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
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
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
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
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
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
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
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
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
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
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
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
Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the stand...Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.展开更多
We derive bilinear forms and Casoratian solutions for two semi-discrete potential Korteweg-de Vries equations. Their continuum limits go to the counterparts of the continuous potential Korteweg-de Vries equation.
With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic fun...With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.展开更多
Based on the Grammian and Pfaffian derivative formulae, Grammian and Pfaffian solutions are obtained for a （3＋1）-dimensional generalized shallow water equation in the Hirota bilinear form. Moreover, a Pfaffian exte...Based on the Grammian and Pfaffian derivative formulae, Grammian and Pfaffian solutions are obtained for a （3＋1）-dimensional generalized shallow water equation in the Hirota bilinear form. Moreover, a Pfaffian extension is made for the equation by means of the Pfaffianization procedure, the Wronski-type and Gramm-type Pfaffian solutions of the resulting coupled system are presented.展开更多
The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutio...The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions （including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions） for the new （2＋1）-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.展开更多
In this paper,we propose a combined form of the bilinear Kadomtsev-Petviashvili equation and the bilinear extended(2+1)-dimensional shallow water wave equation,which is linked with a novel(2+1)-dimensional nonlinear m...In this paper,we propose a combined form of the bilinear Kadomtsev-Petviashvili equation and the bilinear extended(2+1)-dimensional shallow water wave equation,which is linked with a novel(2+1)-dimensional nonlinear model.This model might be applied to describe the evolution of nonlinear waves in the ocean.Under the effect of a novel combination of nonlinearity and dispersion terms,two cases of lump solutions to the(2+1)-dimensional nonlinear model are derived by searching for the quadratic function solutions to the bilinear form.Moreover,the one-lump-multi-stripe solutions are constructed by the test function combining quadratic functions and multiple exponential functions.The one-lump-multi-soliton solutions are derived by the test function combining quadratic functions and multiple hyperbolic cosine functions.Dynamic behaviors of the lump solutions and mixed solutions are analyzed via numerical simulation.The result is of importance to provide efficient expressions to model nonlinear waves and explain some interaction mechanism of nonlinear waves in physics.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 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 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.
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.
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 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 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.
基金the National Natural Science Foundation of China(Grant Nos.11601312,11631007,and 11875040)
文摘Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.
基金Supported by National Natural Science Foundation of China under Grant No.11371241National Natural Science Foundation of Shanghai under Grant No.13ZR1416700
文摘We derive bilinear forms and Casoratian solutions for two semi-discrete potential Korteweg-de Vries equations. Their continuum limits go to the counterparts of the continuous potential Korteweg-de Vries equation.
基金Supported by National Natural Science Foundation of China under Grant Nos.11271211,11275072,and 11435005Ningbo Natural Science Foundation under Grant No.2015A610159+1 种基金the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No.xkzw11502K.C.Wong Magna Fund in Ningbo University
文摘With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10932009 and 11172233)the Northwestern Polytechnical University Foundation for Fundamental Research, China (Grant No. GBKY1034)the State Administration of Foreign Experts Affairs of China, and the Chunhui Plan of the Ministry of Education of China
文摘Based on the Grammian and Pfaffian derivative formulae, Grammian and Pfaffian solutions are obtained for a （3＋1）-dimensional generalized shallow water equation in the Hirota bilinear form. Moreover, a Pfaffian extension is made for the equation by means of the Pfaffianization procedure, the Wronski-type and Gramm-type Pfaffian solutions of the resulting coupled system are presented.
基金Supported by the Natural Science Foundation of China under Grant Nos.10361007,10661002Yunnan Natural Science Foundation under Grant No.2006A0082M
文摘The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions （including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions） for the new （2＋1）-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.
基金supported by the Project of the Fundamental Research Funds for the Central Universities of China(2022JBMC034)the National Natural Science Foundation of China under Grant No.12275017Beijing Laboratory of National Economic Security Early-warning Engineering,Beijing Jiaotong University
文摘In this paper,we propose a combined form of the bilinear Kadomtsev-Petviashvili equation and the bilinear extended(2+1)-dimensional shallow water wave equation,which is linked with a novel(2+1)-dimensional nonlinear model.This model might be applied to describe the evolution of nonlinear waves in the ocean.Under the effect of a novel combination of nonlinearity and dispersion terms,two cases of lump solutions to the(2+1)-dimensional nonlinear model are derived by searching for the quadratic function solutions to the bilinear form.Moreover,the one-lump-multi-stripe solutions are constructed by the test function combining quadratic functions and multiple exponential functions.The one-lump-multi-soliton solutions are derived by the test function combining quadratic functions and multiple hyperbolic cosine functions.Dynamic behaviors of the lump solutions and mixed solutions are analyzed via numerical simulation.The result is of importance to provide efficient expressions to model nonlinear waves and explain some interaction mechanism of nonlinear waves in physics.
