In this study, the exact solutions for the propagation of pulses in optical fibers are obtained.Special values are given in the model used, and two nonlinear differential equations are obtained.Nonlinear equations are...In this study, the exact solutions for the propagation of pulses in optical fibers are obtained.Special values are given in the model used, and two nonlinear differential equations are obtained.Nonlinear equations are reduced to ordinary differential equations with the help of wave transformations. Then, the obtained differential equations are solved by two different methods,namely the modified simplest equation and the modified Kudryashov procedures. The solutions are given by hyperbolic, trigonometric and rational functions and the results are useful for optics,engineering and other related areas. Finally three-dimensional, contour and two-dimensional shapes are given for some solutions. These figures are important for understanding the motion of the wave. The given methods are applied to the equations for the first time. To the best of the authors' knowledge, these results are new and have not been obtained in the literature. The results are useful for applied mathematics, physics and other related areas.展开更多
Schrödinger equations are very common equations in physics and mathematics for nonlinear physics to model the dynamics of wave propagation in waveguides such as power lines, atomic chains, optical fibers, and eve...Schrödinger equations are very common equations in physics and mathematics for nonlinear physics to model the dynamics of wave propagation in waveguides such as power lines, atomic chains, optical fibers, and even in quantum mechanics. But all these equations are most often studied without worrying about what would happen if this equation were maintained, that is to say, had a second member synonymous with an external force. It is true that on a physical level, such equations can be considered as describing the generation of waves on a waveguide using an external force. However, the in-depth analysis of this aspect is not at the center of our reflection in this article, but for us, it is a question of proposing exact solutions to this type of equation and above all proposing the general form of the external force so that the obtaining exact solutions is possible.展开更多
By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solu...By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.展开更多
By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of ...By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries （KdV） equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.展开更多
Using the modified find some new exact solutions to Lie point symmetry groups and also get conservation laws, of the CK＇s direct method, we build the relationship between new solutions and old ones and the （3＋1）-d...Using the modified find some new exact solutions to Lie point symmetry groups and also get conservation laws, of the CK＇s direct method, we build the relationship between new solutions and old ones and the （3＋1）-dimensional potentiaial-YTSF equation. Baaed on the invariant group theory, Lie symmetries of the （3＋1）-dimensional potential-YTSF equation are obtained. We equation with the given Lie symmetry.展开更多
In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, ...In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the （2＋1）-dimensional Lax pair is reduced to （1＋1）-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.展开更多
Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integ...Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.展开更多
In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation...In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.展开更多
New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solu...New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solutionsand triangular periodic wave solutions are obtained.展开更多
In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions ar...In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.展开更多
With the aid of the classical Lie group method and nonclassical Lie group method,we derive the classicalLie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS)equation.Usi...With the aid of the classical Lie group method and nonclassical Lie group method,we derive the classicalLie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS)equation.Usingthe symmetries,we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation.Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.展开更多
The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the si...The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.展开更多
Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equation...Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.展开更多
In this paper we investigate the Einstein＇s hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed an...In this paper we investigate the Einstein＇s hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein＇s hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow.展开更多
In this paper, a variable-coefficient Benjarnin-Bona-Mahony-Burger （BBMB） equation arising as a math- ematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The in...In this paper, a variable-coefficient Benjarnin-Bona-Mahony-Burger （BBMB） equation arising as a math- ematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The inte- grability of such an equation is studied with Painlevd analysis. The Lie symmetry method is performed for the BBMB equation and then similarity reductions and exact solutions are obtained based on the optimal system method. Further- more different types of solitary, periodic and kink waves can be seen with the change of variable coefficients.展开更多
Bosonization approach is applied in solving the most general ;N= 1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a （sKdV-a） equation. By introducing some fermionic parameters in the expansion...Bosonization approach is applied in solving the most general ;N= 1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a （sKdV-a） equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosoNic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonie equations are simply obtained for all values of a, Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.展开更多
The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed...The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV(BSmKdV)equation admits Painlevé property by the standard singularity analysis. The traveling wave solutions of the BSmKdV system are presented by the mapping and deformation method. We also provide other ideas to construct new super integrable systems.展开更多
Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to t...Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations.The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions,hyperbolic functions,and trigonometric functions.Some figures are given to show the properties of the solutions.展开更多
To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are pr...To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.展开更多
A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations...A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.展开更多
文摘In this study, the exact solutions for the propagation of pulses in optical fibers are obtained.Special values are given in the model used, and two nonlinear differential equations are obtained.Nonlinear equations are reduced to ordinary differential equations with the help of wave transformations. Then, the obtained differential equations are solved by two different methods,namely the modified simplest equation and the modified Kudryashov procedures. The solutions are given by hyperbolic, trigonometric and rational functions and the results are useful for optics,engineering and other related areas. Finally three-dimensional, contour and two-dimensional shapes are given for some solutions. These figures are important for understanding the motion of the wave. The given methods are applied to the equations for the first time. To the best of the authors' knowledge, these results are new and have not been obtained in the literature. The results are useful for applied mathematics, physics and other related areas.
文摘Schrödinger equations are very common equations in physics and mathematics for nonlinear physics to model the dynamics of wave propagation in waveguides such as power lines, atomic chains, optical fibers, and even in quantum mechanics. But all these equations are most often studied without worrying about what would happen if this equation were maintained, that is to say, had a second member synonymous with an external force. It is true that on a physical level, such equations can be considered as describing the generation of waves on a waveguide using an external force. However, the in-depth analysis of this aspect is not at the center of our reflection in this article, but for us, it is a question of proposing exact solutions to this type of equation and above all proposing the general form of the external force so that the obtaining exact solutions is possible.
文摘By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.
基金Project supported by the National Natural Science Foundation of China（Grant No 10461006）, the High Education Science Research Program（Grant No NJ02035） of Inner Mongolia Autonomous Region, Natural Science Foundation of Inner Mongolia Autonomous Region（Grant No 2004080201103） and the Youth Research Program of Inner Mongolia Normal University（Grant No QN005023）.
文摘By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries （KdV） equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004zx16 tCorresponding author, E-maih zzlh100@163.com
文摘Using the modified find some new exact solutions to Lie point symmetry groups and also get conservation laws, of the CK＇s direct method, we build the relationship between new solutions and old ones and the （3＋1）-dimensional potentiaial-YTSF equation. Baaed on the invariant group theory, Lie symmetries of the （3＋1）-dimensional potential-YTSF equation are obtained. We equation with the given Lie symmetry.
基金Natural Science Foundation of Shandong Province under Grant Nos.2004zx16 and Q2005A01
文摘In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the （2＋1）-dimensional Lax pair is reduced to （1＋1）-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.
文摘Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant Nos. 2004zx16 and Q2005A01
文摘In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.
文摘New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solutionsand triangular periodic wave solutions are obtained.
基金The project supported by the State Key Basic Research Program of China under Grant No 2004CB318000
文摘In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.
基金Supported by National Natural Science Foundation of China and China Academy of Engineering Physics (NSAF 11076015)
文摘With the aid of the classical Lie group method and nonclassical Lie group method,we derive the classicalLie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS)equation.Usingthe symmetries,we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation.Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.
基金The project supported by National Natural Science Foundation of China under Grant No. 10071033 and the Natural Science Foundation of Jiangsu Province under Grant No. BK2002003. Acknowledgments 0ne of the authors （S.P. Qian） is indebted to Prof. S.Y. Lou for his helpful discussions.
文摘The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.
基金Supported by the Natural Science Foundation of Zhejiang Province(1 0 2 0 3 7)
文摘Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.
基金The project supported in part by the National Natural Science Foundation of China under Grant No. 10671124 and the Program for New Century Excellent Talents in University of China under Grant No. NCET-05-0390 Acknowledgments The author would like to thank the Center of Mathematical Sciences at Zhejiang University for the great support and hospitality and the referee for pertinent comments and valuable suggestions.
文摘In this paper we investigate the Einstein＇s hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein＇s hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow.
文摘In this paper, a variable-coefficient Benjarnin-Bona-Mahony-Burger （BBMB） equation arising as a math- ematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The inte- grability of such an equation is studied with Painlevd analysis. The Lie symmetry method is performed for the BBMB equation and then similarity reductions and exact solutions are obtained based on the optimal system method. Further- more different types of solitary, periodic and kink waves can be seen with the change of variable coefficients.
基金Supported by the National Natural Science Foundation of China under Nos.11175092,11275123 and 10905038Scientific Research Fund of Zhejiang Provincial Education Department under Grant No.Y201017148K.C.Wong Magna Fund in Ningbo University
文摘Bosonization approach is applied in solving the most general ;N= 1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a （sKdV-a） equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosoNic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonie equations are simply obtained for all values of a, Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11775146,11435005,and 11472177Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV(BSmKdV)equation admits Painlevé property by the standard singularity analysis. The traveling wave solutions of the BSmKdV system are presented by the mapping and deformation method. We also provide other ideas to construct new super integrable systems.
文摘Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations.The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions,hyperbolic functions,and trigonometric functions.Some figures are given to show the properties of the solutions.
基金supported by the National Natural Science Foundation of China(Grant No.10862003)the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region,China(Grant No.NJZZ07031)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2010MS0111)
文摘To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.
基金supported by National Natural Science Foundation of China and the Natural Science Foundation of Shandong Province
文摘A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.
