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
Based on the direct method of calculating the periodic wave solution proposed by Nakamura,we give an approximate analytical three-periodic solutions of Korteweg-de Vries(KdV)-type equations by perturbation method for ...Based on the direct method of calculating the periodic wave solution proposed by Nakamura,we give an approximate analytical three-periodic solutions of Korteweg-de Vries(KdV)-type equations by perturbation method for the first time.Limit methods have been used to establish the asymptotic relationships between the three-periodic solution separately and another three solutions,the soliton solution,the one-and the two-periodic solutions.Furthermore,it is found that the asymptotic three-soliton solution presents the same repulsive phenomenon as the asymptotic three-soliton solution during the interaction.展开更多
In the present letter, we get the appropriate bilinear forms of (2 + 1)-dimensional KdV equation, extended (2 + 1)-dimensional shallow water wave equation and (2 + 1)-dimensional Sawada -Kotera equation in a ...In the present letter, we get the appropriate bilinear forms of (2 + 1)-dimensional KdV equation, extended (2 + 1)-dimensional shallow water wave equation and (2 + 1)-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.展开更多
Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutio...Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.展开更多
With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pemp...With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.展开更多
In this paper,multi-periodic (quasi-periodic) wave solutions are constructed for the Boiti-Leon-Manna-Pempinelli(BLMP) equation by using Hirota bilinear method and Riemann theta function.At the same time,weanalyze in ...In this paper,multi-periodic (quasi-periodic) wave solutions are constructed for the Boiti-Leon-Manna-Pempinelli(BLMP) equation by using Hirota bilinear method and Riemann theta function.At the same time,weanalyze in details asymptotic properties of the multi-periodic wave solutions and give their asymptotic relations betweenthe periodic wave solutions and the soliton solutions.展开更多
In this paper, the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials. The N-soliton solution and one periodic wave solution are presented by use of the ...In this paper, the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials. The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function, respectively. And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution. In the end, the bilinear Backlund transformations are derived.展开更多
With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method ...With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.展开更多
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.
基金the National National Science Foundation of China(Grant Nos.52171251,U2106225,and 52231011)the Science and Technology Innovation Fund of Dalian City(Grant No.2022JJ12GX036)。
文摘Based on the direct method of calculating the periodic wave solution proposed by Nakamura,we give an approximate analytical three-periodic solutions of Korteweg-de Vries(KdV)-type equations by perturbation method for the first time.Limit methods have been used to establish the asymptotic relationships between the three-periodic solution separately and another three solutions,the soliton solution,the one-and the two-periodic solutions.Furthermore,it is found that the asymptotic three-soliton solution presents the same repulsive phenomenon as the asymptotic three-soliton solution during the interaction.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11075055,61021004,10735030Shanghai Leading Academic Discipline Project under Grant No.B412Program for Changjiang Scholars and Innovative Research Team in University(IRT0734)
文摘In the present letter, we get the appropriate bilinear forms of (2 + 1)-dimensional KdV equation, extended (2 + 1)-dimensional shallow water wave equation and (2 + 1)-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.
基金The project supported by National Natural Science Foundation of China under Grant No.10771196the Natural Science Foundation of Zhejiang Province under Grant No.Y605044
文摘Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.BUAA-SKLSDE-09KF-04+1 种基金Beijing University of Aeronautics and Astronautics,by the National Basic Research Program of China (973 Program) under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos.20060006024 and 200800130006,Chinese Ministry of Education
文摘With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.
基金Supported by the Natural Science Foundation of Shanghai under Grant No.09ZR1412800 the Innovation Program of Shanghai Municipal Education Commission under Grant No.10ZZ131
文摘In this paper,multi-periodic (quasi-periodic) wave solutions are constructed for the Boiti-Leon-Manna-Pempinelli(BLMP) equation by using Hirota bilinear method and Riemann theta function.At the same time,weanalyze in details asymptotic properties of the multi-periodic wave solutions and give their asymptotic relations betweenthe periodic wave solutions and the soliton solutions.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 10735030 and 11075055Innovative Research Team Program of the National Natural Science Foundation of China under Grant No. 61021004
文摘In this paper, the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials. The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function, respectively. And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution. In the end, the bilinear Backlund transformations are derived.
基金Supported by Shandong Provincial Key Laboratory of Marine Ecology and Environment&Disaster Prevention and Mitigation project under Grant No.2012010National Natural Science Foundation of China under Grant No.11271007+1 种基金Special Funds for Theoretical Physics of the National Natural Science Foundation of China under Grant No.11447205Shandong University of Science and Technology Research Fund under Grant No.2012KYTD105
文摘With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.