We have found two types of important exact solutions, compacton solutions, which are solitary waves with the property that after colliding with their own kind, they re-emerge with the same coherent shape very much as ...We have found two types of important exact solutions, compacton solutions, which are solitary waves with the property that after colliding with their own kind, they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction, in the and even models, and dromion solutions (exponentially decaying solutions in all direction) in many and models. In this paper, symmetry reductions in are considered for the break soliton-type equation with fully nonlinear dispersion (called equation) , which is a generalized model of break soliton equation , by using the extended direct reduction method. As a result, six types of symmetry reductions are obtained. Starting from the reduction equations and some simple transformations, we obtain the solitary wave solutions of equations, compacton solutions of equations and the compacton-like solution of the potential form (called ) . In addition, we show that the variable admits dromion solutions rather than the field itself in equation.展开更多
In this paper, we establish exact solutions for the .R（m,n） equations by using an sn-cn metnou,As a result, abundant new cornpactons, i,e, solitons with the absence of infinite wings, new type of Jacobi elliptic fun...In this paper, we establish exact solutions for the .R（m,n） equations by using an sn-cn metnou,As a result, abundant new cornpactons, i,e, solitons with the absence of infinite wings, new type of Jacobi elliptic function, solitary wave and periodic wave solutions, of this equation are obtained with minimal calculations. The properties of the R（m, n） equations are shown in figures.展开更多
Two Darboux transformations of the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka(CDGKS)equation and(2+1)-dimensional modified Korteweg-de Vries(mKdV) equation are constructed through the Darboux matrix method...Two Darboux transformations of the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka(CDGKS)equation and(2+1)-dimensional modified Korteweg-de Vries(mKdV) equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux transformations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the(2+1)-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the(2+1)-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.展开更多
Utilizing the Wronskian technique, a combined Wronskian condition is established for a （3＋1）-dimensional generalized KP equation. The generating functions for matrix entries satisfy a linear system of new partial d...Utilizing the Wronskian technique, a combined Wronskian condition is established for a （3＋1）-dimensional generalized KP equation. The generating functions for matrix entries satisfy a linear system of new partial differential equations. Moreover, as applications, examples of Wronskian determinant solutions, including N-soliton solutions, periodic solutions and rational solutions, are computed.展开更多
Starting from a simple transformation, and with the aid of symbolic computation, we establish the relationship between the solution of a generalized variable coefficient Kadomtsev–Petviashvili(vKP) equation and the s...Starting from a simple transformation, and with the aid of symbolic computation, we establish the relationship between the solution of a generalized variable coefficient Kadomtsev–Petviashvili(vKP) equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover,we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.展开更多
In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the d...In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the defining function of δΩ.Imposing suitable conditions on the coefficients and f（x,u）,one can get the L^∞-estimates of（＊）via blow up method.展开更多
文摘We have found two types of important exact solutions, compacton solutions, which are solitary waves with the property that after colliding with their own kind, they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction, in the and even models, and dromion solutions (exponentially decaying solutions in all direction) in many and models. In this paper, symmetry reductions in are considered for the break soliton-type equation with fully nonlinear dispersion (called equation) , which is a generalized model of break soliton equation , by using the extended direct reduction method. As a result, six types of symmetry reductions are obtained. Starting from the reduction equations and some simple transformations, we obtain the solitary wave solutions of equations, compacton solutions of equations and the compacton-like solution of the potential form (called ) . In addition, we show that the variable admits dromion solutions rather than the field itself in equation.
文摘In this paper, we establish exact solutions for the .R（m,n） equations by using an sn-cn metnou,As a result, abundant new cornpactons, i,e, solitons with the absence of infinite wings, new type of Jacobi elliptic function, solitary wave and periodic wave solutions, of this equation are obtained with minimal calculations. The properties of the R（m, n） equations are shown in figures.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11075055,11275072Innovative Research Team Program of the National Science Foundation of China under Grant No.61021104+3 种基金National High Technology Research and Development Program under Grant No.2011AA010101Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213Talent FundK.C.Wong Magna Fund in Ningbo University
文摘Two Darboux transformations of the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka(CDGKS)equation and(2+1)-dimensional modified Korteweg-de Vries(mKdV) equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux transformations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the(2+1)-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the(2+1)-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.
基金Supported by the National Natural Science Foundation of China under Grant No.11171312
文摘Utilizing the Wronskian technique, a combined Wronskian condition is established for a （3＋1）-dimensional generalized KP equation. The generating functions for matrix entries satisfy a linear system of new partial differential equations. Moreover, as applications, examples of Wronskian determinant solutions, including N-soliton solutions, periodic solutions and rational solutions, are computed.
基金Supported by the Fundamental Research Funds for the Central Universities under Grant No. BUPT2013RC0902
文摘Starting from a simple transformation, and with the aid of symbolic computation, we establish the relationship between the solution of a generalized variable coefficient Kadomtsev–Petviashvili(vKP) equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover,we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.
基金supported by National Natural Science Foundation of China(Grant No.11131005)
文摘In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a^ ij（x）δij u＋b^ i（x）δiu＋f（x,u）=0,in Ω belong to belong toR^n,（＊）where a^ijδiφδjφ=0 on δΩ,andφis the defining function of δΩ.Imposing suitable conditions on the coefficients and f（x,u）,one can get the L^∞-estimates of（＊）via blow up method.
