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
Utilizing some conservation laws of the(1+1)-dimensional Camassa–Holm(CH) equation and/or its reciprocal forms, some(n+1)-dimensional CH equations for n ≥ 1 are constructed by a modified deformation algorithm.The La...Utilizing some conservation laws of the(1+1)-dimensional Camassa–Holm(CH) equation and/or its reciprocal forms, some(n+1)-dimensional CH equations for n ≥ 1 are constructed by a modified deformation algorithm.The Lax integrability can be proven by applying the same deformation algorithm to the Lax pair of the(1+1)-dimensional CH equation. A novel type of peakon solution is implicitly given and expressed by the Lambert W function.展开更多
The new dimensional deformation approach is proposed to generate higher-dimensional analogues of integrable systems.An arbitrary(K+1)-dimensional integrable Korteweg-de Vries(Kd V)system,as an example,exhibiting symme...The new dimensional deformation approach is proposed to generate higher-dimensional analogues of integrable systems.An arbitrary(K+1)-dimensional integrable Korteweg-de Vries(Kd V)system,as an example,exhibiting symmetry,is illustrated to arise from a reconstructed deformation procedure,starting with a general symmetry integrable(1+1)-dimensional dark Kd V system and its conservation laws.Physically,the dark equation systems may be related to dark matter physics.To describe nonlinear physics,both linear and nonlinear dispersions should be considered.In the original lower-dimensional integrable systems,only liner or nonlinear dispersion is included.The deformation algorithm naturally makes the model also include the linear dispersion and nonlinear dispersion.展开更多
The weak Darboux transformation of the （2＋1） dimensional Euler equation is used to find its exact solutions. Starting from a constant velocity field solution, a set of quite general periodic wave solutions such as ...The weak Darboux transformation of the （2＋1） dimensional Euler equation is used to find its exact solutions. Starting from a constant velocity field solution, a set of quite general periodic wave solutions such as the Rossby waves can be simply obtained from the weak Darboux transformation with zero spectral parameters. The constant vorticity seed solution is utilized to generate Bessel waves.展开更多
Using the nonlinear Schrodinger(NLS)equation,which is used to describe the propagation of the solitons in many real physical systems like fiber and plasma,as a simple example,a direct perturbation method is establishe...Using the nonlinear Schrodinger(NLS)equation,which is used to describe the propagation of the solitons in many real physical systems like fiber and plasma,as a simple example,a direct perturbation method is established.Up to the adiabatic(zero order)approximation,any waves of the NLS equation decay in the same rate.Especially,different from the known claims in literature,the decay rate of the dark soliton in fiber is the same as that of the bright soliton.Starting from any one of the infinitely many adiabatic symmetries(or conservation laws)of the nonperturbative NLS equation,one can get the same adiabatic solutions.An adiabatic symmetry by multiplying a decay factor is just the first order modification.Higher order modifications can be obtained by solving linear equations.展开更多
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.
基金supported by the National Natural Science Foundation of China(Grant Nos.12235007,11975131,11435005,and 12275144)。
文摘Utilizing some conservation laws of the(1+1)-dimensional Camassa–Holm(CH) equation and/or its reciprocal forms, some(n+1)-dimensional CH equations for n ≥ 1 are constructed by a modified deformation algorithm.The Lax integrability can be proven by applying the same deformation algorithm to the Lax pair of the(1+1)-dimensional CH equation. A novel type of peakon solution is implicitly given and expressed by the Lambert W function.
基金supported by the National Natural Science Foundation of China(Grant Nos.12235007,12090020,11975131,12090025)。
文摘The new dimensional deformation approach is proposed to generate higher-dimensional analogues of integrable systems.An arbitrary(K+1)-dimensional integrable Korteweg-de Vries(Kd V)system,as an example,exhibiting symmetry,is illustrated to arise from a reconstructed deformation procedure,starting with a general symmetry integrable(1+1)-dimensional dark Kd V system and its conservation laws.Physically,the dark equation systems may be related to dark matter physics.To describe nonlinear physics,both linear and nonlinear dispersions should be considered.In the original lower-dimensional integrable systems,only liner or nonlinear dispersion is included.The deformation algorithm naturally makes the model also include the linear dispersion and nonlinear dispersion.
基金Supported by the National Natural Science Foundation of China under Grant Nos 90203001, 10475055 and 90503006.
文摘The weak Darboux transformation of the （2＋1） dimensional Euler equation is used to find its exact solutions. Starting from a constant velocity field solution, a set of quite general periodic wave solutions such as the Rossby waves can be simply obtained from the weak Darboux transformation with zero spectral parameters. The constant vorticity seed solution is utilized to generate Bessel waves.
基金the National Natural Science Foundation of China under Grant No.19975025“Scaling Plan”of China,and the Natural Science Foundation of Zhejiang Province.
文摘Using the nonlinear Schrodinger(NLS)equation,which is used to describe the propagation of the solitons in many real physical systems like fiber and plasma,as a simple example,a direct perturbation method is established.Up to the adiabatic(zero order)approximation,any waves of the NLS equation decay in the same rate.Especially,different from the known claims in literature,the decay rate of the dark soliton in fiber is the same as that of the bright soliton.Starting from any one of the infinitely many adiabatic symmetries(or conservation laws)of the nonperturbative NLS equation,one can get the same adiabatic solutions.An adiabatic symmetry by multiplying a decay factor is just the first order modification.Higher order modifications can be obtained by solving linear equations.
