The Burgers-Korteweg-de Vries equation has wide applications in physics, engineering and fluid mechanics. The Poincaré phase plane analysis reveals that the Burgers-Korteweg-de Vries equation has neither nontrivi...The Burgers-Korteweg-de Vries equation has wide applications in physics, engineering and fluid mechanics. The Poincaré phase plane analysis reveals that the Burgers-Korteweg-de Vries equation has neither nontrivial bell-profile traveling solitary waves, nor periodic waves. In the present paper, we show two approaches for the study of traveling solitary waves of the Burgers-Korteweg-de Vries equation: one is a direct method which involves a few coordinate transformations, and the other is the Lie group method. Our study indicates that the Burgers-Korteweg-de Vries equation indirectly admits one-parameter Lie groups of transformations with certain parametric conditions and a traveling solitary wave solution with an arbitrary velocity is obtained accordingly. Some incorrect statements in the recent literature are clarified.展开更多
In this paper,eight types of (1-+1)-dimensional similarity reductions which contain variable coefficient equation,are obtained for the generalized KdV equation in (2+1)-dimensional space arising from the multidimensio...In this paper,eight types of (1-+1)-dimensional similarity reductions which contain variable coefficient equation,are obtained for the generalized KdV equation in (2+1)-dimensional space arising from the multidimensional isospectral flows associated with the second-order scalar operators by using the direct method.In addition,the cnoidal wave solution and dromion-like solution are also derived by using the reduced nonlinear ordinary differential equations.The (1+1) dromion obtained by Lou [J.Phys.A28 (1995) 7227] and Zhang [Chin.Phys.9 (2000) 1] is only a special case of our results.Moreover,some properties of the dromion-like solutions are analyzed.展开更多
In this paper, we introduce the Painlev property of the Burgers-KdV equation. Two types of exact solutions to the equation are obtained by the standard truncated expansion method and the extended standard truncated ex...In this paper, we introduce the Painlev property of the Burgers-KdV equation. Two types of exact solutions to the equation are obtained by the standard truncated expansion method and the extended standard truncated expansion method, respectively.展开更多
In this note, we analyze a few major claims about . As a consequence, we rewrite a major theorem, nullify its proof and one remark of importance, and offer a valid proof for it. The most important gift of this paper i...In this note, we analyze a few major claims about . As a consequence, we rewrite a major theorem, nullify its proof and one remark of importance, and offer a valid proof for it. The most important gift of this paper is probably the reasoning involved in all: We observe that a constant, namely t, has been changed into a variable, and we then tell why such a move could not have been made, we observe the discrepancy between the claimed domain and the actual domain of a supposed function that is created and we then explain why such a function could not, or should not, have been created, along with others.展开更多
The rational solutions for the discrete Painlevé Ⅱ equation are constructed based on the bilinear formalism. It is shown that they are expressed by a determinant whose entries are given by the Laguerre Polynomials.
In this article, we study the string equation of type (2, 2n + 1), which is derived from 2D gravity theory or the string theory. We consider the equation as a 2n-th order analogue of the first Painlevéequation, t...In this article, we study the string equation of type (2, 2n + 1), which is derived from 2D gravity theory or the string theory. We consider the equation as a 2n-th order analogue of the first Painlevéequation, take the autonomous limit, and solve it concretely by use of the Weierstrass’ elliptic function.展开更多
In this article, we study the string equation of type (2,5), which is derived from 2D gravity theory or the string theory. We consider the equation as a 4th order analogue of the first Painlevé equation, take the...In this article, we study the string equation of type (2,5), which is derived from 2D gravity theory or the string theory. We consider the equation as a 4th order analogue of the first Painlevé equation, take the autonomous limit, and solve it concretely by use of the Weierstrass’ elliptic function.展开更多
In this paper, we investigate the value distribution properties for the solution w（z） of higher-order Painlevé equations. We prove that the Nevanlinna＇s second main inequality for w（z） is reduced to an asymp...In this paper, we investigate the value distribution properties for the solution w（z） of higher-order Painlevé equations. We prove that the Nevanlinna＇s second main inequality for w（z） is reduced to an asymptotic equality.展开更多
In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlevé test and we also calculate its one- and two-soli...In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlevé test and we also calculate its one- and two-soliton solutions.展开更多
This article concerns the quantum superintegrable system obtained by Tremblay and Winternitz, which allows the separation of variables in polar coordinates and possesses three conserved quantities with the potential d...This article concerns the quantum superintegrable system obtained by Tremblay and Winternitz, which allows the separation of variables in polar coordinates and possesses three conserved quantities with the potential described by the sixth Painlevé equation. The degeneration procedure from the sixth Painlvé equation to the fifth one yields another new superintegrable system;however, the Hermitian nature is broken.展开更多
基金This work was supported by US National Science Foundation(Grant No.CCF-0514768)partly supported by UTPA Faculty Research Council(Grant No.119100)
文摘The Burgers-Korteweg-de Vries equation has wide applications in physics, engineering and fluid mechanics. The Poincaré phase plane analysis reveals that the Burgers-Korteweg-de Vries equation has neither nontrivial bell-profile traveling solitary waves, nor periodic waves. In the present paper, we show two approaches for the study of traveling solitary waves of the Burgers-Korteweg-de Vries equation: one is a direct method which involves a few coordinate transformations, and the other is the Lie group method. Our study indicates that the Burgers-Korteweg-de Vries equation indirectly admits one-parameter Lie groups of transformations with certain parametric conditions and a traveling solitary wave solution with an arbitrary velocity is obtained accordingly. Some incorrect statements in the recent literature are clarified.
文摘In this paper,eight types of (1-+1)-dimensional similarity reductions which contain variable coefficient equation,are obtained for the generalized KdV equation in (2+1)-dimensional space arising from the multidimensional isospectral flows associated with the second-order scalar operators by using the direct method.In addition,the cnoidal wave solution and dromion-like solution are also derived by using the reduced nonlinear ordinary differential equations.The (1+1) dromion obtained by Lou [J.Phys.A28 (1995) 7227] and Zhang [Chin.Phys.9 (2000) 1] is only a special case of our results.Moreover,some properties of the dromion-like solutions are analyzed.
基金supported by the Research Project of SDUST Spring Bud(No.2009AZZ167)the Graduate Innovation Foundation from Shandong University of Science and Technology(YCA100209)
文摘In this paper, we introduce the Painlev property of the Burgers-KdV equation. Two types of exact solutions to the equation are obtained by the standard truncated expansion method and the extended standard truncated expansion method, respectively.
文摘In this note, we analyze a few major claims about . As a consequence, we rewrite a major theorem, nullify its proof and one remark of importance, and offer a valid proof for it. The most important gift of this paper is probably the reasoning involved in all: We observe that a constant, namely t, has been changed into a variable, and we then tell why such a move could not have been made, we observe the discrepancy between the claimed domain and the actual domain of a supposed function that is created and we then explain why such a function could not, or should not, have been created, along with others.
文摘The rational solutions for the discrete Painlevé Ⅱ equation are constructed based on the bilinear formalism. It is shown that they are expressed by a determinant whose entries are given by the Laguerre Polynomials.
文摘In this article, we study the string equation of type (2, 2n + 1), which is derived from 2D gravity theory or the string theory. We consider the equation as a 2n-th order analogue of the first Painlevéequation, take the autonomous limit, and solve it concretely by use of the Weierstrass’ elliptic function.
文摘In this article, we study the string equation of type (2,5), which is derived from 2D gravity theory or the string theory. We consider the equation as a 4th order analogue of the first Painlevé equation, take the autonomous limit, and solve it concretely by use of the Weierstrass’ elliptic function.
基金Supported by the Mathematical Tianyuan Foundation of China(No.10426007)the National Natural Science Foundation of China(No.10272017,No.10471028)
文摘In this paper, we investigate the value distribution properties for the solution w（z） of higher-order Painlevé equations. We prove that the Nevanlinna＇s second main inequality for w（z） is reduced to an asymptotic equality.
基金Project supported by the National Natural Science Foundation of China (Grant No 10671206)
文摘In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlevé test and we also calculate its one- and two-soliton solutions.
文摘This article concerns the quantum superintegrable system obtained by Tremblay and Winternitz, which allows the separation of variables in polar coordinates and possesses three conserved quantities with the potential described by the sixth Painlevé equation. The degeneration procedure from the sixth Painlvé equation to the fifth one yields another new superintegrable system;however, the Hermitian nature is broken.
