The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger （CNLS） equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass th...The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger （CNLS） equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.展开更多
In this paper, the Painleve properties of the modified C-KdV equation are verified by using the W-K algorithm. Then some exact soliton solutions are obtained by applying the standard truncated expansion method and the...In this paper, the Painleve properties of the modified C-KdV equation are verified by using the W-K algorithm. Then some exact soliton solutions are obtained by applying the standard truncated expansion method and the nonstandard truncated expansion method with the help of Maple software, respectively.展开更多
A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painl...A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.展开更多
The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé...The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.展开更多
The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e prope...The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e property, then it admits certain class of first integrals. We also present some relationships between the Painlev′e property and the structure of the differential Galois group of the corresponding variational equations along some complex integral curve.展开更多
Painleve property and infinite symmetries of the （2＋1）-dimensional higher-order Broer-Kaup （HBK） system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.u...Painleve property and infinite symmetries of the （2＋1）-dimensional higher-order Broer-Kaup （HBK） system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.ups to （2＋1）-dimensional HBK system. Based on our theorem, some new forms of solutions are obtained. We also find infinite number of conservation laws of the （2＋1）-dimensional HBK system.展开更多
The （3＋1）-dimensional Jimbo-Miwa （JM） equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new （3＋1）-dimensional integrable models, ...The （3＋1）-dimensional Jimbo-Miwa （JM） equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new （3＋1）-dimensional integrable models, which are conformal invariant and possess Painlevé property, the approximate solutions are obtained for the JM equation, containing not only one-soliton solutions but also periodic solutions and multi-soliton solutions. Some approximate solutions happen to be exact and some approximate solutions can become exact by choosing relations between the parameters properly.展开更多
Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians H = （p1^2 ＋ p2^2）/2 ＋ V（q1, q2） whose some integer power qj^nj （t）...Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians H = （p1^2 ＋ p2^2）/2 ＋ V（q1, q2） whose some integer power qj^nj （t） of the general solution is a singlevalued function of the complez time t. In addition to the well known rational potentials V of Hénon-Heiles, this selects possible cases with a trigonometric dependence of V on qj. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven （three “cubic” plus four “quartic”） rational Hénon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.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
This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth.As opposed to a logistic growth where the rate of growth of the population decreases from the onset,an autocatalyti...This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth.As opposed to a logistic growth where the rate of growth of the population decreases from the onset,an autocatalytic growth has a point of inflection where the rate of growth switches from an increasing trend to a decreasing trend.Employing the idea of Painleve property,we show that a variety of exact traveling wave solutions can be obtained for this model depending on the choice of the advection term.In particular,due to situations in resource distribution or environmental variations,if the advection is represented as a decaying function in time or an oscillating function in time,we are able to find exact solutions with interesting behavior.We also carry out a computational study of the model using an exponentially upwinded numerical scheme and illustrate the movement of the solutions and their characteristics pictorially.展开更多
In this work, we consider a Fisher and generalized Fisher equations with variable coefficients. Using truncated Painleve expansions of these equations, we obtain exact solutions of these equations with a constraint on...In this work, we consider a Fisher and generalized Fisher equations with variable coefficients. Using truncated Painleve expansions of these equations, we obtain exact solutions of these equations with a constraint on the coefficients a（t） and b（t）.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos. 10735030and 40775069)the Natural Science Foundation of Guangdong Province of China(No. 10452840301004616)the Scientific Research Foundation for the Doctors of University of Electronic Science and Technology of China Zhongshan Institute (No. 408YKQ09)
文摘The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger （CNLS） equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.
基金Project supported by the National Natural Science Foundation of China (Grant No.70971079)the Science Foundation of the Educational Department of Shandong Province of China (Grant No.J07YH01)
文摘In this paper, the Painleve properties of the modified C-KdV equation are verified by using the W-K algorithm. Then some exact soliton solutions are obtained by applying the standard truncated expansion method and the nonstandard truncated expansion method with the help of Maple software, respectively.
基金supported by Chinese National Social Science Foundation(Grant Number:CNSSF:13CJY037)Research on the indemnificatory Apartment Construction Based on Residential Integration.
文摘A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11975131 and 11435005)the K C Wong Magna Fund in Ningbo University。
文摘The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.
基金The NSF (11371166,11301210)of ChinaNational 973 Project (2012CB821200) of ChinaJilin Province Youth Science Foundation
文摘The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e property, then it admits certain class of first integrals. We also present some relationships between the Painlev′e property and the structure of the differential Galois group of the corresponding variational equations along some complex integral curve.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004 zx 16
文摘Painleve property and infinite symmetries of the （2＋1）-dimensional higher-order Broer-Kaup （HBK） system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.ups to （2＋1）-dimensional HBK system. Based on our theorem, some new forms of solutions are obtained. We also find infinite number of conservation laws of the （2＋1）-dimensional HBK system.
基金The project supported by the Natural Science Foundation of Zhejiang Province of China under Grant No. Y604036 and State Key Laboratory of 0il/Gas Reservoir Geology and Exploitation ＂PLN0402＂ The authors would like to thank Prof. Sen-Yue Lou for his help and discussion.
文摘The （3＋1）-dimensional Jimbo-Miwa （JM） equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new （3＋1）-dimensional integrable models, which are conformal invariant and possess Painlevé property, the approximate solutions are obtained for the JM equation, containing not only one-soliton solutions but also periodic solutions and multi-soliton solutions. Some approximate solutions happen to be exact and some approximate solutions can become exact by choosing relations between the parameters properly.
文摘Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians H = （p1^2 ＋ p2^2）/2 ＋ V（q1, q2） whose some integer power qj^nj （t） of the general solution is a singlevalued function of the complez time t. In addition to the well known rational potentials V of Hénon-Heiles, this selects possible cases with a trigonometric dependence of V on qj. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven （three “cubic” plus four “quartic”） rational Hénon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.
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This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth.As opposed to a logistic growth where the rate of growth of the population decreases from the onset,an autocatalytic growth has a point of inflection where the rate of growth switches from an increasing trend to a decreasing trend.Employing the idea of Painleve property,we show that a variety of exact traveling wave solutions can be obtained for this model depending on the choice of the advection term.In particular,due to situations in resource distribution or environmental variations,if the advection is represented as a decaying function in time or an oscillating function in time,we are able to find exact solutions with interesting behavior.We also carry out a computational study of the model using an exponentially upwinded numerical scheme and illustrate the movement of the solutions and their characteristics pictorially.
文摘In this work, we consider a Fisher and generalized Fisher equations with variable coefficients. Using truncated Painleve expansions of these equations, we obtain exact solutions of these equations with a constraint on the coefficients a（t） and b（t）.
