An infinite number of semi-discrete and continuous conservation laws for the differential-difference KP equation were obtained by using a solvable generalized Riccati equation.
New type of variable-coefficient KP equation with self-consistent sources and its Grammian solutions are obtained by using the source generation procedure.
By means of the auxiliary ordinary differential equation method,we have obtained many solitary wave solutions,periodic wave solutions and variable separation solutions for the (2+1)-dimensional KP equation.Using a mix...By means of the auxiliary ordinary differential equation method,we have obtained many solitary wave solutions,periodic wave solutions and variable separation solutions for the (2+1)-dimensional KP equation.Using a mixed method,many exact solutions have been obtained.展开更多
Starting with a decomposition conjecture,we carefully explain the basic decompositions for the Kadomtsev-Petviashvili(KP)equation as well as the necessary calculation procedures,and it is shown that the KP equation al...Starting with a decomposition conjecture,we carefully explain the basic decompositions for the Kadomtsev-Petviashvili(KP)equation as well as the necessary calculation procedures,and it is shown that the KP equation allows the Burgers-STO(BSTO)decomposition,two types of reducible coupled BSTO decompositions and the BSTO-KdV decomposition.Furthermore,we concentrate ourselves on pointing out the main idea and result of Bäcklund transformation of the KP equation based on a special superposition principle in the particular context of the BSTO decompositions.Using the framework of standard Lie point symmetry theory,these decompositions are studied and the problem of computing the corresponding symmetry constraints is treated.展开更多
Using the solution of general Korteweg-de Vries (KdV) equation, the solutions of the generalized variable coefficient Kadomtsev-Petviashvili (KP) equation are constructed, and then its new solitary wave-like solut...Using the solution of general Korteweg-de Vries (KdV) equation, the solutions of the generalized variable coefficient Kadomtsev-Petviashvili (KP) equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.展开更多
The novel multisoliton solutions for the nonlinear lumped self-dual network equations, Toda lattice and KP equation were obtained by using the Hirota direct method.
This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili (KP) equation. By using the Hirota method and Pfaffian technique, it...This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili (KP) equation. By using the Hirota method and Pfaffian technique, it finds the Grammian determinant-type solution for the variable-coefficient KP equation (VCKP), the Wronski-type Pfaffian solution and the Gram-type Pfaffian solutions for the Pfaffianized VCKP equation.展开更多
In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili (cKP) equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structur...In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili (cKP) equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.展开更多
The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painle...The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlevéexpansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE)solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.展开更多
On the bases of N-soliton solutions of Hirota’s bilinear method,high-order rogue wave solutions can be derived by a direct limit method.In this paper,a(3+1)-dimensional Kadomtsev-Petviashvili equation is taken to ill...On the bases of N-soliton solutions of Hirota’s bilinear method,high-order rogue wave solutions can be derived by a direct limit method.In this paper,a(3+1)-dimensional Kadomtsev-Petviashvili equation is taken to illustrate the process of obtaining rogue waves,that is,based on the long-wave limit method,rogue wave solutions are generated by reconstructing the phase parameters of N-solitons.Besides the fundamental pattern of rogue waves,the triangle or pentagon patterns are also obtained.Moreover,the different patterns of these solutions are determined by newly introduced parameters.In the end,the general form of N-order rogue wave solutions are proposed.展开更多
The multiple patterns of internal solitary wave interactions(ISWI)are a complex oceanic phenomenon.Satellite remote sensing techniques indirectly detect these ISWI,but do not provide information on their detailed stru...The multiple patterns of internal solitary wave interactions(ISWI)are a complex oceanic phenomenon.Satellite remote sensing techniques indirectly detect these ISWI,but do not provide information on their detailed structure and dynamics.Recently,the authors considered a three-layer fluid with shear flow and developed a(2+1)Kadomtsev-Petviashvili(KP)model that is capable of describing five types of oceanic ISWI,including O-type,P-type,TO-type,TP-type,and Y-shaped.Deep learning models,particularly physics-informed neural networks(PINN),are widely used in the field of fluids and internal solitary waves.However,the authors find that the amplitude of internal solitary waves is much smaller than the wavelength and the ISWI occur at relatively large spatial scales,and these characteristics lead to an imbalance in the loss function of the PINN model.To solve this problem,the authors introduce two weighted loss function methods,the fixed weighing and the adaptive weighting methods,to improve the PINN model.This successfully simulated the detailed structure and dynamics of ISWI,with simulation results corresponding to the satellite images.In particular,the adaptive weighting method can automatically update the weights of different terms in the loss function and outperforms the fixed weighting method in terms of generalization ability.展开更多
The extended Jacobian elliptic function expansion method is introduced and applied to solve the coupled ZK equations and the coupled KP equations describing two weakly long nonlinear wave models in fluid system. Many ...The extended Jacobian elliptic function expansion method is introduced and applied to solve the coupled ZK equations and the coupled KP equations describing two weakly long nonlinear wave models in fluid system. Many types of doubly periodic traveling wave solutions are obtained. Under limiting conditions these solutions are reduced into solitary wave solutions.展开更多
Most fundamental themes in mathematical physics and modern engineering are investigated by the closed form traveling wave solutions of nonlinear evolution equations.In our research,we ascertain abundant new closed for...Most fundamental themes in mathematical physics and modern engineering are investigated by the closed form traveling wave solutions of nonlinear evolution equations.In our research,we ascertain abundant new closed form traveling wave solution of the nonlinear integro-differential equations via Ito equation,integro-differential Sawada-Kotera equation,first integro-differential KP hierarchy equation and second integro-differential KP hierarchy equation by two variable(G/G,1/G)-expansion method with the help of computer package like Mathematica.Some shape of solutions like,bell profile solution,anti-king profile solution,soliton profile solution,periodic profile solution etc.are obtain in this investigation.Trigonometric function solution,hyperbolic function solution and rational function solution are established by using our eminent method and comparing with our results to all of the well-known results which are given in the literature.By means of free parameters,plentiful solitary solutions are derived from the exact traveling wave solutions.The method can be easier and more applicable to investigate such type of nonlinear evolution models.展开更多
We propose a systematic method to construct the Mel’nikov model of long–short wave interactions,which is a special case of the Kadomtsev–Petviashvili(KP)equation with self-consistent sources(KPSCS).We show details ...We propose a systematic method to construct the Mel’nikov model of long–short wave interactions,which is a special case of the Kadomtsev–Petviashvili(KP)equation with self-consistent sources(KPSCS).We show details how the Cauchy matrix approach applies to Mel’nikov’s model which is derived as a complex reduction of the KPSCS.As a new result wefind that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel’nikov model.This function brings time variant velocity for the long wave and also governs the short-wave packet.The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.展开更多
文摘An infinite number of semi-discrete and continuous conservation laws for the differential-difference KP equation were obtained by using a solvable generalized Riccati equation.
基金Supported by the NSF of Henan Province(112300410109)Supported by the NSF of the Education Department(2010A110022)
文摘New type of variable-coefficient KP equation with self-consistent sources and its Grammian solutions are obtained by using the source generation procedure.
基金supported by the National Natural Science Foundation of China(10672053)
文摘By means of the auxiliary ordinary differential equation method,we have obtained many solitary wave solutions,periodic wave solutions and variable separation solutions for the (2+1)-dimensional KP equation.Using a mixed method,many exact solutions have been obtained.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 12235007, 11975131, and 12275144)the K. C. Wong Magna Fund in Ningbo Universitythe Natural Science Foundation of Zhejiang Province of China (Grant No. LQ20A010009)
文摘Starting with a decomposition conjecture,we carefully explain the basic decompositions for the Kadomtsev-Petviashvili(KP)equation as well as the necessary calculation procedures,and it is shown that the KP equation allows the Burgers-STO(BSTO)decomposition,two types of reducible coupled BSTO decompositions and the BSTO-KdV decomposition.Furthermore,we concentrate ourselves on pointing out the main idea and result of Bäcklund transformation of the KP equation based on a special superposition principle in the particular context of the BSTO decompositions.Using the framework of standard Lie point symmetry theory,these decompositions are studied and the problem of computing the corresponding symmetry constraints is treated.
文摘Using the solution of general Korteweg-de Vries (KdV) equation, the solutions of the generalized variable coefficient Kadomtsev-Petviashvili (KP) equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.
文摘The novel multisoliton solutions for the nonlinear lumped self-dual network equations, Toda lattice and KP equation were obtained by using the Hirota direct method.
基金Project supported by the National Key Basic Research Project of China (2004CB318000), the National Science Foundation of China (Grant No 10371023) and Shanghai Shuguang Project of China (Grant No 02SG02).
文摘This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili (KP) equation. By using the Hirota method and Pfaffian technique, it finds the Grammian determinant-type solution for the variable-coefficient KP equation (VCKP), the Wronski-type Pfaffian solution and the Gram-type Pfaffian solutions for the Pfaffianized VCKP equation.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10747141 and 10735030)National Basic Research Program of China (Grant No 2007CB814800)+2 种基金Natural Science Foundations of Zhejiang Province of China (Grant No605408)Ningbo Natural Science Foundation (Grant Nos 2007A610049 and 2008A610017)K. C.Wong Magna Fund in Ningbo University
文摘In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili (cKP) equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11775047,11775146,and 11865013)the Science and Technology Project Foundation of Zhongshan City,China(Grant No.2017B1016).
文摘The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlevéexpansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE)solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.
基金supported by the National Natural Science Foundation of China under Grant Nos.12175111,12235007 and 11975131K.C.Wong Magna Fund in Ningbo University
文摘On the bases of N-soliton solutions of Hirota’s bilinear method,high-order rogue wave solutions can be derived by a direct limit method.In this paper,a(3+1)-dimensional Kadomtsev-Petviashvili equation is taken to illustrate the process of obtaining rogue waves,that is,based on the long-wave limit method,rogue wave solutions are generated by reconstructing the phase parameters of N-solitons.Besides the fundamental pattern of rogue waves,the triangle or pentagon patterns are also obtained.Moreover,the different patterns of these solutions are determined by newly introduced parameters.In the end,the general form of N-order rogue wave solutions are proposed.
基金supported by the National Natural Science Foundation of China under Grant Nos.12275085,12235007,and 12175069Science and Technology Commission of Shanghai Municipality under Grant Nos.21JC1402500 and 22DZ2229014.
文摘The multiple patterns of internal solitary wave interactions(ISWI)are a complex oceanic phenomenon.Satellite remote sensing techniques indirectly detect these ISWI,but do not provide information on their detailed structure and dynamics.Recently,the authors considered a three-layer fluid with shear flow and developed a(2+1)Kadomtsev-Petviashvili(KP)model that is capable of describing five types of oceanic ISWI,including O-type,P-type,TO-type,TP-type,and Y-shaped.Deep learning models,particularly physics-informed neural networks(PINN),are widely used in the field of fluids and internal solitary waves.However,the authors find that the amplitude of internal solitary waves is much smaller than the wavelength and the ISWI occur at relatively large spatial scales,and these characteristics lead to an imbalance in the loss function of the PINN model.To solve this problem,the authors introduce two weighted loss function methods,the fixed weighing and the adaptive weighting methods,to improve the PINN model.This successfully simulated the detailed structure and dynamics of ISWI,with simulation results corresponding to the satellite images.In particular,the adaptive weighting method can automatically update the weights of different terms in the loss function and outperforms the fixed weighting method in terms of generalization ability.
基金Project supported by the National Natural Science Foundation of China (Grant No.10272071)
文摘The extended Jacobian elliptic function expansion method is introduced and applied to solve the coupled ZK equations and the coupled KP equations describing two weakly long nonlinear wave models in fluid system. Many types of doubly periodic traveling wave solutions are obtained. Under limiting conditions these solutions are reduced into solitary wave solutions.
文摘Most fundamental themes in mathematical physics and modern engineering are investigated by the closed form traveling wave solutions of nonlinear evolution equations.In our research,we ascertain abundant new closed form traveling wave solution of the nonlinear integro-differential equations via Ito equation,integro-differential Sawada-Kotera equation,first integro-differential KP hierarchy equation and second integro-differential KP hierarchy equation by two variable(G/G,1/G)-expansion method with the help of computer package like Mathematica.Some shape of solutions like,bell profile solution,anti-king profile solution,soliton profile solution,periodic profile solution etc.are obtain in this investigation.Trigonometric function solution,hyperbolic function solution and rational function solution are established by using our eminent method and comparing with our results to all of the well-known results which are given in the literature.By means of free parameters,plentiful solitary solutions are derived from the exact traveling wave solutions.The method can be easier and more applicable to investigate such type of nonlinear evolution models.
基金supported by the NSF of China(Nos.11875040 and 11631007)。
文摘We propose a systematic method to construct the Mel’nikov model of long–short wave interactions,which is a special case of the Kadomtsev–Petviashvili(KP)equation with self-consistent sources(KPSCS).We show details how the Cauchy matrix approach applies to Mel’nikov’s model which is derived as a complex reduction of the KPSCS.As a new result wefind that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel’nikov model.This function brings time variant velocity for the long wave and also governs the short-wave packet.The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.