In this paper, a variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann the...In this paper, a variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.展开更多
In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:"">...In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different </span><span style="font-size:10.0pt;font-family:"">coefficient</span><span style="font-size:10.0pt;font-family:"">s</span><span style="font-size:10pt;font-family:""> <i>p</i>, <i>q</i> and <i>r</i> in the</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">elliptic equation. Then these solutions are</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">coupled into an auxiliary equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">and substituted into the (2+1)-dimensional KDV equation. As <span>a result,</span></span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">a large number of complex Jacobi elliptic function solutions are ob</span><span style="font-size:10pt;font-family:"">tained, and many of them have not been found in other documents. As</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10.0pt;font-family:""><span></span></span><span style="font-size:10pt;font-family:"">, some complex solitary solutions are also obtained correspondingly.</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.展开更多
The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behavi...The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly.展开更多
This paper systematically studies the complete integrability of the Newell equation. Using generalized Bell polynomials, the corresponding bilinear equation, bilinear Bäcklund transformation, Lax pair, and mu...This paper systematically studies the complete integrability of the Newell equation. Using generalized Bell polynomials, the corresponding bilinear equation, bilinear Bäcklund transformation, Lax pair, and multi-shock wave solutions are successfully obtained. In addition, using the multidimensional Riemann theta functions, the periodic wave solutions of the Newell equation are constructed. On this basis, the asymptotic behavior of the periodic wave solution is given, which is the relationship between the periodic wave solution and the solitary wave solution.展开更多
In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solu...In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.展开更多
In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for (2+1)-dimensional dispersive long wave equations in mathematical physics, which are...In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for (2+1)-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions (m → 1). This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.展开更多
In this paper, for b ∈ (-∞,∞) and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut + 2kux - uxxt + (1 + b)u2ux =buxuxx + uuxxx, which contains the generaliz...In this paper, for b ∈ (-∞,∞) and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut + 2kux - uxxt + (1 + b)u2ux =buxuxx + uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.展开更多
In this paper, we consider the following nonlinear equation ut+2kux-uxxt+au^2ux=2uxuxx+uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcat...In this paper, we consider the following nonlinear equation ut+2kux-uxxt+au^2ux=2uxuxx+uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcation method of dynamical systems. We also obtain two explicit solitary wave solutions via the limits of the explicit periodic wave solutions. One of the two solitary wave solutions is new.展开更多
In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability prope...In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover,several new solutions have been constructed for the gvcmKdV.Additionally,the classical direct similarity reduction method is used to re-duce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation,many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.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
In this paper. a new kind of generalized BBM equation is introduced and discussed Some existence theorems of periodic traveling wave solutions for this kind generalized BBM equation are given.
By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are al...By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are also presented.展开更多
In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling...In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.展开更多
In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under d...In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.展开更多
By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parame...By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.展开更多
By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave...By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.展开更多
The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type. In different regions of the parametric space, sufficient conditi...The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type. In different regions of the parametric space, sufficient conditions to guarantee the existence of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions are given. All possible exact explicit parametric representations are obtained for these waves.展开更多
This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are ...This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.展开更多
By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-sm...By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-smooth periodic waves. Under the given parametric conditions, we present the sufficient conditions to guarantee the existence of the above solutions.展开更多
By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations, the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travellin...By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations, the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown. Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10771196 and 10831003)the Innovation Project of Zhejiang Province of China(Grant No.T200905)
文摘In this paper, a variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.
文摘In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different </span><span style="font-size:10.0pt;font-family:"">coefficient</span><span style="font-size:10.0pt;font-family:"">s</span><span style="font-size:10pt;font-family:""> <i>p</i>, <i>q</i> and <i>r</i> in the</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">elliptic equation. Then these solutions are</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">coupled into an auxiliary equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">and substituted into the (2+1)-dimensional KDV equation. As <span>a result,</span></span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">a large number of complex Jacobi elliptic function solutions are ob</span><span style="font-size:10pt;font-family:"">tained, and many of them have not been found in other documents. As</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10.0pt;font-family:""><span></span></span><span style="font-size:10pt;font-family:"">, some complex solitary solutions are also obtained correspondingly.</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.
基金supported by the National Natural Science Foundation of China (Grant No.11461022)。
文摘The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly.
文摘This paper systematically studies the complete integrability of the Newell equation. Using generalized Bell polynomials, the corresponding bilinear equation, bilinear Bäcklund transformation, Lax pair, and multi-shock wave solutions are successfully obtained. In addition, using the multidimensional Riemann theta functions, the periodic wave solutions of the Newell equation are constructed. On this basis, the asymptotic behavior of the periodic wave solution is given, which is the relationship between the periodic wave solution and the solitary wave solution.
基金Supported by National Natural Science Foundation of China(Grant Nos.11361069 and 11171115)
文摘In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.
基金Project supported by the Anhui Key Laboratory of Information Materials and Devices (Anhui University),China
文摘In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for (2+1)-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions (m → 1). This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.
基金Supported by the National Natural Science Foundation of China(No.11401222)Natural Science Foundation of Guangdong Province(No.S2012040007959)+1 种基金The Fundamental Research Funds for the Central Universities(No.2014ZZ0064)Pearl River Science and Technology Nova Program of Guangzhou
文摘In this paper, for b ∈ (-∞,∞) and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut + 2kux - uxxt + (1 + b)u2ux =buxuxx + uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.
基金Supported by the National Natural Science Foundation of China (No. 10871073)Guangdong Province(No.07006552)
文摘In this paper, we consider the following nonlinear equation ut+2kux-uxxt+au^2ux=2uxuxx+uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcation method of dynamical systems. We also obtain two explicit solitary wave solutions via the limits of the explicit periodic wave solutions. One of the two solitary wave solutions is new.
基金The author would like to thank the Deanship of Scientific Re-search,Majmaah University,Saudi Arabia,for funding this work under project No.R-2021-222.
文摘In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover,several new solutions have been constructed for the gvcmKdV.Additionally,the classical direct similarity reduction method is used to re-duce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation,many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.
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.
文摘In this paper. a new kind of generalized BBM equation is introduced and discussed Some existence theorems of periodic traveling wave solutions for this kind generalized BBM equation are given.
基金Supported by National Nature Science Foundation of China under Grant No.11102076Natural Science Fund for Colleges and Universities in Jiangsu Province under Grant No.15KJB110005
文摘By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are also presented.
文摘In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.
基金Supported by the NNSF of China(60464001) Guangxi Science Foundation(0575092).
文摘In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.
基金the National Natural Science Foundation of China(Nos.10671179 and 10772158)
文摘By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.
基金Project supported by the National Natural Science Foundation of China(Nos.10671179,10772158)
文摘By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.
基金Project supported by the National Natural Science Foundation of China (No.10231020)the Natural Science Foundation of Yunnan Province of China (No.2003A0018M)Key Project of the Science Foundation of Yunnan Education Department of China (No.5Z0071A)
文摘The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type. In different regions of the parametric space, sufficient conditions to guarantee the existence of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions are given. All possible exact explicit parametric representations are obtained for these waves.
基金supported by the Natural Science Foundation of Ningbo City of China(No.2008A610029)
文摘This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.
基金supported by the National Natural Science Foundation of China (No. 10971085)
文摘By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-smooth periodic waves. Under the given parametric conditions, we present the sufficient conditions to guarantee the existence of the above solutions.
文摘By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations, the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown. Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.