First, two tanh-coth type solutions of a class of nonlinear wave equation are derived by using a simplified modified extended tanh-function method. Then, further analysis to some tanh-coth type solutions of nonlinear ...First, two tanh-coth type solutions of a class of nonlinear wave equation are derived by using a simplified modified extended tanh-function method. Then, further analysis to some tanh-coth type solutions of nonlinear evolution equations are given. The results show that when balance number m is one or two, the tanh-eoth type solutions obtained by the modified extended tanh-funetion method can be obtained by using the hyperbolic-function method.展开更多
In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its app...In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the system of shallow water wave equations and modified Liouville equation which play an important role in mathematical physics.展开更多
In this paper, a new extended complex tanh-function method is presented for constructing traveling wave, non-traveling wave, and coefficient functions＇ soliton-like solutions of nonlinear equations. This method is mo...In this paper, a new extended complex tanh-function method is presented for constructing traveling wave, non-traveling wave, and coefficient functions＇ soliton-like solutions of nonlinear equations. This method is more powerful than the complex tanh-function method [Chaos, Solitons and Fractals 20 （2004） 1037]. Abundant new solutions o[ （2q-1）-dimensional Hirota equation are obtained by using this method and symbolic computation system Maple.展开更多
In this paper, the Auto-B?cklund transformation connected with the homogeneous balance method (HB) and the extended tanh-function method are used to construct new exact solutions for the time-dependent coefficients Ca...In this paper, the Auto-B?cklund transformation connected with the homogeneous balance method (HB) and the extended tanh-function method are used to construct new exact solutions for the time-dependent coefficients Calogero-Degasperis (VCCD) equation. New soliton and periodic solutions of many types are obtained. Furthermore, the soliton propagation is discussed under the effect of the variable coefficients.展开更多
In this paper, we investigate some new traveling wave solutions to Vakhnenko-Parkes equation via three modified mathematical methods. The derived solutions have been obtained including periodic and solitons solutions ...In this paper, we investigate some new traveling wave solutions to Vakhnenko-Parkes equation via three modified mathematical methods. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic, and rational function solutions. The graphical representations of some solutions by assigning particular values to the parameters under prescribed conditions in each solutions and comparing of solutions with those gained by other authors indicate that these employed techniques are more effective, efficient and applicable mathematical tools for solving nonlinear problems in applied science.展开更多
The(3+1)-dimensional Zakharov–Kuznetsov(ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy,ion plasma, quantum acoustic wave...The(3+1)-dimensional Zakharov–Kuznetsov(ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy,ion plasma, quantum acoustic waves, and quantum Langmuir waves. The enhanced modified simple equation(EMSE) method is a substantial approach to determine competent solutions and in this article, we have constructed standard, illustrative, rich structured and further comprehensive soliton solutions via this method. The solutions are ascertained as the integration of exponential, hyperbolic,trigonometric and rational functions and formulate the bright solitons, periodic, compacton, bellshape, parabolic shape, singular periodic, plane shape and some new type of solitons. It is worth noting that the wave profile varies as the physical and subsidiary parameters change. The standard and advanced soliton solutions may be useful to assist in describing the physical phenomena previously mentioned. To open out the inward structure of the tangible incidents, we have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the EMSE technique for extracting soliton solutions to nonlinear evolution equations is efficient, compatible and reliable in nonlinear science and engineering.展开更多
In this paper, the modified extended tanh method is used to construct more general exact solutions of a （2＋1）-dimensional nonlinear SchrSdinger equation. With the aid of Maple and Matlab software, we obtain exact e...In this paper, the modified extended tanh method is used to construct more general exact solutions of a （2＋1）-dimensional nonlinear SchrSdinger equation. With the aid of Maple and Matlab software, we obtain exact explicit kink wave solutions, peakon wave solutions, periodic wave solutions and their 3D images.展开更多
Using the modified extended tanh-function method,explicit and exact traveling wave solutions for the(2+1)-dimensional higher-order Broer-Kaup(HBK)system,comprising new soliton-like and period-form solutions,are obtained.
Using a further modified extended tanh-function method, rich new families of the exact solutions for the (2+ 1)-dimensional Broer-Kaup (BK) system, comprising the non-traveling wave and coefficient functions' soli...Using a further modified extended tanh-function method, rich new families of the exact solutions for the (2+ 1)-dimensional Broer-Kaup (BK) system, comprising the non-traveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained.展开更多
Starting from the extended tanh-function method （ETM） based on the mapping method, the variable separation solutions of the （2＋1）-dimensional asymmetric Nizhnik Novikov Veselov （ANNV） system are derived. By fur...Starting from the extended tanh-function method （ETM） based on the mapping method, the variable separation solutions of the （2＋1）-dimensional asymmetric Nizhnik Novikov Veselov （ANNV） system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semifoldon, are investigated.展开更多
In the present study,the solitary wave solutions of modified Degasperis-Procesi equation are developed.Unlike the standard Degasperis-Procesi equation,where multi-peakon solutions arise,the modification caused a chang...In the present study,the solitary wave solutions of modified Degasperis-Procesi equation are developed.Unlike the standard Degasperis-Procesi equation,where multi-peakon solutions arise,the modification caused a change in the characteristic of these peakon solutions and changed it to bell-shaped solitons.By using the extended auxiliary equation method,we deduced some new soliton solutions of the fourthorder nonlinear modified Degasperis-Procesi equation with constant coefficient.These solutions include symmetrical,non-symmetrical kink solutions,solitary pattern solutions,weiestrass elliptic function solutions and triangular function solutions.We discuss the stability analysis for these solutions.展开更多
The searching exact solutions in the solitary wave form of non-linear partial differential equations （PDEs） play a significant role to understand the internal mechanism of complex physical phenomena. In this paper w...The searching exact solutions in the solitary wave form of non-linear partial differential equations （PDEs） play a significant role to understand the internal mechanism of complex physical phenomena. In this paper we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the （2-1-1）-dimensional cubic Klein-Gordon （K-G） equation. The Klein-Gordon equations are relativistic version of Schr6dinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which several solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions of PDEs arise in mathematical physics.展开更多
基金Supported by the Natural Science Foundation of China under Grant No.11071209
文摘First, two tanh-coth type solutions of a class of nonlinear wave equation are derived by using a simplified modified extended tanh-function method. Then, further analysis to some tanh-coth type solutions of nonlinear evolution equations are given. The results show that when balance number m is one or two, the tanh-eoth type solutions obtained by the modified extended tanh-funetion method can be obtained by using the hyperbolic-function method.
文摘In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the system of shallow water wave equations and modified Liouville equation which play an important role in mathematical physics.
基金The project supported by National Natural Science Foundation of China and the Natural Science Foundation of Shandong Province of China
文摘In this paper, a new extended complex tanh-function method is presented for constructing traveling wave, non-traveling wave, and coefficient functions＇ soliton-like solutions of nonlinear equations. This method is more powerful than the complex tanh-function method [Chaos, Solitons and Fractals 20 （2004） 1037]. Abundant new solutions o[ （2q-1）-dimensional Hirota equation are obtained by using this method and symbolic computation system Maple.
文摘In this paper, the Auto-B?cklund transformation connected with the homogeneous balance method (HB) and the extended tanh-function method are used to construct new exact solutions for the time-dependent coefficients Calogero-Degasperis (VCCD) equation. New soliton and periodic solutions of many types are obtained. Furthermore, the soliton propagation is discussed under the effect of the variable coefficients.
文摘In this paper, we investigate some new traveling wave solutions to Vakhnenko-Parkes equation via three modified mathematical methods. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic, and rational function solutions. The graphical representations of some solutions by assigning particular values to the parameters under prescribed conditions in each solutions and comparing of solutions with those gained by other authors indicate that these employed techniques are more effective, efficient and applicable mathematical tools for solving nonlinear problems in applied science.
文摘The(3+1)-dimensional Zakharov–Kuznetsov(ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy,ion plasma, quantum acoustic waves, and quantum Langmuir waves. The enhanced modified simple equation(EMSE) method is a substantial approach to determine competent solutions and in this article, we have constructed standard, illustrative, rich structured and further comprehensive soliton solutions via this method. The solutions are ascertained as the integration of exponential, hyperbolic,trigonometric and rational functions and formulate the bright solitons, periodic, compacton, bellshape, parabolic shape, singular periodic, plane shape and some new type of solitons. It is worth noting that the wave profile varies as the physical and subsidiary parameters change. The standard and advanced soliton solutions may be useful to assist in describing the physical phenomena previously mentioned. To open out the inward structure of the tangible incidents, we have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the EMSE technique for extracting soliton solutions to nonlinear evolution equations is efficient, compatible and reliable in nonlinear science and engineering.
文摘In this paper, the modified extended tanh method is used to construct more general exact solutions of a （2＋1）-dimensional nonlinear SchrSdinger equation. With the aid of Maple and Matlab software, we obtain exact explicit kink wave solutions, peakon wave solutions, periodic wave solutions and their 3D images.
文摘Using the modified extended tanh-function method,explicit and exact traveling wave solutions for the(2+1)-dimensional higher-order Broer-Kaup(HBK)system,comprising new soliton-like and period-form solutions,are obtained.
文摘Using a further modified extended tanh-function method, rich new families of the exact solutions for the (2+ 1)-dimensional Broer-Kaup (BK) system, comprising the non-traveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained.
基金Project supported by the National Natural Science Foundation of China （Grant No 10672147） and Natural Science Foundation of Zhejiang Forestry University, China （Grant No 2006FR035）. Acknowledgments The authors are indebted to Professor Zhang J F for his helpful suggestions and fruitful discussions, and also express their sincere thanks to the editors and the anonymous referees for their constructive suggestions and kind help.
文摘Starting from the extended tanh-function method （ETM） based on the mapping method, the variable separation solutions of the （2＋1）-dimensional asymmetric Nizhnik Novikov Veselov （ANNV） system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semifoldon, are investigated.
文摘In the present study,the solitary wave solutions of modified Degasperis-Procesi equation are developed.Unlike the standard Degasperis-Procesi equation,where multi-peakon solutions arise,the modification caused a change in the characteristic of these peakon solutions and changed it to bell-shaped solitons.By using the extended auxiliary equation method,we deduced some new soliton solutions of the fourthorder nonlinear modified Degasperis-Procesi equation with constant coefficient.These solutions include symmetrical,non-symmetrical kink solutions,solitary pattern solutions,weiestrass elliptic function solutions and triangular function solutions.We discuss the stability analysis for these solutions.
文摘The searching exact solutions in the solitary wave form of non-linear partial differential equations （PDEs） play a significant role to understand the internal mechanism of complex physical phenomena. In this paper we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the （2-1-1）-dimensional cubic Klein-Gordon （K-G） equation. The Klein-Gordon equations are relativistic version of Schr6dinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which several solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions of PDEs arise in mathematical physics.
