In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the (1+1)-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which...In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the (1+1)-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.展开更多
This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational funct...This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.展开更多
In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new typ...In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As <em>m</em> → 0 and <em>m</em> → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.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
The Sasa-satsuma(SS)dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers.This dynamical model has important physical significance.In this article,two mathematical technique...The Sasa-satsuma(SS)dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers.This dynamical model has important physical significance.In this article,two mathematical techniques namely,improved F-expansion and improved aux-iliary methods are utilized to construct the several types of solitons such as dark soliton,bright soliton,periodic soliton,Elliptic function and solitary waves solutions of Sasa-satsuma dynamical equation.These results have imperative applications in sciences and other fields,and construc-tive to recognize the physical structure of this complex dynamical model.The computing work and obtained results show the infuence and effectiveness of current methods.展开更多
In this research article,the(3+1)-dimensional nonlinear Gardner-Kadomtsov-Petviashvili(Gardner-KP)equation which depicts the nonlinear modulation of periodic waves,is analyzed through the Lie group-theoretic technique...In this research article,the(3+1)-dimensional nonlinear Gardner-Kadomtsov-Petviashvili(Gardner-KP)equation which depicts the nonlinear modulation of periodic waves,is analyzed through the Lie group-theoretic technique.Considering the Lie invariance condition,we find the symmetry generators.The pro-posed model yields eight-dimensional Lie algebra.Moreover,an optimal system of sub-algebras is com-puted,and similarity reductions are made.The considered nonlinear partial differential equation is re-duced into ordinary differential equations(ODEs)by utilizing the similarity transformation method(STM),which has the benefit of yielding a large number of accurate traveling wave solutions.These ODEs are further solved to get closed-form solutions of the Gardner-KP equation in some cases,while in other cases,we use the new auxiliary equation method to get its soliton solutions.The evolution profiles of the obtained solutions are examined graphically under the appropriate selection of arbitrary parameters.展开更多
This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differenti...This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differential equation.The proposed scheme has been successfully applied on two very important evolution equations,the space-time fractional differential equation governing wave propagation in low-pass electrical transmission lines equation and the time fractional Burger’s equation.The obtained results show that the proposed method is more powerful,promising and convenient for solving nonlinear fractional differential equations(NFPDEs).To our knowledge,the solutions obtained by the proposed method have not been reported in former literature.展开更多
It is commonly recognized that,despite current analytical approaches,many physical aspects of nonlinear models remain unknown.It is critical to build more efficient integration methods to design and construct numerous...It is commonly recognized that,despite current analytical approaches,many physical aspects of nonlinear models remain unknown.It is critical to build more efficient integration methods to design and construct numerous other unknown solutions and physical attributes for the nonlinear models,as well as for the benefit of the largest audience feasible.To achieve this goal,we propose a new extended unified auxiliary equation technique,a brand-new analytical method for solving nonlinear partial differential equations.The proposed method is applied to the nonlinear Schrödinger equation with a higher dimension in the anomalous dispersion.Many interesting solutions have been obtained.Moreover,to shed more light on the features of the obtained solutions,the figures for some obtained solutions are graphed.The propagation characteristics of the generated solutions are shown.The results show that the proper physical quantities and nonlinear wave qualities are connected to the parameter values.It is worth noting that the new method is very effective and efficient,and it may be applied in the realisation of novel solutions.展开更多
Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equat...Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other (l+l)-dimensional and higher dimensional systems.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
In this paper, the higher order NLS equation with cubic-quintic nonlinear terms is studied, new abundant solitary solutions with traveling-wave envelope of this equation are obtained with the aid of a generalized auxi...In this paper, the higher order NLS equation with cubic-quintic nonlinear terms is studied, new abundant solitary solutions with traveling-wave envelope of this equation are obtained with the aid of a generalized auxiliary equation method and complex envelope non-traveling transform approach.展开更多
基金supported by the National Natural Science Foundation of China (No.10461005)the Ph.D.Programs Foundation of Ministry of Education of China (No.20070128001)the High Education Science Research Program of Inner Mongolia (No.NJZY08057)
文摘In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the (1+1)-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.
基金Project supported by the Fundamental Research Funds for the Central Universities (Grant No. 2010B17914) and the National Natural Science Foundation of China (Grant No. 10926162).
文摘This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.
文摘In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As <em>m</em> → 0 and <em>m</em> → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.
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.
文摘The Sasa-satsuma(SS)dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers.This dynamical model has important physical significance.In this article,two mathematical techniques namely,improved F-expansion and improved aux-iliary methods are utilized to construct the several types of solitons such as dark soliton,bright soliton,periodic soliton,Elliptic function and solitary waves solutions of Sasa-satsuma dynamical equation.These results have imperative applications in sciences and other fields,and construc-tive to recognize the physical structure of this complex dynamical model.The computing work and obtained results show the infuence and effectiveness of current methods.
基金The authors would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project R-2022-178.
文摘In this research article,the(3+1)-dimensional nonlinear Gardner-Kadomtsov-Petviashvili(Gardner-KP)equation which depicts the nonlinear modulation of periodic waves,is analyzed through the Lie group-theoretic technique.Considering the Lie invariance condition,we find the symmetry generators.The pro-posed model yields eight-dimensional Lie algebra.Moreover,an optimal system of sub-algebras is com-puted,and similarity reductions are made.The considered nonlinear partial differential equation is re-duced into ordinary differential equations(ODEs)by utilizing the similarity transformation method(STM),which has the benefit of yielding a large number of accurate traveling wave solutions.These ODEs are further solved to get closed-form solutions of the Gardner-KP equation in some cases,while in other cases,we use the new auxiliary equation method to get its soliton solutions.The evolution profiles of the obtained solutions are examined graphically under the appropriate selection of arbitrary parameters.
文摘This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differential equation.The proposed scheme has been successfully applied on two very important evolution equations,the space-time fractional differential equation governing wave propagation in low-pass electrical transmission lines equation and the time fractional Burger’s equation.The obtained results show that the proposed method is more powerful,promising and convenient for solving nonlinear fractional differential equations(NFPDEs).To our knowledge,the solutions obtained by the proposed method have not been reported in former literature.
文摘It is commonly recognized that,despite current analytical approaches,many physical aspects of nonlinear models remain unknown.It is critical to build more efficient integration methods to design and construct numerous other unknown solutions and physical attributes for the nonlinear models,as well as for the benefit of the largest audience feasible.To achieve this goal,we propose a new extended unified auxiliary equation technique,a brand-new analytical method for solving nonlinear partial differential equations.The proposed method is applied to the nonlinear Schrödinger equation with a higher dimension in the anomalous dispersion.Many interesting solutions have been obtained.Moreover,to shed more light on the features of the obtained solutions,the figures for some obtained solutions are graphed.The propagation characteristics of the generated solutions are shown.The results show that the proper physical quantities and nonlinear wave qualities are connected to the parameter values.It is worth noting that the new method is very effective and efficient,and it may be applied in the realisation of novel solutions.
基金Supported by the National Natural Science Foundation of China (No. 10647112)the Foundation of Donghua University
文摘Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other (l+l)-dimensional and higher dimensional systems.
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.
基金Supported by the National Natural Science Foundation of China(No.11361048)
文摘In this paper, the higher order NLS equation with cubic-quintic nonlinear terms is studied, new abundant solitary solutions with traveling-wave envelope of this equation are obtained with the aid of a generalized auxiliary equation method and complex envelope non-traveling transform approach.