The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With th...The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.展开更多
Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the com...Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg-de Vries （KdV） and modified KdV （mKdV） equation and （2＋1）-dimensional Broer-Kaup-Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.展开更多
The mesoscale eddy and internal wave both are phenomena commonly observed in oceans. It is aimed to investigate how the presence of a mesoscale eddy in the ocean affects wave form deformation of the internal solitary ...The mesoscale eddy and internal wave both are phenomena commonly observed in oceans. It is aimed to investigate how the presence of a mesoscale eddy in the ocean affects wave form deformation of the internal solitary wave propagation. An ocean eddy is produced by a quasi-geostrophic model in f-plane, and the one-dimensional nonlinear variable-coefficient extended Korteweg-de Vries （eKdV） equation is used to simulate an internal solitary wave passing through the mesoscale eddy field. The results suggest that the mode structures of the linear internal wave are modified due to the presence of the mesoscale eddy field. A cyclonic eddy and an anticyclonic eddy have different influences on the background environment of the internal solitary wave propagation. The existence of a mesoscale eddy field has almost no prominent impact on the propagation of a smallamplitude internal solitary wave only based on the first mode vertical structure, but the mesoscale eddy background field exerts a considerable influence on the solitary wave propagation if considering high-mode vertical structures. Furthermore, whether an internal solitary wave first passes through anticyclonic eddy or cyclonic eddy, the deformation of wave profiles is different. Many observations of solitary internal waves in the real oceans suggest the formation of the waves. Apart from topography effect, it is shown that the mesoscale eddy background field is also a considerable factor which influences the internal solitary wave propagation and deformation.展开更多
Amongst the important phenomena in neurophysiology, nerve pulse generation and propagation is fundamental. Scientists have studied this phenomena using mathematical models based on experimental observations on the phy...Amongst the important phenomena in neurophysiology, nerve pulse generation and propagation is fundamental. Scientists have studied this phenomena using mathematical models based on experimental observations on the physiological processes in the nerve cell. Widely used models include: the Hodgkin-Huxley (H-H) model, which is based entirely on the electrical activity of the nerve cell;and the Heimburg and Jackson (H-J), model based on the thermodynamic activity of the nerve cell. These classes of models do not, individually, give a complete picture of the processes that lead to nerve pulse generation and propagation. Recently, a hybrid model proposed by Mengnjo, Dikandé and Ngwa (M-D-N), takes into consideration both the electrical and thermodynamic activities of the nerve cell. In their work, the first three bound states of the model are analytically computed and they showed great resemblance to some of the experimentally observed pulse profiles. With these bound states, the M-D-N model reduces to an initial value problem of a linear parabolic partial differential equation with variable coefficients. In this work we consider the resulting initial value problem and, using the theory of function spaces, propose and prove conditions under which such equations will admit unique solutions. We then verify that the resulting initial value problem from the M-D-N model satisfies these conditions and so has a unique solution. Given that the derived initial value problem is complex and there are no known analytic techniques that can be deployed to obtain its solution, we designed a numerical experiment to estimate the solutions. The simulations revealed that the unique solution is a stable pulse that propagates in the x-t plane with constant velocity and maintains the shape of the initial profile.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 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
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
In this paper, based on the concatenating method, we present a unified framework to construct a series of local structure-preserving algorithms for the Korteweg-de Vries （KdV） equation, including eight multi-symplec...In this paper, based on the concatenating method, we present a unified framework to construct a series of local structure-preserving algorithms for the Korteweg-de Vries （KdV） equation, including eight multi-symplectic algorithms, eight local energy-conserving algo- rithms and eight local momentum-conserving algorithms. Among these algorithms, some have been discussed and widely used while the most are new. The outstanding advantage of these proposed algorithms is that they conserve the local structures in any time-space re- gion exactly. Therefore, the local structure-preserving algorithms overcome the restriction of global structure-preserving algorithms on the boundary conditions. Numerical experiments are conducted to show the performance of the proposed methods. Moreover, the unified framework can be easily applied to many other equations.展开更多
The major concern of this work is to propose new prototypes of surface hybrid waves, in particular waves propagating without sprawl or deformation on the surface of a fluid. The model considered for this purpose is th...The major concern of this work is to propose new prototypes of surface hybrid waves, in particular waves propagating without sprawl or deformation on the surface of a fluid. The model considered for this purpose is the modified KdV (Korteweg-de Vries) equation. A peculiarity of the obtained solutions is that they form packages constituted by combinations of waves belonging to the two main families of well-known bright and dark solitary waves. This putting together creates competitions between the different components of the considered packages which, following the values assigned to the parameters of the considered system and in relation to those of the wave parameters, generate hybrid or multi-form structures. The direct method of resolution which made possible the obtained results is that of Bogning-Djeumen Tchaho-Kofane extended to the new implicit Bogning functions. The existence conditions of some solutions are obtained. The numerical simulations carried out with a view to testing the observable and applicable characters of the obtained solutions revealed their stabilities over a relatively long time, and at the same time, confirmed the recommended theoretical forecasts. We are convinced that the solutions proposed as part of this work will make it possible to detect, understand and explain some physical phenomena linked to fluid molecular interactions, former or new, which constantly occur on the fluid surfaces, mainly at the shallow water surface.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
This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave i...This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.展开更多
A high-order finite difference Pade scheme also called compact scheme for solving Korteweg-de Vries （KdV） equations, which preserve energy and mass conservations, was developed in this paper. This structure-preservi...A high-order finite difference Pade scheme also called compact scheme for solving Korteweg-de Vries （KdV） equations, which preserve energy and mass conservations, was developed in this paper. This structure-preserving algorithm has been widely applied in these years for its advantage of maintaining the inherited properties. For spatial discretization, the authors obtained an implicit compact scheme by which spatial derivative terms may be approximated through combining a few knots. By some numerical examples including propagation of single soliton and interaction of two solitons, the scheme is proved to be effective.展开更多
In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can...In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ.The jump matrix L(x,t,θ)has an explicit representation dependent on x,t and it can be represented exactly by the two pairs of spectral functions y(θ),z(θ)(obtained from the initial value u0(x))and Y(θ),Z(θ)(obtained from the boundary conditions v0(t),{vk(t)}_(1)^(4)).Furthermore,the two pairs of spectral functions y(θ),z(θ)and Y(θ),Z(θ)are not independent of each other,but are related to the compatibility condition,the so-called global relation.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
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
A perturbation method is introduced in the context of dynamical system for solving the nonlinear Korteweg-de Vries (KdV) equation. Best efficiency is obtained for few perturbative corrections. It is shown that, the qu...A perturbation method is introduced in the context of dynamical system for solving the nonlinear Korteweg-de Vries (KdV) equation. Best efficiency is obtained for few perturbative corrections. It is shown that, the question of convergence of this approach is completely guaranteed here, because a limited number of term included in the series can describe a sufficient exact solution. Comparisons with the solutions of the quintic spline, and finite difference are presented.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
In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions ...In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.展开更多
基金Supported by the National Natural Science Foundation of China under Grant No. 60772023by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. BUAA-SKLSDE-09KF-04+2 种基金Beijing University of Aeronautics and Astronautics, by the National Basic Research Program of China (973 Program) under Grant No. 2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 200800130006Chinese Ministry of Education, and Scientific Research Common Program of Beijing Municipal Commission of Education under Grant No. KM201010772020
文摘The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.
基金Project supported by the National Natural Science Foundation of China （Grant No 10472029）.
文摘Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg-de Vries （KdV） and modified KdV （mKdV） equation and （2＋1）-dimensional Broer-Kaup-Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.
基金The National Basic Research Program of China under contract Nos 2011CB403503 and 2012CB955601the Scientific Research Fund of the Second Institute of Oceanography, the State Oceanic Administration of China under contract Nos JG1009, JT1006 and JT0905
文摘The mesoscale eddy and internal wave both are phenomena commonly observed in oceans. It is aimed to investigate how the presence of a mesoscale eddy in the ocean affects wave form deformation of the internal solitary wave propagation. An ocean eddy is produced by a quasi-geostrophic model in f-plane, and the one-dimensional nonlinear variable-coefficient extended Korteweg-de Vries （eKdV） equation is used to simulate an internal solitary wave passing through the mesoscale eddy field. The results suggest that the mode structures of the linear internal wave are modified due to the presence of the mesoscale eddy field. A cyclonic eddy and an anticyclonic eddy have different influences on the background environment of the internal solitary wave propagation. The existence of a mesoscale eddy field has almost no prominent impact on the propagation of a smallamplitude internal solitary wave only based on the first mode vertical structure, but the mesoscale eddy background field exerts a considerable influence on the solitary wave propagation if considering high-mode vertical structures. Furthermore, whether an internal solitary wave first passes through anticyclonic eddy or cyclonic eddy, the deformation of wave profiles is different. Many observations of solitary internal waves in the real oceans suggest the formation of the waves. Apart from topography effect, it is shown that the mesoscale eddy background field is also a considerable factor which influences the internal solitary wave propagation and deformation.
文摘Amongst the important phenomena in neurophysiology, nerve pulse generation and propagation is fundamental. Scientists have studied this phenomena using mathematical models based on experimental observations on the physiological processes in the nerve cell. Widely used models include: the Hodgkin-Huxley (H-H) model, which is based entirely on the electrical activity of the nerve cell;and the Heimburg and Jackson (H-J), model based on the thermodynamic activity of the nerve cell. These classes of models do not, individually, give a complete picture of the processes that lead to nerve pulse generation and propagation. Recently, a hybrid model proposed by Mengnjo, Dikandé and Ngwa (M-D-N), takes into consideration both the electrical and thermodynamic activities of the nerve cell. In their work, the first three bound states of the model are analytically computed and they showed great resemblance to some of the experimentally observed pulse profiles. With these bound states, the M-D-N model reduces to an initial value problem of a linear parabolic partial differential equation with variable coefficients. In this work we consider the resulting initial value problem and, using the theory of function spaces, propose and prove conditions under which such equations will admit unique solutions. We then verify that the resulting initial value problem from the M-D-N model satisfies these conditions and so has a unique solution. Given that the derived initial value problem is complex and there are no known analytic techniques that can be deployed to obtain its solution, we designed a numerical experiment to estimate the solutions. The simulations revealed that the unique solution is a stable pulse that propagates in the x-t plane with constant velocity and maintains the shape of the initial profile.
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.
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In this paper, based on the concatenating method, we present a unified framework to construct a series of local structure-preserving algorithms for the Korteweg-de Vries （KdV） equation, including eight multi-symplectic algorithms, eight local energy-conserving algo- rithms and eight local momentum-conserving algorithms. Among these algorithms, some have been discussed and widely used while the most are new. The outstanding advantage of these proposed algorithms is that they conserve the local structures in any time-space re- gion exactly. Therefore, the local structure-preserving algorithms overcome the restriction of global structure-preserving algorithms on the boundary conditions. Numerical experiments are conducted to show the performance of the proposed methods. Moreover, the unified framework can be easily applied to many other equations.
文摘The major concern of this work is to propose new prototypes of surface hybrid waves, in particular waves propagating without sprawl or deformation on the surface of a fluid. The model considered for this purpose is the modified KdV (Korteweg-de Vries) equation. A peculiarity of the obtained solutions is that they form packages constituted by combinations of waves belonging to the two main families of well-known bright and dark solitary waves. This putting together creates competitions between the different components of the considered packages which, following the values assigned to the parameters of the considered system and in relation to those of the wave parameters, generate hybrid or multi-form structures. The direct method of resolution which made possible the obtained results is that of Bogning-Djeumen Tchaho-Kofane extended to the new implicit Bogning functions. The existence conditions of some solutions are obtained. The numerical simulations carried out with a view to testing the observable and applicable characters of the obtained solutions revealed their stabilities over a relatively long time, and at the same time, confirmed the recommended theoretical forecasts. We are convinced that the solutions proposed as part of this work will make it possible to detect, understand and explain some physical phenomena linked to fluid molecular interactions, former or new, which constantly occur on the fluid surfaces, mainly at the shallow water surface.
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.
基金Supported by National Natural Science Foundation of China under Grant No.11505090Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No.BS2015SF009
文摘This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.
文摘A high-order finite difference Pade scheme also called compact scheme for solving Korteweg-de Vries （KdV） equations, which preserve energy and mass conservations, was developed in this paper. This structure-preserving algorithm has been widely applied in these years for its advantage of maintaining the inherited properties. For spatial discretization, the authors obtained an implicit compact scheme by which spatial derivative terms may be approximated through combining a few knots. By some numerical examples including propagation of single soliton and interaction of two solitons, the scheme is proved to be effective.
基金supported by the National Natural Science Foundation of China under Grant Nos.12147115 and 11835011the Natural Science Foundation of Anhui Province under Grant No.2108085QA09+3 种基金the University Natural Science Research Project of Anhui Province under Grant No.KJ2021A1094China Postdoctoral Science Foundation under Grant No.2022M712833the Program for Science and Technology Innovation Talents in Universities of Henan Province under Grant No.22HASTIT019the Natural Science Foundation of Henan Province under Grant No.202300410524
文摘In this paper,we investigate the fifth-order modified Korteweg-de Vries(mKdV)equation on the half-line via the Fokas unified transformation approach.We show that the solution u(x,t)of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ.The jump matrix L(x,t,θ)has an explicit representation dependent on x,t and it can be represented exactly by the two pairs of spectral functions y(θ),z(θ)(obtained from the initial value u0(x))and Y(θ),Z(θ)(obtained from the boundary conditions v0(t),{vk(t)}_(1)^(4)).Furthermore,the two pairs of spectral functions y(θ),z(θ)and Y(θ),Z(θ)are not independent of each other,but are related to the compatibility condition,the so-called global relation.
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.
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Project (No. D0701/01/05) supported by Ministry of the Educationand Scientific Research (M.E.S.R), Algeria
文摘A perturbation method is introduced in the context of dynamical system for solving the nonlinear Korteweg-de Vries (KdV) equation. Best efficiency is obtained for few perturbative corrections. It is shown that, the question of convergence of this approach is completely guaranteed here, because a limited number of term included in the series can describe a sufficient exact solution. Comparisons with the solutions of the quintic spline, and finite difference are presented.
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In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.
