Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation i...Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.展开更多
Abraham Lempel et al made a connection between linear codes and systems of bilinear forms over finite fields. In this correspondence, a new simple proof of a theorem in [1] is presented; in addition, the encoding proc...Abraham Lempel et al made a connection between linear codes and systems of bilinear forms over finite fields. In this correspondence, a new simple proof of a theorem in [1] is presented; in addition, the encoding process and the decoding procedure of RS codes are simplified via circulant matrices. Finally, the results show that the correspondence between bilinear forms and linear codes is not unique.展开更多
The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-d...The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-de Vriesmodified Korteweg-de Vries equation for the atmosphere,oceanic fluids and plasmas.With symbolic computation,beginning with a presumption,we work out certain scaling transformations,bilinear forms through the binary Bell polynomials and our scaling transformations,N solitons(with N being a positive integer)via the aforementioned bilinear forms and bilinear auto-Bäcklund transformations through the Hirota method with some solitons.In addition,Painlevé-type auto-Bäcklund transformations with some solitons are symbolically computed out.Respective dependences and constraints on the variable/constant coefficients are discussed,while those coefficients correspond to the quadratic-nonlinear,cubic-nonlinear,dispersive,dissipative and line-damping effects in the atmosphere,oceanic fluids and plasmas.展开更多
In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We fin...In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.展开更多
An improved algorithm for symbolic computation of Hirota bilinear form of nonlinear equations by a logarithm transformation is presented. The improved algorithm is more efficient by using the property of Hirota-D oper...An improved algorithm for symbolic computation of Hirota bilinear form of nonlinear equations by a logarithm transformation is presented. The improved algorithm is more efficient by using the property of Hirota-D operator. The software package HBFTrans2 is written in Maple and its running efficiency is tested by a variety of soliton equations.展开更多
Investigation in this paper is given to the reduced Maxwell-Bloeh equations with variable coetcients, describing the propagation of the intense ultra-short optical pulses through an inhomogeneous two-level dielectric ...Investigation in this paper is given to the reduced Maxwell-Bloeh equations with variable coetcients, describing the propagation of the intense ultra-short optical pulses through an inhomogeneous two-level dielectric medium. We apply the Hirota method and symbolic computation to study such equations. With the help of the dependent variable transformations, we present the variable-coetteient-dependent bilinear forms. Then, we construct the one-, two- and N- soliton solutions in analytic forms for them.展开更多
An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations withlogarithmic transformations is presented.In the algorithm,the general assumption of Hirota bilinear form is successfull...An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations withlogarithmic transformations is presented.In the algorithm,the general assumption of Hirota bilinear form is successfullyreduced based on the property of uniformity in rank.Furthermore,we discard the integral operation in the traditionalalgorithm.The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety solitonequations.展开更多
In this paper, we proved a theorem on the orthogonal decomposition of a flatsymmetric bilinear form. If the nullity space of a flat symmetric bilinear formis not too large, it can be decomposed into two bilinear forms...In this paper, we proved a theorem on the orthogonal decomposition of a flatsymmetric bilinear form. If the nullity space of a flat symmetric bilinear formis not too large, it can be decomposed into two bilinear forms, one of which isnull, the other is flat and has a large nullity.展开更多
The author gives a characterization of the Fourier transforms of bounded bilinear forms on C＊（S1）×C＊（S2） of two foundation semigroups S1 and S2 in terms of Jordan ＊-representations, hemimogeneous random fi...The author gives a characterization of the Fourier transforms of bounded bilinear forms on C＊（S1）×C＊（S2） of two foundation semigroups S1 and S2 in terms of Jordan ＊-representations, hemimogeneous random fields, and as well as weakly harmonizable random fields of S1 and S2 into Hilbert spaces.展开更多
A bilinear form f on a nonassociative triple system T is said to be invariant if and only if f( abc ,d) = f(a, dcb ) = f(c, bad ) for all a,b,c,d ∈ T . (T ,f) is called a pseudo-metric triple system if f is non-degen...A bilinear form f on a nonassociative triple system T is said to be invariant if and only if f( abc ,d) = f(a, dcb ) = f(c, bad ) for all a,b,c,d ∈ T . (T ,f) is called a pseudo-metric triple system if f is non-degenerate and invariant. A decomposition theory for triple systems and pseudo-metric triple systems is established. Moreover, the ?nite-dimensional metric Lie triple systems are characterized in terms of the structure of the non-degenerate, invariant and symmetric bilinear forms on them.展开更多
Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which ...Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field F.展开更多
We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers ...We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers the Polya-Vinogradov range,and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves.In the case of hyper-Kloosterman sums,we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski,Michel and Sawin(2017) on sum-products of Kloosterman sheaves.Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.展开更多
In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plas...In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plasma in three spatial dimensions. In order to study the integrability property of such an equation, the Painlevé analysis is performed on it. And then, based on the truncated Painlevé expansion, the bilinear form of the （3＋1）-dimensionaJ vcKP equation is obtained under certain coefficients constraint, and its solution in the Wronskian determinant form is constructed and verified by virtue of the Wronskian technique. Besides the Wronskian determinant solution, it is shown that the （3＋1）-dimensional vcKP equation also possesses a solution in the form of the Grammian determinant.展开更多
Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bgcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bgcklund ...Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bgcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bgcklund transformation, we can give transformations among many kinds of solutions.展开更多
In this paper, two types of the （2＋1）-dimensional breaking soliton equations axe investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilinea...In this paper, two types of the （2＋1）-dimensional breaking soliton equations axe investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilineax forms and Bgcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilineax equations. Via the Wronskian technique, it is proved that the Bgcklund transformations obtained are the ones between the （ N - 1）- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonie properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.展开更多
By truncating the Painleve expansion at the constant level term,the Hirota bilinear form is obtainedfor a (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation.Based on its bilinear form,solitary-wave...By truncating the Painleve expansion at the constant level term,the Hirota bilinear form is obtainedfor a (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation.Based on its bilinear form,solitary-wavesolutions are constructed via the ε-expansion method and the corresponding graphical analysis is given.Furthermore,the exact solution in the Wronskian form is presented and proved by direct substitution into the bilinear equation.展开更多
In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation （mKdV） via the Backlund transformation （BT） and a generalized Wronskian condition is given, which allows us to substitute an a...In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation （mKdV） via the Backlund transformation （BT） and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the GN （t） for the original diagonal one.展开更多
To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations.There are also some other methods that are based on integrable scala...To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations.There are also some other methods that are based on integrable scalar nonlinear partial differential equations.We show that some systems of integrable equations published recently are the M_(2)-extension of integrable such scalar equations.For illustration,we give Korteweg-de Vries,Kaup-Kupershmidt,and SawadaKotera equations as examples.By the use of such an extension of integrable scalar equations,we obtain some new integrable systems with recursion operators.We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.展开更多
基金supported by the National Nature Science Foundation of China under Grant No.11871116Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11。
文摘Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.
基金She was with the Department of Mathematics in Wuhan University while writting this paper.
文摘Abraham Lempel et al made a connection between linear codes and systems of bilinear forms over finite fields. In this correspondence, a new simple proof of a theorem in [1] is presented; in addition, the encoding process and the decoding procedure of RS codes are simplified via circulant matrices. Finally, the results show that the correspondence between bilinear forms and linear codes is not unique.
基金the National Natural Science Foundation of China(Grant No.11871116)the Fundamental Research Funds for the Central Universities of China(Grant No.2019XD-A11)the BUPT Innovation and Entrepreneurship Support Program,Beijing University of Posts and Telecommunications,and the National Scholarship for Doctoral Students of China.
文摘The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-de Vriesmodified Korteweg-de Vries equation for the atmosphere,oceanic fluids and plasmas.With symbolic computation,beginning with a presumption,we work out certain scaling transformations,bilinear forms through the binary Bell polynomials and our scaling transformations,N solitons(with N being a positive integer)via the aforementioned bilinear forms and bilinear auto-Bäcklund transformations through the Hirota method with some solitons.In addition,Painlevé-type auto-Bäcklund transformations with some solitons are symbolically computed out.Respective dependences and constraints on the variable/constant coefficients are discussed,while those coefficients correspond to the quadratic-nonlinear,cubic-nonlinear,dispersive,dissipative and line-damping effects in the atmosphere,oceanic fluids and plasmas.
基金the National Natural Science Foundation of China under Grant No.11772017the Fundamental Research Funds for the Central Universities
文摘In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.
基金Supported by Scientific Research Fund of Zhejiang Provincial Education Department under Grant No.Y201017148the National Natural Science Foundations of China under Grant No.10735030+2 种基金the Natural Science Fund of Ningbo under Grant No.2009B21003the Scientific Research Fund of Ningbo University under Grant No.XKL09059the K.C.Wong Magana Fund in Ningbo University
文摘An improved algorithm for symbolic computation of Hirota bilinear form of nonlinear equations by a logarithm transformation is presented. The improved algorithm is more efficient by using the property of Hirota-D operator. The software package HBFTrans2 is written in Maple and its running efficiency is tested by a variety of soliton equations.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11772017,11272023,11471050the Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications),China(IPOC:2017ZZ05)the Fundamental Research Funds for the Central Universities of China under Grant No.2011BUPTYB02
文摘Investigation in this paper is given to the reduced Maxwell-Bloeh equations with variable coetcients, describing the propagation of the intense ultra-short optical pulses through an inhomogeneous two-level dielectric medium. We apply the Hirota method and symbolic computation to study such equations. With the help of the dependent variable transformations, we present the variable-coetteient-dependent bilinear forms. Then, we construct the one-, two- and N- soliton solutions in analytic forms for them.
基金Supported by Scientific Research Fund of Zhejiang Provincial Education Department under Grant No.20070979the National Natural Science Foundations of China under Grant Nos.10675065 and 10735030+1 种基金the Scientific Research Found of Ningbo University under Grant No.XKL09059the K.C.Wong Magana Fund in Ningbo University
文摘An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations withlogarithmic transformations is presented.In the algorithm,the general assumption of Hirota bilinear form is successfullyreduced based on the property of uniformity in rank.Furthermore,we discard the integral operation in the traditionalalgorithm.The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety solitonequations.
文摘In this paper, we proved a theorem on the orthogonal decomposition of a flatsymmetric bilinear form. If the nullity space of a flat symmetric bilinear formis not too large, it can be decomposed into two bilinear forms, one of which isnull, the other is flat and has a large nullity.
基金the Research Project No. 830104the Center of Excellence for Mathematics of the University of Isfahan for their financial supports
文摘The author gives a characterization of the Fourier transforms of bounded bilinear forms on C＊（S1）×C＊（S2） of two foundation semigroups S1 and S2 in terms of Jordan ＊-representations, hemimogeneous random fields, and as well as weakly harmonizable random fields of S1 and S2 into Hilbert spaces.
文摘A bilinear form f on a nonassociative triple system T is said to be invariant if and only if f( abc ,d) = f(a, dcb ) = f(c, bad ) for all a,b,c,d ∈ T . (T ,f) is called a pseudo-metric triple system if f is non-degenerate and invariant. A decomposition theory for triple systems and pseudo-metric triple systems is established. Moreover, the ?nite-dimensional metric Lie triple systems are characterized in terms of the structure of the non-degenerate, invariant and symmetric bilinear forms on them.
文摘Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field F.
基金supported by National Natural Science Foundation of China (Grant Nos. 12025106 and 11971370)。
文摘We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers the Polya-Vinogradov range,and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves.In the case of hyper-Kloosterman sums,we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski,Michel and Sawin(2017) on sum-products of Kloosterman sheaves.Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.
基金Supported by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 20080013006Chinese Ministry of Education, by the National Natural Science Foundation of China under Grant No. 60772023+2 种基金by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE-07-001Beijing University of Aeronautics and Astronauticsby the National Basic Research Program of China (973 Program) under Grant No. 2005CB321901
文摘In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plasma in three spatial dimensions. In order to study the integrability property of such an equation, the Painlevé analysis is performed on it. And then, based on the truncated Painlevé expansion, the bilinear form of the （3＋1）-dimensionaJ vcKP equation is obtained under certain coefficients constraint, and its solution in the Wronskian determinant form is constructed and verified by virtue of the Wronskian technique. Besides the Wronskian determinant solution, it is shown that the （3＋1）-dimensional vcKP equation also possesses a solution in the form of the Grammian determinant.
基金Supported by National Natural Science Foundation of China under Grant No. 10671121Shanghai Leading Academic Discipline Project under Grant No. J50101
文摘Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bgcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bgcklund transformation, we can give transformations among many kinds of solutions.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023 the Open Fund under Grant No.BUAASKLSDE-09KF-04l+2 种基金Supported Project under Grant No.SKLSDE-2010ZX-07 of the State Key Laboratory of Software Development Environment,Beijing University of Aeronautics and Astronauticsthe National Basic Research Program of China (973 Program) under Grant No.2005CB321901 the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.200800130006,Chinese Ministry of Education
文摘In this paper, two types of the （2＋1）-dimensional breaking soliton equations axe investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilineax forms and Bgcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilineax equations. Via the Wronskian technique, it is proved that the Bgcklund transformations obtained are the ones between the （ N - 1）- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonie properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.
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,the Ministry of Education
文摘By truncating the Painleve expansion at the constant level term,the Hirota bilinear form is obtainedfor a (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation.Based on its bilinear form,solitary-wavesolutions are constructed via the ε-expansion method and the corresponding graphical analysis is given.Furthermore,the exact solution in the Wronskian form is presented and proved by direct substitution into the bilinear equation.
基金National Natural Science Foundation of China under Grant Nos.10371070 and 10671121the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers
文摘In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation （mKdV） via the Backlund transformation （BT） and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the GN （t） for the original diagonal one.
基金partially supported by the Scientific and Technological Research Council of Turkey(TüBITAK)。
文摘To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations.There are also some other methods that are based on integrable scalar nonlinear partial differential equations.We show that some systems of integrable equations published recently are the M_(2)-extension of integrable such scalar equations.For illustration,we give Korteweg-de Vries,Kaup-Kupershmidt,and SawadaKotera equations as examples.By the use of such an extension of integrable scalar equations,we obtain some new integrable systems with recursion operators.We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.
