We presents a generalized(2+1)-dimensional Sharma-Tasso-Olver-Burgers(STOB)equation,unifying dissipative and dispersive wave dynamics.By introducing an auxiliary potential𝑦as a new space variable and employing...We presents a generalized(2+1)-dimensional Sharma-Tasso-Olver-Burgers(STOB)equation,unifying dissipative and dispersive wave dynamics.By introducing an auxiliary potential𝑦as a new space variable and employing a simpler deformation algorithm,we deform the(1+1)-dimensional STOB model to higher dimensions.The resulting equation is proven Lax-integrable via introducing strong and weak Lax pairs.Traveling wave solutions of the(2+1)-dimensional STOB equation are derived through an ordinary differential equation reduction,with implicit solutions obtained for a special case.Crucially,we demonstrate that the system admits dispersionless decompositions into two types:Case 1 yields non-traveling twisted kink and bell solitons,while Case 2 involves complex implicit functions governed by cubic-algebraic constraints.Numerical visualizations reveal novel anisotropic soliton structures,and the decomposition methodology is shown to generalize broadly to other higher dimensional dispersionless decomposition solvable integrable systems.展开更多
In this paper, the dispersionless D-type Drinfeld–Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. Thes...In this paper, the dispersionless D-type Drinfeld–Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. These flows form an infinite-dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type.展开更多
By using the Jacobi elliptic-function method, this paper obtains the periodic solutions for coupled integrable dispersionless equations. The periodic solutions include some kink and anti-kink solitons.
The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integ...The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Biicklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new （2 ＋ 1）-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.展开更多
A new coupled integrable dispersionless equation is presented by considering a spectral problem. A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of sp...A new coupled integrable dispersionless equation is presented by considering a spectral problem. A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of spectral problems. As an application, the N-soliton solution of the coupled integrable dispersionless equation is explicitly given.展开更多
Under investigation in this paper are two coupled integrable dispersionless (CID) equations modelingthe dynamics of the current-fed string within an external magnetic field.Through a set of the dependent variabletrans...Under investigation in this paper are two coupled integrable dispersionless (CID) equations modelingthe dynamics of the current-fed string within an external magnetic field.Through a set of the dependent variabletransformations, the bilinear forms for the CID equations are derived.Based on the Hirota method and symboliccomputation, the analytic N-soliton solutions are presented.Infinitely many conservation laws for the CID equationsare given through the known spectral problem.Propagation characteristics and interaction behaviors of the solitons areanalyzed graphically.展开更多
Using improved homogeneous balance method, we obtain new exact solutions for the coupled integrable dispersionless equation. On the basis of these exact solutions, we find some new interesting coherent structures by s...Using improved homogeneous balance method, we obtain new exact solutions for the coupled integrable dispersionless equation. On the basis of these exact solutions, we find some new interesting coherent structures by selecting arbitrary functions appropriately.展开更多
We present a matrix coupled dispersionless(CD)system.A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair.We ex...We present a matrix coupled dispersionless(CD)system.A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair.We express an N soliton formula for the solutions of the matrix CD system in terms of quasideterminants.By using properties of the quasideterminants,we obtain some exact solutions,including bright and dark-type solitons,rogue wave and breather solutions of the matrix CD system.Furthermore,it has been shown that the solutions of the matrix CD system are expressed in terms of solutions to the usual CD system,sine-Gordon equation and Maxwell-Bloch system.展开更多
This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equatio...This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more general variable separation solutions with two arbitrary functions of （1＋1）-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p（x） and q（t） in our solutions, the annihilation phenomena of the fiat-basin soliton, arch-basin soliton, and fiat-top soliton are discussed.展开更多
Based on the corresponding theorem between dispersionless KP （dKP） hierarchy and h-dependent KP （hKP） hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is gi...Based on the corresponding theorem between dispersionless KP （dKP） hierarchy and h-dependent KP （hKP） hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding hKP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.展开更多
In this paper, we investigate the Rotating N Loop-Soliton solution of the coupled integrable dispersionless equation (CIDE) that describes a current-fed string within an external magnetic field in 2D-space. Through a ...In this paper, we investigate the Rotating N Loop-Soliton solution of the coupled integrable dispersionless equation (CIDE) that describes a current-fed string within an external magnetic field in 2D-space. Through a set of independent variable transformation, we derive the bilinear form of the CIDE Equation. Based on the Hirota’s method, Perturbation technique and Symbolic computation, we present the analytic N-rotating loop soliton solution and proceed to some illustrations by presenting the cases of three- and four-soliton solutions.展开更多
In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard H...In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions axe derived.展开更多
The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation i...The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation （in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data）, or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.展开更多
We apply the （G＇/G）-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational...We apply the （G＇/G）-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. We highlight the power of the （G＇/G）-expansion method in providing generalized solitary wave solutions of different physical structures. It is shown that the （G＇/G）-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.展开更多
The algorithm for constructing conservation laws of Euler Lagvange type equations via Noether-type symmetry operators associated with partial Lagrangian has been presented. As applications, many new conservation laws ...The algorithm for constructing conservation laws of Euler Lagvange type equations via Noether-type symmetry operators associated with partial Lagrangian has been presented. As applications, many new conservation laws of some important systems of nonlinear partial differential equations have been obtained.展开更多
基金supported by the National Natural Science Foundations of China(Grant Nos.12235007,12375003,and 11975131).
文摘We presents a generalized(2+1)-dimensional Sharma-Tasso-Olver-Burgers(STOB)equation,unifying dissipative and dispersive wave dynamics.By introducing an auxiliary potential𝑦as a new space variable and employing a simpler deformation algorithm,we deform the(1+1)-dimensional STOB model to higher dimensions.The resulting equation is proven Lax-integrable via introducing strong and weak Lax pairs.Traveling wave solutions of the(2+1)-dimensional STOB equation are derived through an ordinary differential equation reduction,with implicit solutions obtained for a special case.Crucially,we demonstrate that the system admits dispersionless decompositions into two types:Case 1 yields non-traveling twisted kink and bell solitons,while Case 2 involves complex implicit functions governed by cubic-algebraic constraints.Numerical visualizations reveal novel anisotropic soliton structures,and the decomposition methodology is shown to generalize broadly to other higher dimensional dispersionless decomposition solvable integrable systems.
基金Supported by the National Natural Science Foundation of China under Grant No.11201251the National Natural Science Foundation of China under Grant No.11271210+5 种基金Zhejiang Provincial Natural Science Foundation under Grant No.LY12A01007the Natural Science Foundation of Ningbo under Grant No.2013A610105K.C.Wong Magna Fund in Ningbo Universitythe National Science Foundation of China under Grant No.11371278the Shanghai Municipal Science and Technology Commission under Grant No.12XD1405000the Fundamental Research Funds for the Central Universities of China
文摘In this paper, the dispersionless D-type Drinfeld–Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. These flows form an infinite-dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type.
基金supported by the National Natural Science Foundation of China (Grant Nos.40975028 and 40805022)
文摘By using the Jacobi elliptic-function method, this paper obtains the periodic solutions for coupled integrable dispersionless equations. The periodic solutions include some kink and anti-kink solitons.
文摘The prolongation structure methodologies of Wahlquist-Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 （1975） 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Biicklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new （2 ＋ 1）-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.
基金supported by the National Key Basic Research Project of China(Grant No 2004CB318000)the National Natural Science Foundation of China(Grant No 10771072)+2 种基金the Research Fund far the Doctoral Program of Higher Education of China(Grant No 20060269006)the PhD Program Scholarship Fund of East China Normal University(ECNU)2008,China(Grant No 20080052)the Natural Science Foundation of Inner Mongolia Normal University,China(Grant No ZRYB08017)
文摘A new coupled integrable dispersionless equation is presented by considering a spectral problem. A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of spectral problems. As an application, the N-soliton solution of the coupled integrable dispersionless equation is explicitly given.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023the Open Fund No.BUAA-SKLSDE-09KF-04+2 种基金Supported Project No.SKLSDE-2010ZX-07 of the State Key Laboratory of Software Development Environment,Beijing University of Aeronautics and Astronauticsthe National Basic Research Program of China (973 Program) under Grant No.2005CB321901 the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.200800130006,Chinese Ministry of Education
文摘Under investigation in this paper are two coupled integrable dispersionless (CID) equations modelingthe dynamics of the current-fed string within an external magnetic field.Through a set of the dependent variabletransformations, the bilinear forms for the CID equations are derived.Based on the Hirota method and symboliccomputation, the analytic N-soliton solutions are presented.Infinitely many conservation laws for the CID equationsare given through the known spectral problem.Propagation characteristics and interaction behaviors of the solitons areanalyzed graphically.
基金Supported by Colleges and Universities Scientific Research Foundation of Inner Mongolia Autonomous Region under Grant N0. NJZY07139Natural Science Foundation of Inner Mongolia Autonomous Region under Grant No. 200408020113
文摘Using improved homogeneous balance method, we obtain new exact solutions for the coupled integrable dispersionless equation. On the basis of these exact solutions, we find some new interesting coherent structures by selecting arbitrary functions appropriately.
基金the National Natural Science Foundation of China(Grant Nos.11871471,11331008 and 11931017)Foreign Experts Scientific Cooperation Fund。
文摘We present a matrix coupled dispersionless(CD)system.A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair.We express an N soliton formula for the solutions of the matrix CD system in terms of quasideterminants.By using properties of the quasideterminants,we obtain some exact solutions,including bright and dark-type solitons,rogue wave and breather solutions of the matrix CD system.Furthermore,it has been shown that the solutions of the matrix CD system are expressed in terms of solutions to the usual CD system,sine-Gordon equation and Maxwell-Bloch system.
基金Supported by the National Natural Science Foundation of China under Grant No.11005092the Program for Innovative Research Team of Young Teachers under Grant No.2009RC01Scientific Research,and Developed Fund under Grant No.2009FK42 of Zhejiang A&F University
文摘This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more general variable separation solutions with two arbitrary functions of （1＋1）-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p（x） and q（t） in our solutions, the annihilation phenomena of the fiat-basin soliton, arch-basin soliton, and fiat-top soliton are discussed.
基金Supported by the Natural Science Foundation of China under Grant No.10971109
文摘Based on the corresponding theorem between dispersionless KP （dKP） hierarchy and h-dependent KP （hKP） hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding hKP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.
文摘In this paper, we investigate the Rotating N Loop-Soliton solution of the coupled integrable dispersionless equation (CIDE) that describes a current-fed string within an external magnetic field in 2D-space. Through a set of independent variable transformation, we derive the bilinear form of the CIDE Equation. Based on the Hirota’s method, Perturbation technique and Symbolic computation, we present the analytic N-rotating loop soliton solution and proceed to some illustrations by presenting the cases of three- and four-soliton solutions.
基金National Natural Science Foundation of China under Grant No.10726063
文摘In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions axe derived.
基金supported by the National Science Foundation under grant DMS-0807653
文摘The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation （in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data）, or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.
基金Project supported by the National Natural Science Foundations of China(11001250,11331008)the Foundation for Young Teachers in Colleges and Universities of Henan Province(2013GGJS-010)the Soft Science Foundation of Science and Technology Department of Henan Province(142400410274)
基金Project supported by the Scientific Research Project of Eskisehir Osmangazi University, Turkey (Grant No. 201019031)
文摘We apply the （G＇/G）-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. We highlight the power of the （G＇/G）-expansion method in providing generalized solitary wave solutions of different physical structures. It is shown that the （G＇/G）-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.
基金Supported by the Natural Science Foundation of Inner Mongolia(Grant No 2009 MS0108)the High Education Science Research of Inner Mongolia Autonomous Region(Grant No.NJ10045)~~
基金supported by the State Key Basic Research Program of China under Grant No.2004CB318000
文摘The algorithm for constructing conservation laws of Euler Lagvange type equations via Noether-type symmetry operators associated with partial Lagrangian has been presented. As applications, many new conservation laws of some important systems of nonlinear partial differential equations have been obtained.
