This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either...This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.展开更多
The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-d...The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-de Vriesmodified Korteweg-de Vries equation for the atmosphere,oceanic fluids and plasmas.With symbolic computation,beginning with a presumption,we work out certain scaling transformations,bilinear forms through the binary Bell polynomials and our scaling transformations,N solitons(with N being a positive integer)via the aforementioned bilinear forms and bilinear auto-Bäcklund transformations through the Hirota method with some solitons.In addition,Painlevé-type auto-Bäcklund transformations with some solitons are symbolically computed out.Respective dependences and constraints on the variable/constant coefficients are discussed,while those coefficients correspond to the quadratic-nonlinear,cubic-nonlinear,dispersive,dissipative and line-damping effects in the atmosphere,oceanic fluids and plasmas.展开更多
In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used...In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.展开更多
In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation (mKdV) via theBcklund transformation (BT) and a generalized Wronskian condition is given,which allows us to substitute an arbitrar...In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation (mKdV) via theBcklund transformation (BT) and a generalized Wronskian condition is given,which allows us to substitute an arbitrarycoefficient matrix in the G_N(t) for the original diagonal one.展开更多
The extended symmetry approach is used to study the general Korteweg-de Vries-type (KdV-type) equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be construc...The extended symmetry approach is used to study the general Korteweg-de Vries-type (KdV-type) equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation.展开更多
The travelling solitary wave solutions to the higher order Korteweg-de Vries equation are obtained by using tanh-polynomial method. The method is effective and concise, which is also applied to various partial differe...The travelling solitary wave solutions to the higher order Korteweg-de Vries equation are obtained by using tanh-polynomial method. The method is effective and concise, which is also applied to various partial differential equations to obtain traveling wave solutions. The numerical simulation of the solutions is given for completeness. Numerical results show that the tanh-polynomial method works quite well.展开更多
A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) s...A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.展开更多
The deterministic extended Korteweg-de Vries equation plays an essential role in the description of the creation and propagation of nonlinear waves in many fields. We study a stochastic extended Korteweg-de Vries equa...The deterministic extended Korteweg-de Vries equation plays an essential role in the description of the creation and propagation of nonlinear waves in many fields. We study a stochastic extended Korteweg-de Vries equation driven by a multiplicative noise in the form of a cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied for all physically relevant initial conditions. The proof of the solution is based on two approximations of the problem considered and the compactness method.展开更多
In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton ph...In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator<span style="white-space:nowrap;">—</span>Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations;however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities.展开更多
Consider the following system of coupled Korteweg-de Vries equations, <img src="Edit_81ea1215-e696-403f-9d6c-1449e107359f.bmp" alt="" /><span style="white-space:nowrap;">where...Consider the following system of coupled Korteweg-de Vries equations, <img src="Edit_81ea1215-e696-403f-9d6c-1449e107359f.bmp" alt="" /><span style="white-space:nowrap;">where<em> u</em>, <em>v </em><span style="white-space:nowrap;">⊆</span> <em>W</em><sup>2,2</sup>, 2≤<em>N</em>≤7 and <em>λ</em><sub><em>i</em></sub>,<em>β</em> > 0, <em>β</em> </span>denotes a real coupling parameter. Firstly, we prove the existence of the solutions of a coupled system of Korteweg-de Vries equations using variation approach and minimization techniques on Nehari manifold. Then, we show the multiplicity of the equations by a bifurcation theory which is rare for studying higher order equations.展开更多
In this study, we used Double Elzaki Transform (DET) coupled with Adomian polynomial to produce a new method to solve Third Order Korteweg-De Vries Equations (KdV) equations. We will provide the necessary explanation ...In this study, we used Double Elzaki Transform (DET) coupled with Adomian polynomial to produce a new method to solve Third Order Korteweg-De Vries Equations (KdV) equations. We will provide the necessary explanation for this method with addition some examples to demonstrate the effectiveness of this method.展开更多
A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation ...A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation that is changed to a nonlinear systems and can be solved by the usual fixed point iteration.Numerical results demonstrate the efficiency of the method and spectral accuracy.展开更多
In this paper,we present Lax pairs and solutions for a nonsymmetric lattice equation,which is a torqued version of the lattice potential Korteweg-de Vries equation.This nonsymmetric equation is special in the sense th...In this paper,we present Lax pairs and solutions for a nonsymmetric lattice equation,which is a torqued version of the lattice potential Korteweg-de Vries equation.This nonsymmetric equation is special in the sense that it contains only one spacing parameter but consists of two consistent cubes with other integrable lattice equations.Using such a multidimensionally consistent property we are able to derive its two Lax pairs and also construct solutions using B?cklund transformations.展开更多
A soliton is a packet of self-reinforcing waves that maintains its structure when moving at a constant speed.Solitons are caused by the cancellation of the medium’s nonlinear and dispersive effects.In plas-mas,the bi...A soliton is a packet of self-reinforcing waves that maintains its structure when moving at a constant speed.Solitons are caused by the cancellation of the medium’s nonlinear and dispersive effects.In plas-mas,the bilinear form of Hirota will be utilized to investigate the(2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential.Solutions for complexiton lump interaction have been devel-oped.To throw further light on the physical qualities of the recorded data,certain 3-dimensional and contour plots are presented to illustrate the interaction elements of these solutions.展开更多
In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by ap...In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including;single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, <em>L</em><sub>2</sub> and <em>L</em><sub>∞</sub> and invariants <em>I</em><sub>1</sub>, <em>I</em><sub>2</sub> and <em>I</em><sub>3</sub> have been calculated. Our numerical results are compared with some of those available in the literature.展开更多
A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presen...A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.展开更多
Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an a...Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.展开更多
The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data i...The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data in H^s(R)(s 〉 -3/4). For k = 2, local result for data in H^S(R)(s 〉1/4) is obtained. For k = 3, local result for data in H^S(R)(s 〉 -1/6) is obtained. Moreover, the solutions of generalized Korteweg-de Vries-Benjamin-Ono equation converge to the solutions of KdV equation if the term of Benjamin-Ono equation tends to zero.展开更多
The aim of this paper is to study the unsteady korteweg-de vries equation that plays an important role in describing the shallow water.Two analytical techniques namely the Sardar-subequation method and the energy bala...The aim of this paper is to study the unsteady korteweg-de vries equation that plays an important role in describing the shallow water.Two analytical techniques namely the Sardar-subequation method and the energy balance method are employed to seek the abundant traveling wave solutions for the first time.By these two methods,plenty of traveling wave solutions such as the bright solitary wave solutions,dark solitary wave solutions,singular periodic wave solutions and perfect periodic wave solution that expressed in terms of the generalized hyperbolic functions,generalized trigonometric functions and the cosine function are obtained.Finally,the dynamic behaviors of the solutions are described through the 3D plot and 2D curve.The results in this paper demonstrate that the proposed methods are powerful and effective to construct the traveling wave solutions of the nonlinear evolution equations in ocean engineering and science.展开更多
基金supported by the NSF under Grant DMS-2208391sponsored by the NSF under Grant DMS-1753581.
文摘This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
基金the National Natural Science Foundation of China(Grant No.11871116)the Fundamental Research Funds for the Central Universities of China(Grant No.2019XD-A11)the BUPT Innovation and Entrepreneurship Support Program,Beijing University of Posts and Telecommunications,and the National Scholarship for Doctoral Students of China.
文摘The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-de Vriesmodified Korteweg-de Vries equation for the atmosphere,oceanic fluids and plasmas.With symbolic computation,beginning with a presumption,we work out certain scaling transformations,bilinear forms through the binary Bell polynomials and our scaling transformations,N solitons(with N being a positive integer)via the aforementioned bilinear forms and bilinear auto-Bäcklund transformations through the Hirota method with some solitons.In addition,Painlevé-type auto-Bäcklund transformations with some solitons are symbolically computed out.Respective dependences and constraints on the variable/constant coefficients are discussed,while those coefficients correspond to the quadratic-nonlinear,cubic-nonlinear,dispersive,dissipative and line-damping effects in the atmosphere,oceanic fluids and plasmas.
基金the National Basic Research Program of China(Grant No.2012CB025903)
文摘In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.
基金National Natural Science Foundation of China under Grant Nos.10371070 and 10671121the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers
文摘In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation (mKdV) via theBcklund transformation (BT) and a generalized Wronskian condition is given,which allows us to substitute an arbitrarycoefficient matrix in the G_N(t) for the original diagonal one.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10675065)the Scientific Research Fundof the Education Department of Zhejiang Province of China (Grant No. 20070979)
文摘The extended symmetry approach is used to study the general Korteweg-de Vries-type (KdV-type) equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation.
文摘The travelling solitary wave solutions to the higher order Korteweg-de Vries equation are obtained by using tanh-polynomial method. The method is effective and concise, which is also applied to various partial differential equations to obtain traveling wave solutions. The numerical simulation of the solutions is given for completeness. Numerical results show that the tanh-polynomial method works quite well.
文摘A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.
文摘The deterministic extended Korteweg-de Vries equation plays an essential role in the description of the creation and propagation of nonlinear waves in many fields. We study a stochastic extended Korteweg-de Vries equation driven by a multiplicative noise in the form of a cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied for all physically relevant initial conditions. The proof of the solution is based on two approximations of the problem considered and the compactness method.
文摘In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator<span style="white-space:nowrap;">—</span>Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations;however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities.
文摘Consider the following system of coupled Korteweg-de Vries equations, <img src="Edit_81ea1215-e696-403f-9d6c-1449e107359f.bmp" alt="" /><span style="white-space:nowrap;">where<em> u</em>, <em>v </em><span style="white-space:nowrap;">⊆</span> <em>W</em><sup>2,2</sup>, 2≤<em>N</em>≤7 and <em>λ</em><sub><em>i</em></sub>,<em>β</em> > 0, <em>β</em> </span>denotes a real coupling parameter. Firstly, we prove the existence of the solutions of a coupled system of Korteweg-de Vries equations using variation approach and minimization techniques on Nehari manifold. Then, we show the multiplicity of the equations by a bifurcation theory which is rare for studying higher order equations.
文摘In this study, we used Double Elzaki Transform (DET) coupled with Adomian polynomial to produce a new method to solve Third Order Korteweg-De Vries Equations (KdV) equations. We will provide the necessary explanation for this method with addition some examples to demonstrate the effectiveness of this method.
基金Supported by National Natural Science Foundation of China(Grant Nos.11771299,11371123)Natural Science Foundation of Henan Province(Grant No.202300410156).
文摘A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation that is changed to a nonlinear systems and can be solved by the usual fixed point iteration.Numerical results demonstrate the efficiency of the method and spectral accuracy.
基金supported by the NSF of China(Nos.12271334,12071432)。
文摘In this paper,we present Lax pairs and solutions for a nonsymmetric lattice equation,which is a torqued version of the lattice potential Korteweg-de Vries equation.This nonsymmetric equation is special in the sense that it contains only one spacing parameter but consists of two consistent cubes with other integrable lattice equations.Using such a multidimensionally consistent property we are able to derive its two Lax pairs and also construct solutions using B?cklund transformations.
文摘A soliton is a packet of self-reinforcing waves that maintains its structure when moving at a constant speed.Solitons are caused by the cancellation of the medium’s nonlinear and dispersive effects.In plas-mas,the bilinear form of Hirota will be utilized to investigate the(2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential.Solutions for complexiton lump interaction have been devel-oped.To throw further light on the physical qualities of the recorded data,certain 3-dimensional and contour plots are presented to illustrate the interaction elements of these solutions.
文摘In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including;single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, <em>L</em><sub>2</sub> and <em>L</em><sub>∞</sub> and invariants <em>I</em><sub>1</sub>, <em>I</em><sub>2</sub> and <em>I</em><sub>3</sub> have been calculated. Our numerical results are compared with some of those available in the literature.
基金Project supported by the National Natural Science Foundation of China (No. 60874039)Shanghai Leading Academic Discipline Project (No. J50101)
文摘A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.
基金Project supported by the National Natural Scinece Foundation of China(Grant Nos.11671219,11871446,12071304,and 12071451).
文摘Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.
文摘The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data in H^s(R)(s 〉 -3/4). For k = 2, local result for data in H^S(R)(s 〉1/4) is obtained. For k = 3, local result for data in H^S(R)(s 〉 -1/6) is obtained. Moreover, the solutions of generalized Korteweg-de Vries-Benjamin-Ono equation converge to the solutions of KdV equation if the term of Benjamin-Ono equation tends to zero.
基金the Key Programs of Universities in Henan Province of China(22A140006)the Fundamental Research Funds for the Universities of Henan Province(NSFRF210324)+1 种基金Pro-gram of Henan Polytechnic University(B2018-40)Innovative Sci-entists and Technicians Team of Henan Provincial High Education(21IRTSTHN016).
文摘The aim of this paper is to study the unsteady korteweg-de vries equation that plays an important role in describing the shallow water.Two analytical techniques namely the Sardar-subequation method and the energy balance method are employed to seek the abundant traveling wave solutions for the first time.By these two methods,plenty of traveling wave solutions such as the bright solitary wave solutions,dark solitary wave solutions,singular periodic wave solutions and perfect periodic wave solution that expressed in terms of the generalized hyperbolic functions,generalized trigonometric functions and the cosine function are obtained.Finally,the dynamic behaviors of the solutions are described through the 3D plot and 2D curve.The results in this paper demonstrate that the proposed methods are powerful and effective to construct the traveling wave solutions of the nonlinear evolution equations in ocean engineering and science.