Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct dou...Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.展开更多
In this paper, a new generalized Jacobi elliptic function expansion method based upon four new Jacobi elliptic functions is described and abundant solutions of new Jacobi elliptic functions for the generalized Nizhnik...In this paper, a new generalized Jacobi elliptic function expansion method based upon four new Jacobi elliptic functions is described and abundant solutions of new Jacobi elliptic functions for the generalized Nizhnik-Novikov-Veselov equations are obtained. It is shown that the new method is much more powerful in finding new exact solutions to various kinds of nonlinear evolution equations in mathematical physics.展开更多
The Zakharov equation to describe the laser plasma interaction process has very important sense, this paper gives the solitary wave solutions for Zakharov equation by using Jacobi elliptic function method.
Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function ...Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number.The inverse hyperbolic function arsinher(r)■ro 1/√1+t^(2)dt p1tt2 dt is similar to the inverse trigonometric function arcsiner(r)■ro 1/√1+t^(2)dt p1t2 dt,such as the second degree of a polynomial and the constant term 1,except for the sign−and+.Such an analogy holds not only when the degree of the polynomial is 2,but also for higher degrees.As such,a function exists with respect to the leaf function through the imaginary number i,such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number.In this study,we refer to this function as the hyperbolic leaf function.By making such a definition,the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas,such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions.Using the addition formulas,we can also derive the double-angle and half-angle formulas.We then verify the consistency of these formulas by constructing graphs and numerical data.展开更多
This paper presents a closed form solution to the dynamic stability problem of a beam-column system with hinged ends loaded by an axial periodically time-varying compressive force of an elliptic type,i.e.,a1cn 2(τ,...This paper presents a closed form solution to the dynamic stability problem of a beam-column system with hinged ends loaded by an axial periodically time-varying compressive force of an elliptic type,i.e.,a1cn 2(τ,k 2)+a2sn 2(τ,k 2)+a3dn 2(τ,k 2).The solution to the governing equation is obtained in the form of Fourier sine series.The resulting ordinary differential equation is solved analytically.Finding the exact analytical solutions to the dynamic buckling problems is difficult.However,the availability of exact solutions can provide adequate understanding for the physical characteristics of the system.In this study,the frequency-response characteristics of the system,the effects of the static load,the driving forces,and the frequency ratio on the critical buckling load are also investigated.展开更多
New exact solutions, expressed in terms of the Jacobi elliptic functions, to the nonlinear Klein-Gordon equation are obtained by using a modified mapping method. The solutions include the conditions for equation's pa...New exact solutions, expressed in terms of the Jacobi elliptic functions, to the nonlinear Klein-Gordon equation are obtained by using a modified mapping method. The solutions include the conditions for equation's parameters and travelling wave transformation parameters. Some figures for a specific kind of solution are also presented.展开更多
An extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics is presented, which can be thought of as a concentration of extended Jacobi elliptic function...An extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics is presented, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by Jacobi elliptic functions for the coupled KdV equations are derived. In the limit cases, the solitary wave solutions and the other type of travelling wave solutions for the system are also obtained.展开更多
In this paper, based on the generalized Jacobi elliptic function expansion method, we obtain abundant new explicit and exact solutions of the Klein-Gordon- Zakharov equations, which degenerate to solitary wave solutio...In this paper, based on the generalized Jacobi elliptic function expansion method, we obtain abundant new explicit and exact solutions of the Klein-Gordon- Zakharov equations, which degenerate to solitary wave solutions and triangle function solutions in the limit cases, showing that this new method is more powerful to seek exact solutions of nonlinear partial differential equations in mathematical physics.展开更多
In this work,di erent kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional(3-D)nonlinear evolution equations(NEEs)through the implementation of the modi ed ext...In this work,di erent kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional(3-D)nonlinear evolution equations(NEEs)through the implementation of the modi ed extended direct algebraic method.Bright-singular and dark-singular combo solitons,Jacobi's elliptic functions,Weierstrass elliptic functions,constant wave solutions and so on are attained beside their existing conditions.Physical interpretation of the solutions to the 3-D modi ed KdV-Zakharov-Kuznetsov equation are also given.展开更多
The equation of motion for a large-deflection beam in the Lagrangian description are derived using the coupling of flexural deformation and midplane stretching as a key source of nonlinearity and taking into account t...The equation of motion for a large-deflection beam in the Lagrangian description are derived using the coupling of flexural deformation and midplane stretching as a key source of nonlinearity and taking into account the transverse, axial and rotary inertia effects. Assuming a traveling wave solution, the nonlinear partial differential equations are then transformed into ordinary differential equations. Qualitative analysis indicates that the system can have either a homoclinic orbit or a heteroclinic orbit, depending on whether the rotary inertia effect is taken into account. Furthermore, exact periodic solutions of the nonlinear wave equations are obtained by means of the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function m→1 in the degenerate case, either a solitary wave solution or a shock wave solution can be obtained.展开更多
A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated ...A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.展开更多
Jacobi elliptic function expansion method is extended to construct the exact solutions to another kind of KdV equations, which have variable coefficients or forcing terms. And new periodic solutions obtained by this m...Jacobi elliptic function expansion method is extended to construct the exact solutions to another kind of KdV equations, which have variable coefficients or forcing terms. And new periodic solutions obtained by this method can be reduced to the soliton-typed solutions under the limited condition.展开更多
The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. ...The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.展开更多
Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling w...Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov's method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.展开更多
In this paper, the Jacobi elliptic function expansion method provides an effective approach to obtain the exact periodic wave solutions of two-component Bose-Einstein condensates. Exact combined bright-bright and dark...In this paper, the Jacobi elliptic function expansion method provides an effective approach to obtain the exact periodic wave solutions of two-component Bose-Einstein condensates. Exact combined bright-bright and dark-dark soliton wave solutions can be achieved in their limit conditions. We also obtain the different formation regions of combined solitons. Our results show that the intraspecies (interspecies) interaction strengths clearly affect the formation of dar^dark, bright-bright and dark-bright soliton solutions in different regions.展开更多
In the present paper, a general solution involv- ing three arbitrary functions for the generalized (2+1)- dimensional KdV-mKdV equation, which is derived from the generalized (1+1)-dimensional KdV-mKdV equa- tio...In the present paper, a general solution involv- ing three arbitrary functions for the generalized (2+1)- dimensional KdV-mKdV equation, which is derived from the generalized (1+1)-dimensional KdV-mKdV equa- tion, is first introduced by means of the Wiess, Tabor, Carnevale (WTC) truncation method. And then multi- symplectic formulations with several conservation laws taken into account are presented for the generalized (2+1)- dimensional KdV-mKdV equation based on the multi- symplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from thegeneral solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent 1:o the Preissmann scheme. From the results of the numerical experiments, we can con- clude that the multi-symplectic schemes can accurately sim- ulate the periodic wave solutions of the generalized (2+1)- dimensional KdV-mKdV equation while preserve approxi- mately the conservation laws.展开更多
In this paper,some new periodic solutions of nonlinear evolution equations and corresponding travelling wave solutions are obtained by using the double function method and Jacobi elliptic functions.
Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung flui...Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equation.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
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
文摘Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.
基金The Scientific Research Foundation (QKJA2010011) of Nanjing Institute of Technology
文摘In this paper, a new generalized Jacobi elliptic function expansion method based upon four new Jacobi elliptic functions is described and abundant solutions of new Jacobi elliptic functions for the generalized Nizhnik-Novikov-Veselov equations are obtained. It is shown that the new method is much more powerful in finding new exact solutions to various kinds of nonlinear evolution equations in mathematical physics.
文摘The Zakharov equation to describe the laser plasma interaction process has very important sense, this paper gives the solitary wave solutions for Zakharov equation by using Jacobi elliptic function method.
文摘Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number.The inverse hyperbolic function arsinher(r)■ro 1/√1+t^(2)dt p1tt2 dt is similar to the inverse trigonometric function arcsiner(r)■ro 1/√1+t^(2)dt p1t2 dt,such as the second degree of a polynomial and the constant term 1,except for the sign−and+.Such an analogy holds not only when the degree of the polynomial is 2,but also for higher degrees.As such,a function exists with respect to the leaf function through the imaginary number i,such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number.In this study,we refer to this function as the hyperbolic leaf function.By making such a definition,the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas,such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions.Using the addition formulas,we can also derive the double-angle and half-angle formulas.We then verify the consistency of these formulas by constructing graphs and numerical data.
文摘This paper presents a closed form solution to the dynamic stability problem of a beam-column system with hinged ends loaded by an axial periodically time-varying compressive force of an elliptic type,i.e.,a1cn 2(τ,k 2)+a2sn 2(τ,k 2)+a3dn 2(τ,k 2).The solution to the governing equation is obtained in the form of Fourier sine series.The resulting ordinary differential equation is solved analytically.Finding the exact analytical solutions to the dynamic buckling problems is difficult.However,the availability of exact solutions can provide adequate understanding for the physical characteristics of the system.In this study,the frequency-response characteristics of the system,the effects of the static load,the driving forces,and the frequency ratio on the critical buckling load are also investigated.
文摘New exact solutions, expressed in terms of the Jacobi elliptic functions, to the nonlinear Klein-Gordon equation are obtained by using a modified mapping method. The solutions include the conditions for equation's parameters and travelling wave transformation parameters. Some figures for a specific kind of solution are also presented.
基金Project supported by the Natural Science Foundation of Henan Province of China (Grant No 0111050200) and the Science Foundation of Henan University of Science and Technology (Grant Nos 2004ZY040 and 2004ZD002).
文摘An extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics is presented, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by Jacobi elliptic functions for the coupled KdV equations are derived. In the limit cases, the solitary wave solutions and the other type of travelling wave solutions for the system are also obtained.
基金The Scientific Research Foundation (KXJ08047) of NanJing Institute of Technology
文摘In this paper, based on the generalized Jacobi elliptic function expansion method, we obtain abundant new explicit and exact solutions of the Klein-Gordon- Zakharov equations, which degenerate to solitary wave solutions and triangle function solutions in the limit cases, showing that this new method is more powerful to seek exact solutions of nonlinear partial differential equations in mathematical physics.
文摘In this work,di erent kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional(3-D)nonlinear evolution equations(NEEs)through the implementation of the modi ed extended direct algebraic method.Bright-singular and dark-singular combo solitons,Jacobi's elliptic functions,Weierstrass elliptic functions,constant wave solutions and so on are attained beside their existing conditions.Physical interpretation of the solutions to the 3-D modi ed KdV-Zakharov-Kuznetsov equation are also given.
基金supported by the National Natural Science Foundation of China(Nos.10772129 and 10702047).
文摘The equation of motion for a large-deflection beam in the Lagrangian description are derived using the coupling of flexural deformation and midplane stretching as a key source of nonlinearity and taking into account the transverse, axial and rotary inertia effects. Assuming a traveling wave solution, the nonlinear partial differential equations are then transformed into ordinary differential equations. Qualitative analysis indicates that the system can have either a homoclinic orbit or a heteroclinic orbit, depending on whether the rotary inertia effect is taken into account. Furthermore, exact periodic solutions of the nonlinear wave equations are obtained by means of the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function m→1 in the degenerate case, either a solitary wave solution or a shock wave solution can be obtained.
基金Project supported by the National Natural Science Foundation of China (No. 10472076).
文摘A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.
文摘Jacobi elliptic function expansion method is extended to construct the exact solutions to another kind of KdV equations, which have variable coefficients or forcing terms. And new periodic solutions obtained by this method can be reduced to the soliton-typed solutions under the limited condition.
基金Project supported by the National Nature Science Foundation of China (Grant No 49894190) of the Chinese Academy of Science (Grant No KZCXI-sw-18), and Knowledge Innovation Program.
文摘The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.
基金Project supported by the National Natural Science Foundation of China (No. 10772129)
文摘Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov's method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.
基金supported by the Key Program of Chinese Ministry of Education(Grant No.2011015)the Hundred Innovation Talents Supporting Project of Hebei Province of China(Grant No.CPRC014)
文摘In this paper, the Jacobi elliptic function expansion method provides an effective approach to obtain the exact periodic wave solutions of two-component Bose-Einstein condensates. Exact combined bright-bright and dark-dark soliton wave solutions can be achieved in their limit conditions. We also obtain the different formation regions of combined solitons. Our results show that the intraspecies (interspecies) interaction strengths clearly affect the formation of dar^dark, bright-bright and dark-bright soliton solutions in different regions.
基金supported by the National Natural Science Foundation of China (11002115,10972182,11172239)the Science Foundation of Aviation of China (2010ZB53021)+5 种基金the China Postdoctoral Science Special Foundation (201003682)111 project(B07050) to the Northwestern Polytechnical Universitythe NPU Foundation for Fundamental Research (JC200938,JC20110259)the Doctoral Program Foundation of Education Ministry of China(20106102110019)the Open Foundation of State Key Laboratory of Mechanical System & Vibration (MSV-2011-21)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (GZ0802)
文摘In the present paper, a general solution involv- ing three arbitrary functions for the generalized (2+1)- dimensional KdV-mKdV equation, which is derived from the generalized (1+1)-dimensional KdV-mKdV equa- tion, is first introduced by means of the Wiess, Tabor, Carnevale (WTC) truncation method. And then multi- symplectic formulations with several conservation laws taken into account are presented for the generalized (2+1)- dimensional KdV-mKdV equation based on the multi- symplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from thegeneral solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent 1:o the Preissmann scheme. From the results of the numerical experiments, we can con- clude that the multi-symplectic schemes can accurately sim- ulate the periodic wave solutions of the generalized (2+1)- dimensional KdV-mKdV equation while preserve approxi- mately the conservation laws.
文摘In this paper,some new periodic solutions of nonlinear evolution equations and corresponding travelling wave solutions are obtained by using the double function method and Jacobi elliptic functions.
文摘Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equation.
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.
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.