The modified Kadomtsev-Petviashvili hierarchy of B type(mBKP hierarchy)and its constrained cases are investigated from the aspect of the Lax formulation.Starting from the Lax equation of the mBKP hierarchy,we firstly ...The modified Kadomtsev-Petviashvili hierarchy of B type(mBKP hierarchy)and its constrained cases are investigated from the aspect of the Lax formulation.Starting from the Lax equation of the mBKP hierarchy,we firstly derive the bilinear equations and show the existence of the tau functions.Then the additional symmetries are constructed,and the corresponding generator is showed to be the squared eigenfunction symmetry.After that,the actions of the additional symmetries on the tau functions are given.At last,the Lax formulation of the constrained mBKP hierarchy is investigated and the corresponding bilinear equations are also discussed.展开更多
Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the ...Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the velocity resonance mechanism.The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper.With particular selections of the involved arbitrary parameters,especially the wave numbers,it is confirmed that,besides the usual multi-bright soliton molecules,the multi-dark soliton molecules and the mixed multibright-dark soliton molecules can also be obtained.In addition,we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions.The results demonstrate that when N≥4,the combined mKdVtype bilinear equation no longer admits soliton molecules comprising more than four solitons.展开更多
A set of symmetries of a generalized (2+l)-dimensional bilinear equation is given by a formal serins formula. There exist four truncated symmetries for the KdV-Ito model. These trun-cated symmetries with four arourary...A set of symmetries of a generalized (2+l)-dimensional bilinear equation is given by a formal serins formula. There exist four truncated symmetries for the KdV-Ito model. These trun-cated symmetries with four arourary functions of time t constitute an infinnite-dirmensional Lin algebra which contains two types of the Virasoro subalgebra.展开更多
Kortweg-de Vries (KdV)-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV (GCCKdV) equations are investigated in this paper. Through the d...Kortweg-de Vries (KdV)-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV (GCCKdV) equations are investigated in this paper. Through the dependent variable transformations and symbolic computation, GCCKdV equations are transformed into their bilinear forms, based on which the one- and two-soliton solutions are obtained. Through the interactions of two solitons, the regular elastic collision are shown. When the wave numbers are complex, three kinds of solitonie collisions are presented: (i) two solitons merge and separate from each other periodically; (ii) two solitons exhibit the attraction and repulsion nearly twice, and finally separate from each other after such type of interaction; (iii) two solitons are ftuctuant in the central region of the collision. Propagation features of solitons are investigated with the effects of the coefficients in the GCCKdV equations considered. Velocity of soliton increase with the a increasing. Amplitude of v increase with the a increasing and decrease with the β increasing.展开更多
A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painl...A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.展开更多
It is common knowledge that the soliton solutions u(x, t) defined by the bell-shape form is required to satisfy the following condition lira u(x, t) = u(±∞, t) = 0. However, we think that the above conditi...It is common knowledge that the soliton solutions u(x, t) defined by the bell-shape form is required to satisfy the following condition lira u(x, t) = u(±∞, t) = 0. However, we think that the above condition can be modified as lim u(x, t) = u(±∞, t)^x→ = c, where c is a constant, which is called as a stationary height of u(x, t) in the present paper.^x→∞ If u(x, t) is a bell-shape solitary solution, then the stationary height of each solitary wave is just c. Under the constraint c = 0, all the solitary waves coming from the N-bell-shape-sollton solutions of the KdV equation are the same-oriented travelling. A new type of N-soliton solution with the bell shape is obtained in the paper, whose stationary height is an arbitrary constant c. Taking c ≥ 0, the resulting solitary wave is bound to be the same-oriented travelling. Otherwise, the resulting solitary wave may travel at the same orientation, and also at the opposite orientation. In addition, another type of singular rational travelling solution to the KdV equation is worked out.展开更多
A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance method.Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneousbala...A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance method.Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneousbalance method.Generalized Boussinesq equation,KP equation,and mKdV equation are chosen as examples to illustrateour method.This approach is also applicable to a large variety of nonlinear evolution equations.展开更多
A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show ...A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero.The particular lump solutions with specific values of the involved parameters are plotted,as illustrative examples.展开更多
In this paper,we construct the addition formulae for several integrable hierarchies,including the discrete KP,the q-deformed KP,the two-component BKP and the D type Drinfeld–Sokolov hierarchies.With the help of the H...In this paper,we construct the addition formulae for several integrable hierarchies,including the discrete KP,the q-deformed KP,the two-component BKP and the D type Drinfeld–Sokolov hierarchies.With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies,we prove that these addition formulae are equivalent to these hierarchies.These studies show that the addition formula in the research of the integrable systems has good universality.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12171472)。
文摘The modified Kadomtsev-Petviashvili hierarchy of B type(mBKP hierarchy)and its constrained cases are investigated from the aspect of the Lax formulation.Starting from the Lax equation of the mBKP hierarchy,we firstly derive the bilinear equations and show the existence of the tau functions.Then the additional symmetries are constructed,and the corresponding generator is showed to be the squared eigenfunction symmetry.After that,the actions of the additional symmetries on the tau functions are given.At last,the Lax formulation of the constrained mBKP hierarchy is investigated and the corresponding bilinear equations are also discussed.
基金the National Natural Science Foundation of China(Grant Nos.11975204 and 12075208)the Project of Zhoushan City Science and Technology Bureau(Grant No.2021C21015)the Training Program for Leading Talents in Universities of Zhejiang Province。
文摘Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the velocity resonance mechanism.The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper.With particular selections of the involved arbitrary parameters,especially the wave numbers,it is confirmed that,besides the usual multi-bright soliton molecules,the multi-dark soliton molecules and the mixed multibright-dark soliton molecules can also be obtained.In addition,we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions.The results demonstrate that when N≥4,the combined mKdVtype bilinear equation no longer admits soliton molecules comprising more than four solitons.
文摘A set of symmetries of a generalized (2+l)-dimensional bilinear equation is given by a formal serins formula. There exist four truncated symmetries for the KdV-Ito model. These trun-cated symmetries with four arourary functions of time t constitute an infinnite-dirmensional Lin algebra which contains two types of the Virasoro subalgebra.
基金*Supported by the National Natural Science Foundation of China under Grant No. 60772023, by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE-07-001, Beijing University of Aeronautics and Astronautics, by the National Basic Research Program of China (973 Program) under Grant No. 2005CB321901, and by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 200800130006, Chinese Ministry of Education.
文摘Kortweg-de Vries (KdV)-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV (GCCKdV) equations are investigated in this paper. Through the dependent variable transformations and symbolic computation, GCCKdV equations are transformed into their bilinear forms, based on which the one- and two-soliton solutions are obtained. Through the interactions of two solitons, the regular elastic collision are shown. When the wave numbers are complex, three kinds of solitonie collisions are presented: (i) two solitons merge and separate from each other periodically; (ii) two solitons exhibit the attraction and repulsion nearly twice, and finally separate from each other after such type of interaction; (iii) two solitons are ftuctuant in the central region of the collision. Propagation features of solitons are investigated with the effects of the coefficients in the GCCKdV equations considered. Velocity of soliton increase with the a increasing. Amplitude of v increase with the a increasing and decrease with the β increasing.
基金supported by Chinese National Social Science Foundation(Grant Number:CNSSF:13CJY037)Research on the indemnificatory Apartment Construction Based on Residential Integration.
文摘A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.
文摘It is common knowledge that the soliton solutions u(x, t) defined by the bell-shape form is required to satisfy the following condition lira u(x, t) = u(±∞, t) = 0. However, we think that the above condition can be modified as lim u(x, t) = u(±∞, t)^x→ = c, where c is a constant, which is called as a stationary height of u(x, t) in the present paper.^x→∞ If u(x, t) is a bell-shape solitary solution, then the stationary height of each solitary wave is just c. Under the constraint c = 0, all the solitary waves coming from the N-bell-shape-sollton solutions of the KdV equation are the same-oriented travelling. A new type of N-soliton solution with the bell shape is obtained in the paper, whose stationary height is an arbitrary constant c. Taking c ≥ 0, the resulting solitary wave is bound to be the same-oriented travelling. Otherwise, the resulting solitary wave may travel at the same orientation, and also at the opposite orientation. In addition, another type of singular rational travelling solution to the KdV equation is worked out.
基金Supported by the National Natural Science Foundation of China under Grant No. 11071209the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province under Grant No. 10KJBll0011
文摘A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance method.Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneousbalance method.Generalized Boussinesq equation,KP equation,and mKdV equation are chosen as examples to illustrateour method.This approach is also applicable to a large variety of nonlinear evolution equations.
基金Supported by the National Natural Science Foundation of China under Grant No.10647112the Fund of Science and Technology Commission of Shanghai Municipality under Grant No.ZX201307000014
文摘A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero.The particular lump solutions with specific values of the involved parameters are plotted,as illustrative examples.
基金Supported by the Zhejiang Provincial Natural Science Foundation under Grant No.LY15A010004the National Natural Science Foundation of China under Grant Nos.11201251,11571192+2 种基金the Natural Science Foundation of Ningbo under Grant No.2015A610157supported by the National Natural Science Foundation of China under Grant No.11271210K.C.Wong Magna Fund in Ningbo University
文摘In this paper,we construct the addition formulae for several integrable hierarchies,including the discrete KP,the q-deformed KP,the two-component BKP and the D type Drinfeld–Sokolov hierarchies.With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies,we prove that these addition formulae are equivalent to these hierarchies.These studies show that the addition formula in the research of the integrable systems has good universality.