In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta funct...In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.展开更多
We investigate dark solitons lying on elliptic function background in the defocusing Hirota equation with third-order dispersion and self-steepening terms.By means of the modified squared wavefunction method,we obtain...We investigate dark solitons lying on elliptic function background in the defocusing Hirota equation with third-order dispersion and self-steepening terms.By means of the modified squared wavefunction method,we obtain the Jacobi's elliptic solution of the defocusing Hirota equation,and solve the related linear matrix eigenvalue problem on elliptic function background.The elliptic N-dark soliton solution in terms of theta functions is constructed by the Darboux transformation and limit technique.The asymptotic dynamical behaviors for the elliptic N-dark soliton solution as t→±∞are studied.Through numerical plots of the elliptic one-,two-and three-dark solitons,the amplification effect on the velocity of elliptic dark solitons,and the compression effect on the soliton spatiotemporal distributions produced by the third-order dispersion and self-steepening terms are discussed.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
In this paper, we establish a general theta function identity. It is a common origin of many theta function identities. From which many classical and new modular equations are derived. All the proofs are elementary.
Jacobi theta functions with rational characteristics can be viewed as vector-valued Jacobi forms.Theta relations usually correspond to different constructions of certain Jacobi forms. From this observation, we extract...Jacobi theta functions with rational characteristics can be viewed as vector-valued Jacobi forms.Theta relations usually correspond to different constructions of certain Jacobi forms. From this observation, we extract a new approach, which is called orbits of Jacobi forms, to produce identities on Jacobi theta functions.Our approach not only provides simple proofs of many known theta relations but also produces a large number of new identities, which can be considered as generalizations of Riemann's theta relations.展开更多
Recently,Andrews and Paule established the generating functions for the k-elongated plane partition function d_(k)(n)and proved a large number of results on d_(k)(n)with k=2,3.In particular,they posed some conjectures...Recently,Andrews and Paule established the generating functions for the k-elongated plane partition function d_(k)(n)and proved a large number of results on d_(k)(n)with k=2,3.In particular,they posed some conjectures on congruences modulo powers of 3 for d_(2)(n).Their work has attracted the attention of da Silva,Hirschhorn,Sellers and Smoot.Very recently,Smoot proved a congruence family for d_(2)(n)which implies one conjecture due to Andrews and Paule by using the localization method.In this paper,we prove the rest two conjectures given by Andrews and Paule.展开更多
By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1...By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1 /6)(u0 1) + 2[ln f(x,y,t)] xx,four kinds of triply periodic wave solutions are derived,and their long wave limit are discussed.The properties of one of the solutions are shown in Fig.1.展开更多
The main purpose of this paper is to generalize the study of the Hecke-Rogers type series,which are the extensions of truncated theorems obtained by Andrews,Merca,Wang and Yee.Our proofs rely heavily on the theory of ...The main purpose of this paper is to generalize the study of the Hecke-Rogers type series,which are the extensions of truncated theorems obtained by Andrews,Merca,Wang and Yee.Our proofs rely heavily on the theory of Bailey pairs.展开更多
Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.
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, 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
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
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
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
Orbits of Jacobi forms and theta relations Valery Gritsenko&Haowu Wang Abstract Jacobi theta functions with rational characteristics can be viewed as vector-valued Jacobi forms.Theta relations usually correspond t...Orbits of Jacobi forms and theta relations Valery Gritsenko&Haowu Wang Abstract Jacobi theta functions with rational characteristics can be viewed as vector-valued Jacobi forms.Theta relations usually correspond to different constructions of certain Jacobi forms.From this observation,we extract a new approach,which is called orbits of Jacobi forms,to produce identities on Jacobi theta functions.Our approach not only provides simple proofs of many known theta relations but also produces a large number of new identities,which can be considered as generalizations of Riemann's theta relations.Keywords Jacobi forms,Jacobi theta functions,theta relations MSC(2020)11F50,14K25 https://doi.org/10.1007/s11425-023-2324-4.展开更多
Let G be the group of the fractional linear transformations generated by T（τ）=τ ＋ λ, S（τ）=（τ cos π/n ＋ sin π/n）/（-τ sin π/n ＋ cos π/n）;where λ=2（cos π/m ＋ cos π/n）/sin π/n;m, n is a pair of...Let G be the group of the fractional linear transformations generated by T（τ）=τ ＋ λ, S（τ）=（τ cos π/n ＋ sin π/n）/（-τ sin π/n ＋ cos π/n）;where λ=2（cos π/m ＋ cos π/n）/sin π/n;m, n is a pair of integers with either n ≥ 2, m ≥ 3 or n ≥ 3, m ≥ 2; τ lies in the upper half plane H.A fundamental set of functions f0, fi and f∞ automorphic with respect to G will be constructed from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan＇s triple differential equations associated with the group G and establish the connection of f0, fi and f∞ with a family of hypergeometric functions.展开更多
In this paper a kind of theta function is constructed by means of spherical function. And we also obtain some Hilbert modular forms of half integral weight.
基金Supported by Innovation Program of Shanghai Municipal Education Commission and PCSIRT
文摘In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.
基金supported by the National Natural Science Foundation of China(Grant No.12326304,12326305,12071304)the Shenzhen Natural Science Fund(the Stable Support Plan Program)(Grant No.20220809163103001)+2 种基金the Natural Science Foundation of Henan Province(Grant No.232300420119)the Excellent Science and Technology Innovation Talent Support Program of ZUT(Grant No.K2023YXRC06)Funding for the Enhancement Program of Advantageous Discipline Strength of ZUT(2022)。
文摘We investigate dark solitons lying on elliptic function background in the defocusing Hirota equation with third-order dispersion and self-steepening terms.By means of the modified squared wavefunction method,we obtain the Jacobi's elliptic solution of the defocusing Hirota equation,and solve the related linear matrix eigenvalue problem on elliptic function background.The elliptic N-dark soliton solution in terms of theta functions is constructed by the Darboux transformation and limit technique.The asymptotic dynamical behaviors for the elliptic N-dark soliton solution as t→±∞are studied.Through numerical plots of the elliptic one-,two-and three-dark solitons,the amplification effect on the velocity of elliptic dark solitons,and the compression effect on the soliton spatiotemporal distributions produced by the third-order dispersion and self-steepening terms are discussed.
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Supported by the National Natural Science Foundation of China（11071107, 11371184）
文摘In this paper, we establish a general theta function identity. It is a common origin of many theta function identities. From which many classical and new modular equations are derived. All the proofs are elementary.
基金supported by the project “International Academic Cooperation” National Research University Higher School of Economics。
文摘Jacobi theta functions with rational characteristics can be viewed as vector-valued Jacobi forms.Theta relations usually correspond to different constructions of certain Jacobi forms. From this observation, we extract a new approach, which is called orbits of Jacobi forms, to produce identities on Jacobi theta functions.Our approach not only provides simple proofs of many known theta relations but also produces a large number of new identities, which can be considered as generalizations of Riemann's theta relations.
基金Supported by the National Natural Science Foundation of China(Grant No.12371334)the Natural Science Foundation of Jiangsu Province(Grant No.BK20221383)。
文摘Recently,Andrews and Paule established the generating functions for the k-elongated plane partition function d_(k)(n)and proved a large number of results on d_(k)(n)with k=2,3.In particular,they posed some conjectures on congruences modulo powers of 3 for d_(2)(n).Their work has attracted the attention of da Silva,Hirschhorn,Sellers and Smoot.Very recently,Smoot proved a congruence family for d_(2)(n)which implies one conjecture due to Andrews and Paule by using the localization method.In this paper,we prove the rest two conjectures given by Andrews and Paule.
基金Supported by the National Natural Science Foundation of China under Grant No. 11101382the Natural Science Foundation of Henan Province under Grant No. 2010A110001the Basic and Advanced Technology Project of Henan Province under Grant No. 112300410199
文摘By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1 /6)(u0 1) + 2[ln f(x,y,t)] xx,four kinds of triply periodic wave solutions are derived,and their long wave limit are discussed.The properties of one of the solutions are shown in Fig.1.
基金Supported by the National Natural Science Foundation of China(11871370 and 12001182)the Fundamental Research Funds for the Central Universities(531118010411)。
文摘The main purpose of this paper is to generalize the study of the Hecke-Rogers type series,which are the extensions of truncated theorems obtained by Andrews,Merca,Wang and Yee.Our proofs rely heavily on the theory of Bailey pairs.
文摘Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.
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 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 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 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.
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.
文摘Orbits of Jacobi forms and theta relations Valery Gritsenko&Haowu Wang Abstract Jacobi theta functions with rational characteristics can be viewed as vector-valued Jacobi forms.Theta relations usually correspond to different constructions of certain Jacobi forms.From this observation,we extract a new approach,which is called orbits of Jacobi forms,to produce identities on Jacobi theta functions.Our approach not only provides simple proofs of many known theta relations but also produces a large number of new identities,which can be considered as generalizations of Riemann's theta relations.Keywords Jacobi forms,Jacobi theta functions,theta relations MSC(2020)11F50,14K25 https://doi.org/10.1007/s11425-023-2324-4.
文摘Let G be the group of the fractional linear transformations generated by T（τ）=τ ＋ λ, S（τ）=（τ cos π/n ＋ sin π/n）/（-τ sin π/n ＋ cos π/n）;where λ=2（cos π/m ＋ cos π/n）/sin π/n;m, n is a pair of integers with either n ≥ 2, m ≥ 3 or n ≥ 3, m ≥ 2; τ lies in the upper half plane H.A fundamental set of functions f0, fi and f∞ automorphic with respect to G will be constructed from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan＇s triple differential equations associated with the group G and establish the connection of f0, fi and f∞ with a family of hypergeometric functions.
文摘In this paper a kind of theta function is constructed by means of spherical function. And we also obtain some Hilbert modular forms of half integral weight.
