic Research Program of Shaanxi Province,China(Grant No.2021JZ-21)+1 种基金the Chinese Post Doctoral Science Foundation(Grant No.2020M673332)the Three-year Action Plan Project of Xi’an University(Grant No.2021XDJH01)。
文摘Based on the Hirota bilinear and long wave limit methods,the hybrid solutions of m-lump with n-soliton and nbreather wave for generalized Hirota–Satsuma–Ito(GHSI)equation are constructed.Then,by approximating solutions of the GHSI equation along some parallel orbits at infinity,the trajectory equation of a lump wave before and after collisions with n-soliton and n-breather wave are studied,and the expressions of phase shift for lump wave before and after collisions are given.Furthermore,it is revealed that collisions between the lump wave and other waves are elastic,the corresponding collision diagrams are used to further explain.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11505090)Research Award Foundation for Outstanding Young Scientists of Shandong Province(Grant No.BS2015SF009)+2 种基金the Doctoral Foundation of Liaocheng University(Grant No.318051413)Liaocheng University Level Science and Technology Research Fund(Grant No.318012018)Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology(Grant No.319462208).
文摘This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.
基金Foundation item: Supported by the Natural Science Foundation of China(61072147, 11071159) Supported by the Shanghai Leading Academic Discipline Project(J50101) Supported by the Youth Foundation of Zhoukou Normal University(zknuqn200917)
文摘In this work,using the Hirota bilinear method,N-soliton solution is obtained for Hirota-Satsuma nonlinear evolution equation:u_t - u_(xxt) - 3u_xu_t + u_x = 0.
基金Project supported by the National Natural Science Foundation of China (Grant No.40175014)
文摘The mixed AKNS nonlinear evolution equation in equation, which contains an isospectral term the AKNS system. So searching for its exact and a nonisospectral term, is an important solutions is vital both for the AKNS system and in mathematical sense. In this paper, the corresponding Lax pair was given, the bilinear forms of the mixed AKNS equation were obtained through introducing the transformation of dependent variables. By using Hirota's bilinear method, the N-soliton solutions were obtained.
文摘Instead of the usual Hirota ansatz,i.e.,the functions in bilinear equations being chosen as exponentialtypes,a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation.Based on theresulting generalized Hirota ansatz,a family of new explicit solutions for the equation are derived.
基金the National Natural Science Foundation of China(Grant No.12061054)Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region of China(Grant No.NJYT-20A06)。
文摘We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive quadratic functions,the exponential and hyperbolic functions.According to the known lump solution in the outset,we obtained the superposition formulas of positive quadratic functions by plausible reasoning.Next,we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory.These two kinds of solutions contained superposition formulas of positive quadratic functions,which were turned into general ternary quadratic functions,the coefficients of which were all rational operation of vector inner product.Then we obtained linear superposition formulas of exponential and hyperbolic function solutions.Finally,for aforementioned various solutions,their dynamic properties were showed by choosing specific values for parameters.From concrete plots,we observed wave characteristics of three kinds of solutions.Especially,we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.12102205 and 12161065)the Scientific Research Ability of Youth Teachers of Inner Mongolia Agricultural University(Grant Nos.JC2021001 and BR220126)+1 种基金the Natural Science Foundation of Inner Mongolia Autonomous Region of China(Grant No.2022QN01003)the Research Program of Inner Mongolia Autonomous Region Education Department(Grant Nos.NJYT23099 and NMGIRT2208).
文摘We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equation,we obtain the Manakov model of all-focusing,all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions.Especially,we find that the all-defocusing type Manakov model admits bright-bright soliton solutions.Furthermore,we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions.Therefrom,we go over how the free parameters affect the Manakov model’s allfocusing type’s two-soliton and three-soliton solutions’collision locations,propagation directions,and wave amplitudes.These findings are useful for setting a simulation scene in Rossby waves research.The answers we have found are helpful for studying physical properties of the equation in Rossby waves.
文摘Based on the Hirota bilinear method,the second extended(3+1)-dimensional Jimbo–Miwa equation is established.By Maple symbolic calculation,lump and lump-kink soliton solutions are obtained.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and triangular periodic soliton are derived by combining a multi-exponential function or trigonometric sine and cosine functions with quadratic functions.Furthermore,periodiclump wave solution is derived via the ansatz including hyperbolic and trigonometric functions.Finally,3D plots,2D curves,density plots,and contour plots with particular choices of the suitable parameters are depicted to illustrate the dynamical features of these solutions.
基金the National National Science Foundation of China(Grant Nos.52171251,U2106225,and 52231011)the Science and Technology Innovation Fund of Dalian City(Grant No.2022JJ12GX036)。
文摘Based on the direct method of calculating the periodic wave solution proposed by Nakamura,we give an approximate analytical three-periodic solutions of Korteweg-de Vries(KdV)-type equations by perturbation method for the first time.Limit methods have been used to establish the asymptotic relationships between the three-periodic solution separately and another three solutions,the soliton solution,the one-and the two-periodic solutions.Furthermore,it is found that the asymptotic three-soliton solution presents the same repulsive phenomenon as the asymptotic three-soliton solution during the interaction.
基金supported by the National Natural Science Foundation of China (Grant Nos.11975204 and 12075208)the Project of Zhoushan City Science and Technology Bureau (Grant No.2021C21015)the Training Program for Leading Talents in Universities of Zhejiang Province。
文摘We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.