In this paper, some exact solutions of the (3 + 1)-dimensional variable-coefficient Yu-Toda-Sasa-Fukuyama equation are investigated. By using Hirota’s direct method and symbolic computation, we obtained N-soliton sol...In this paper, some exact solutions of the (3 + 1)-dimensional variable-coefficient Yu-Toda-Sasa-Fukuyama equation are investigated. By using Hirota’s direct method and symbolic computation, we obtained N-soliton solution. By using the long wave limit method, the N-order rational solution can be obtained from N-order soliton solution. Then, through the paired complexification of parameters, the lump solution is obtained from N-order rational solution. Meanwhile, we obtained a hybrid solution between 1-lump solution and N-soliton (N=1,2) by using the long wave limit method and parameter complex. Furthermore, four different sets of three-dimensional graphs of solitons, lump solutions and hybrid solutions are drawn by selecting four different sets of coefficient functions which include one set of constant coefficient function and three sets of variable coefficient functions.展开更多
The mesoscale eddy and internal wave both are phenomena commonly observed in oceans. It is aimed to investigate how the presence of a mesoscale eddy in the ocean affects wave form deformation of the internal solitary ...The mesoscale eddy and internal wave both are phenomena commonly observed in oceans. It is aimed to investigate how the presence of a mesoscale eddy in the ocean affects wave form deformation of the internal solitary wave propagation. An ocean eddy is produced by a quasi-geostrophic model in f-plane, and the one-dimensional nonlinear variable-coefficient extended Korteweg-de Vries （eKdV） equation is used to simulate an internal solitary wave passing through the mesoscale eddy field. The results suggest that the mode structures of the linear internal wave are modified due to the presence of the mesoscale eddy field. A cyclonic eddy and an anticyclonic eddy have different influences on the background environment of the internal solitary wave propagation. The existence of a mesoscale eddy field has almost no prominent impact on the propagation of a smallamplitude internal solitary wave only based on the first mode vertical structure, but the mesoscale eddy background field exerts a considerable influence on the solitary wave propagation if considering high-mode vertical structures. Furthermore, whether an internal solitary wave first passes through anticyclonic eddy or cyclonic eddy, the deformation of wave profiles is different. Many observations of solitary internal waves in the real oceans suggest the formation of the waves. Apart from topography effect, it is shown that the mesoscale eddy background field is also a considerable factor which influences the internal solitary wave propagation and deformation.展开更多
In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We fin...In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.展开更多
Recent theoretical physics efforts have been focused on the probes for nonlinear pulse waves in,for example,variable-radius arteries.With respect to the nonlinear waves in an artery full of blood with certain aneurysm...Recent theoretical physics efforts have been focused on the probes for nonlinear pulse waves in,for example,variable-radius arteries.With respect to the nonlinear waves in an artery full of blood with certain aneurysm,pulses in a blood vessel,or features in a circulatory system,this paper symbolically computes out an auto-B?cklund transformation via a noncharacteristic movable singular manifold,certain families of the solitonic solutions,as well as a family of the similarity reductions for a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers equation.Aiming,e.g.,at the dynamical radial displacement superimposed on the original static deformation from an arterial wall,our results rely on the axial stretch of the injured artery,blood as an incompressible Newtonian fluid,radius variation along the axial direction or aneurysmal geometry,viscosity of the fluid,thickness of the artery,mass density of the membrane material,mass density of the fluid,strain energy density of the artery,shear modulus,stretch ratio,etc.We also highlight that the shock-wave structures from our solutions agree well with those dusty-plasma-experimentally reported.展开更多
This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili （KP） equation. By using the Hirota method and Pfaffian technique, it...This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili （KP） equation. By using the Hirota method and Pfaffian technique, it finds the Grammian determinant-type solution for the variable-coefficient KP equation （VCKP）, the Wronski-type Pfaffian solution and the Gram-type Pfaffian solutions for the Pfaffianized VCKP equation.展开更多
Based on the generalized dressing method, we propose integrable variable coefficient coupled cylin-drical nonlinear SchrSdinger equations and their Lax pairs. As applications, their explicit solutions and their reduct...Based on the generalized dressing method, we propose integrable variable coefficient coupled cylin-drical nonlinear SchrSdinger equations and their Lax pairs. As applications, their explicit solutions and their reductions are constructed.展开更多
A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, th...A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.展开更多
In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann the...In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.展开更多
The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, t...The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, the nonsingular positon solutions of the variable-coefficient modified KdV equation are firstly discovered analytically and graphically.展开更多
The extended symmetry approach is used to study the general Korteweg-de Vries-type （KdV-type） equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be construc...The extended symmetry approach is used to study the general Korteweg-de Vries-type （KdV-type） equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation.展开更多
In this paper, we put our focus on a variable-coe^cient fifth-order Korteweg-de Vries （fKdV） equation, which possesses a great number of excellent properties and is of current importance in physical and engineering ...In this paper, we put our focus on a variable-coe^cient fifth-order Korteweg-de Vries （fKdV） equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out.展开更多
In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an ex...In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an exact solution of this equation through the Wronskian technique. In addition, we testify that this equation can be reduced to a Jacobi identity by considering its solution as a Grammian determinant by means of Pfaffian derivative formulae.展开更多
文摘In this paper, some exact solutions of the (3 + 1)-dimensional variable-coefficient Yu-Toda-Sasa-Fukuyama equation are investigated. By using Hirota’s direct method and symbolic computation, we obtained N-soliton solution. By using the long wave limit method, the N-order rational solution can be obtained from N-order soliton solution. Then, through the paired complexification of parameters, the lump solution is obtained from N-order rational solution. Meanwhile, we obtained a hybrid solution between 1-lump solution and N-soliton (N=1,2) by using the long wave limit method and parameter complex. Furthermore, four different sets of three-dimensional graphs of solitons, lump solutions and hybrid solutions are drawn by selecting four different sets of coefficient functions which include one set of constant coefficient function and three sets of variable coefficient functions.
基金The National Basic Research Program of China under contract Nos 2011CB403503 and 2012CB955601the Scientific Research Fund of the Second Institute of Oceanography, the State Oceanic Administration of China under contract Nos JG1009, JT1006 and JT0905
文摘The mesoscale eddy and internal wave both are phenomena commonly observed in oceans. It is aimed to investigate how the presence of a mesoscale eddy in the ocean affects wave form deformation of the internal solitary wave propagation. An ocean eddy is produced by a quasi-geostrophic model in f-plane, and the one-dimensional nonlinear variable-coefficient extended Korteweg-de Vries （eKdV） equation is used to simulate an internal solitary wave passing through the mesoscale eddy field. The results suggest that the mode structures of the linear internal wave are modified due to the presence of the mesoscale eddy field. A cyclonic eddy and an anticyclonic eddy have different influences on the background environment of the internal solitary wave propagation. The existence of a mesoscale eddy field has almost no prominent impact on the propagation of a smallamplitude internal solitary wave only based on the first mode vertical structure, but the mesoscale eddy background field exerts a considerable influence on the solitary wave propagation if considering high-mode vertical structures. Furthermore, whether an internal solitary wave first passes through anticyclonic eddy or cyclonic eddy, the deformation of wave profiles is different. Many observations of solitary internal waves in the real oceans suggest the formation of the waves. Apart from topography effect, it is shown that the mesoscale eddy background field is also a considerable factor which influences the internal solitary wave propagation and deformation.
基金the National Natural Science Foundation of China under Grant No.11772017the Fundamental Research Funds for the Central Universities
文摘In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.
基金supported by the National Natural Science Foundation of China under Grant Nos.11871116 and 11772017the Fundamental Research Funds for the Central Universities of China under Grant No.2019XD-A11.
文摘Recent theoretical physics efforts have been focused on the probes for nonlinear pulse waves in,for example,variable-radius arteries.With respect to the nonlinear waves in an artery full of blood with certain aneurysm,pulses in a blood vessel,or features in a circulatory system,this paper symbolically computes out an auto-B?cklund transformation via a noncharacteristic movable singular manifold,certain families of the solitonic solutions,as well as a family of the similarity reductions for a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers equation.Aiming,e.g.,at the dynamical radial displacement superimposed on the original static deformation from an arterial wall,our results rely on the axial stretch of the injured artery,blood as an incompressible Newtonian fluid,radius variation along the axial direction or aneurysmal geometry,viscosity of the fluid,thickness of the artery,mass density of the membrane material,mass density of the fluid,strain energy density of the artery,shear modulus,stretch ratio,etc.We also highlight that the shock-wave structures from our solutions agree well with those dusty-plasma-experimentally reported.
基金Project supported by the National Key Basic Research Project of China （2004CB318000）, the National Science Foundation of China （Grant No 10371023） and Shanghai Shuguang Project of China （Grant No 02SG02）.
文摘This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili （KP） equation. By using the Hirota method and Pfaffian technique, it finds the Grammian determinant-type solution for the variable-coefficient KP equation （VCKP）, the Wronski-type Pfaffian solution and the Gram-type Pfaffian solutions for the Pfaffianized VCKP equation.
基金Supported by a grant from City University of Hong Kong(Project No:7002366)the support by National Natural Science Foundation of China(Project No:11301149)+1 种基金Henan Natural Science Foundation For Basic Research under Grant No:132300410310Doctor Foundation of Henan Institute of Engeering under Grant No:D2010007
文摘Based on the generalized dressing method, we propose integrable variable coefficient coupled cylin-drical nonlinear SchrSdinger equations and their Lax pairs. As applications, their explicit solutions and their reductions are constructed.
文摘A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10771196 and 10831003)the Innovation Project of Zhejiang Province of China(Grant No.T200905)
文摘In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.
文摘The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, the nonsingular positon solutions of the variable-coefficient modified KdV equation are firstly discovered analytically and graphically.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10675065)the Scientific Research Fundof the Education Department of Zhejiang Province of China (Grant No. 20070979)
文摘The extended symmetry approach is used to study the general Korteweg-de Vries-type （KdV-type） equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation.
基金The project supported by the Key Project of the Chinese Ministry of Education under Grant No.106033the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20060006024+2 种基金Chinese Ministry of Education,the National Natural Science Foundation of China under Grant Nos.60772023 and 60372095the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE-07-001Beijing University of Aeronautics and Astronautics,and by the National Basic Research Program of China(973 Program)under Grant No.2005CB321901
文摘In this paper, we put our focus on a variable-coe^cient fifth-order Korteweg-de Vries （fKdV） equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out.
基金The project supported by the Key Project of the Ministry of Education under Grant No.106033the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20060006024+2 种基金National Natural Science Foundation of China under Grant Nos.60372095 and 60772023the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE07-001Beijing University of Aeronautics and Astronautics,and the National Basic Research Program of China(973 Program)under Grant No.2005CB321901
文摘In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an exact solution of this equation through the Wronskian technique. In addition, we testify that this equation can be reduced to a Jacobi identity by considering its solution as a Grammian determinant by means of Pfaffian derivative formulae.
