We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of ...We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.展开更多
This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipatio...This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.展开更多
By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic functio...By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.展开更多
One-dimensional generalized Boussinesq equation u tt-u xx＋（f（u）＋u xx）xx=0.with periodic boundary condition is considered, where f（u） = u3. First, the above equation is written as a Hamiltonian system, and then...One-dimensional generalized Boussinesq equation u tt-u xx＋（f（u）＋u xx）xx=0.with periodic boundary condition is considered, where f（u） = u3. First, the above equation is written as a Hamiltonian system, and then by choosing the eigenfunctions of the linear operator as bases, the Hamiltonian system in the coordinates is expressed. Because of the intricate resonance between the tangential frequencies and normal frequencies, some quasi-periodic solutions with special structures are considered. Secondly, the regularity of the Hamiltonian vector field is verified and then the fourth-order terms are normalized. By the Birkhoff normal form, the non- degeneracy and non-resonance conditions are obtained. Applying the infinite dimensional Kolmogorov-Arnold-Moser （KAM） theorem, the existence of finite dimensional invariant tori for the equivalent Hamiltonian system is proved. Hence many small-amplitude quasi-periodic solutions for the above equation are obtained.展开更多
The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton ...The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.展开更多
We study the Cauchy problem of damped generalized Boussinesq equation utt - uxx ＋ （uxx＋ f（u））xx - αuxxt = 0. First we give the local existence of weak solution and smooth solution. Then by using potential well ...We study the Cauchy problem of damped generalized Boussinesq equation utt - uxx ＋ （uxx＋ f（u））xx - αuxxt = 0. First we give the local existence of weak solution and smooth solution. Then by using potential well method and convexity method we prove the global existence and finite time blow up of solution, then we obtain some sharp conditions for the well-posedness problem.展开更多
In this paper,we obtained the topological soliton solution of the(1+1)-dimensional generalized modified Benjamin-Bona-Mahony equation and shock wave solution of the generalized Boussinesq equation.We get that solution...In this paper,we obtained the topological soliton solution of the(1+1)-dimensional generalized modified Benjamin-Bona-Mahony equation and shock wave solution of the generalized Boussinesq equation.We get that solutions by using solitary wave ansatz in terms of tanh^(p) functions.The velocity and the free parameters are the physical parameters in the soliton solutions.They can be obtained as functions of the dependent model coefficients.The domain restriction were also identified in the process.we hope that in nonlinear dynamical system these solutions will be explain some nonlinear physical phenomena.展开更多
This paper consider the Cauchy problem for a class of 1D generalized Boussinesq equations Utt-Uxx-Uxxtt＋Uxxxx＋Uxxxxtt=f（U）xx. By utilizing the potential well method and giving some conditions on f（u）, we obtain ...This paper consider the Cauchy problem for a class of 1D generalized Boussinesq equations Utt-Uxx-Uxxtt＋Uxxxx＋Uxxxxtt=f（U）xx. By utilizing the potential well method and giving some conditions on f（u）, we obtain the invariance of some sets and obtain the threshold result of global existence and nonexistence of solutions.展开更多
In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obt...In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.展开更多
In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the...In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.展开更多
By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalize...By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.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 generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transform...A generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and space.Using this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves,the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves,the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon.展开更多
We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry （GCS） method is applied in order to obtain the conditi...We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry （GCS） method is applied in order to obtain the conditions that enable the equation to admit a special class of second-order GCSs. For the case of quadratic nonlinearities, we outline a new class of invariant solutions.展开更多
文摘We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.
基金Project supported by the National Natural Science Foundation of China(No.11471215)。
文摘This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.
基金Project supported by the State Key Program for Basic Research of China (Grant No 2004CB418304)the National Natural Science Foundation of China (Grant No 40405010)
文摘By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.
基金The National Natural Science Foundation of China(No.11301072)the Natural Science Foundation of Jiangsu Province(No.BK20131285)the Research and Innovation Project for College Graduates of Jiangsu Province(No.CXZZ12-0083,CXLX13-074)
文摘One-dimensional generalized Boussinesq equation u tt-u xx＋（f（u）＋u xx）xx=0.with periodic boundary condition is considered, where f（u） = u3. First, the above equation is written as a Hamiltonian system, and then by choosing the eigenfunctions of the linear operator as bases, the Hamiltonian system in the coordinates is expressed. Because of the intricate resonance between the tangential frequencies and normal frequencies, some quasi-periodic solutions with special structures are considered. Secondly, the regularity of the Hamiltonian vector field is verified and then the fourth-order terms are normalized. By the Birkhoff normal form, the non- degeneracy and non-resonance conditions are obtained. Applying the infinite dimensional Kolmogorov-Arnold-Moser （KAM） theorem, the existence of finite dimensional invariant tori for the equivalent Hamiltonian system is proved. Hence many small-amplitude quasi-periodic solutions for the above equation are obtained.
基金Project supported by the National Natural Science Foundation of China (Nos.10572119,10772147,10632030)the Doctoral Program Foundation of Education Ministry of China (No.20070699028)+1 种基金the Natural Science Foundation of Shaanxi Province of China (No.2006A07)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment,Dalian University of Technology.
文摘The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.
基金Supported by the National Natural Science Foundation of China(11471087,11101102)Ph.D.Programs Foundation of Ministry of Education of China(20102304120022)+3 种基金the Support Plan for the Young College Academic Backbone of Heilongjiang Province(1252G020)the Natural Science Foundation of Heilongjiang Province(A201014)Science and Technology Research Project of Department of Education of Heilongjiang Province(12521401)Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities
文摘We study the Cauchy problem of damped generalized Boussinesq equation utt - uxx ＋ （uxx＋ f（u））xx - αuxxt = 0. First we give the local existence of weak solution and smooth solution. Then by using potential well method and convexity method we prove the global existence and finite time blow up of solution, then we obtain some sharp conditions for the well-posedness problem.
文摘In this paper,we obtained the topological soliton solution of the(1+1)-dimensional generalized modified Benjamin-Bona-Mahony equation and shock wave solution of the generalized Boussinesq equation.We get that solutions by using solitary wave ansatz in terms of tanh^(p) functions.The velocity and the free parameters are the physical parameters in the soliton solutions.They can be obtained as functions of the dependent model coefficients.The domain restriction were also identified in the process.we hope that in nonlinear dynamical system these solutions will be explain some nonlinear physical phenomena.
基金This work was supported by The National Natural Science Foundation （61503091）.
文摘This paper consider the Cauchy problem for a class of 1D generalized Boussinesq equations Utt-Uxx-Uxxtt＋Uxxxx＋Uxxxxtt=f（U）xx. By utilizing the potential well method and giving some conditions on f（u）, we obtain the invariance of some sets and obtain the threshold result of global existence and nonexistence of solutions.
文摘In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.
基金Supported by the National Natural Science Foundation of China under Grant No. 10974160the Scientific Research Foundation of the Education Department of Sichuan Province of China under Grant No. 10ZA004
文摘In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.
基金Supported by the Shanghai Leading Academic Discipline Project(No.T0502)the Science Foundation of the Education Commission of Shanghai(No.07ZZ83).
文摘By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.
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Project supported by the Shandong Provincial Key Laboratory of Marine Ecology and Environment and Disaster Prevention and Mitigation Project,China(Grant No.2012010)the National Natural Science Foundation of China(Grant Nos.41205082 and 41476019)+1 种基金the Special Funds for Theoretical Physics of the National Natural Science Foundation of China(Grant No.11447205)the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD),China
文摘A generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and space.Using this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves,the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves,the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon.
文摘We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry （GCS） method is applied in order to obtain the conditions that enable the equation to admit a special class of second-order GCSs. For the case of quadratic nonlinearities, we outline a new class of invariant solutions.
