Nonlinear Schrodinger equation (NSE) arises in many physical problems. It is a very important equation. A lot of works studied the wellposed, the existence of solution of NSE etc. And there are many works studied the ...Nonlinear Schrodinger equation (NSE) arises in many physical problems. It is a very important equation. A lot of works studied the wellposed, the existence of solution of NSE etc. And there are many works studied the numerical methods for it. Recently, since the development of infinite dimensional dynamic system the dynamical behavior of NSE has been investigated. The paper [1] studied the long time wellposedness, the existence of universal attractor and the estimate of Lyapunov exponent for NSE with weakly damped. At the same time it was need to study the large time new computational methods and to discuss its convergence error estimate, the existence of approximate attractors etc. In this pape we study the NSE with weakly damped (1.1). We assume,where 0【λ【2 is a constant. If we wish to construct the higher accuracy computational scheme, it will be difficult that staigh from the equation (1.1). Therefore we start with (1. 4) and use fully discrete Fourier spectral method with time difference to展开更多
This paper aims to propose an improved numerical model for wave breaking in the nearshore region based on the fully nonlinear form of Boussinesq equations. The model uses the κ equation turbulence scheme to determine...This paper aims to propose an improved numerical model for wave breaking in the nearshore region based on the fully nonlinear form of Boussinesq equations. The model uses the κ equation turbulence scheme to determine the eddy viscosity in the Boussinesq equations. To calculate the turbulence production term in the equation, a new formula is derived based on the concept of surface roller. By use of this formula, the turbulence production in the one-equation turbulence scheme is directly related to the difference between the water particle velocity and the wave celerity. The model is verified by Hansen and Svendsen’s experimental data (1979) in terms of wave height and setup and setdown. The comparison between the model and experimental results of wave height and setup and setdown shows satisfactory agreement. The modeled turbulence energy decreases as waves attenuate in the surf zone. The modeled production term peaks at the breaking point and decreases as waves propagate shoreward. It is also suggested that both convection and diffusion play their important roles in the transport of turbulence energy immediately after wave breaking. When waves approach to the shoreline, the production and dissipation of turbulence energy are almost balanced. By use of the slot technique for the simulation of the movable shoreline boundary, wave runup in the swash zone is well simulated by the present model.展开更多
In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is use...In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is used to study the asymptotic behaviour of Kuramoto-Sivashinsky equation and to construct the bifurcation diagrams.展开更多
We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions.When these conditions ...We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions.When these conditions are violated,there can be blow up of the gradient in the interior or on the boundary of the domain.In particular we derive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before.Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense,where detachment can occur.Another consequence is this:if interior gradient blow up occurs,Perron-type solutions can in general become discontinuous,so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.展开更多
In this paper, coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations with variable coefficients are studied, which can be used to describe the interaction amon...In this paper, coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations with variable coefficients are studied, which can be used to describe the interaction among the modes in nonlinear optics and Bose-Einstein condensation. Some novel bright-dark solitons and dark-dark solitons are obtained by modified Sine-Gordon equation method. Moreover, some figures are provided to illustrate how the soliton solutions propagation is determined by the different values of the variable group velocity dispersion terms, which can be used to model various phenomena.展开更多
In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is ellipt...In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is elliptic,for example,in the class of convex functions.We show that the notion of Monge-Ampere measures and Aleksandrov generalized solutions extends to this equation,subject to a weaker notion of convexity which we call bi-planar convexity.While the equation is also elliptic in the class of bi-planar convex functions,the contrary is not necessarily true.This is a substantial difference compared to the classical Monge-Ampere equation where ellipticity and convexity coincide.We provide explicit counter-examples:classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced.We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.展开更多
Interfacial waves propagating along the interface between a three-dimensional two-fluid system with a rigid upper boundary and an uneven bottom are considered. There is a light fluid layer overlying a heavier one in t...Interfacial waves propagating along the interface between a three-dimensional two-fluid system with a rigid upper boundary and an uneven bottom are considered. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. A set of higher-order Boussinesq-type equations in terms of the depth-averaged velocities accounting for stronger nonlinearity are derived. When the small parameter measuring frequency dispersion keeping up to lower-order and full nonlinearity are considered, the equations include the Choi and Camassa's results (1999). The enhanced equations in terms of the depth-averaged velocities are obtained by applying the enhancement technique introduced by Madsen et al. (1991) and Schaffer and Madsen (1995a). It is noted that the equations derived from the present study include, as special cases, those obtained by Madsen and Schaffer (1998). By comparison with the dispersion relation of the linear Stokes waves, we found that the dispersion relation is more improved than Choi and Camassa's (1999) results, and the applicable scope of water depth is deeper.展开更多
In this paper, we extend the results established by Caffarelli, Nirenberg and Spruck for elliptic case to parabolic case and study the classical solvability for fully nonlinear and non-uniformly parabolic equation-Dtu...In this paper, we extend the results established by Caffarelli, Nirenberg and Spruck for elliptic case to parabolic case and study the classical solvability for fully nonlinear and non-uniformly parabolic equation-Dtuf(K(u))=ψ (x,t) subject to the initial-boundary condition u=(x, t). Here K(u) denotes the principal curvatures of the graph (x. u(x)).展开更多
In this study,the sine-Gordon equation method is modified to deal with variable-coefficient systems containing imaginary parts,such as nonlinear Schrödinger systems.These are of considerable importance in many fi...In this study,the sine-Gordon equation method is modified to deal with variable-coefficient systems containing imaginary parts,such as nonlinear Schrödinger systems.These are of considerable importance in many fields of research,including ocean engineering and optics.As an example,we apply the modified method to variable-coefficient coupled nonlinear Schrödinger equations and Davey-Stewartson system with variable coefficients,treating them as one-dimensional and two-dimensional systems,respectively.As a result of this application,novel solitary wave solutions are obtained for both cases.Moreover,some figures are provided to illustrate how the solitary wave propagation is determined by the different values of the variable group velocity dispersion terms,which can be used to model various phenomena.展开更多
The lowest order H^1-Galerkin mixed finite element method(for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order RaviartThomas element....The lowest order H^1-Galerkin mixed finite element method(for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order RaviartThomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h^2)/O(h^2+ τ~2) in H^1-norm and H(div; ?)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively,which improve the results in the previous literature.展开更多
Interior regularity results for viscosity solutions of fully nonlinear uniformly parabolicequations under the Dini condition, which improve and generalize a result due to Kovats, are obtainedby the use of the approxim...Interior regularity results for viscosity solutions of fully nonlinear uniformly parabolicequations under the Dini condition, which improve and generalize a result due to Kovats, are obtainedby the use of the approximation lemma.展开更多
This paper is devoted to the investigation of C<sup>1,α</sup> regularity of viscosity solutions for aclass of fully nonlinear partial differential equations. The Dirichlet problem of elliptic equa-tions a...This paper is devoted to the investigation of C<sup>1,α</sup> regularity of viscosity solutions for aclass of fully nonlinear partial differential equations. The Dirichlet problem of elliptic equa-tions and the first boundary value problem of parabolic equations are treated in this paper.展开更多
We discuss the existence of global classical solution for the uniformly parabolicequation■ut=a(x,t,u,u<sub>x</sub>,u<sub>xx</sub>)+b(x,t,u,u<sub>x</sub>),(x,t)∈(-1,1)×(...We discuss the existence of global classical solution for the uniformly parabolicequation■ut=a(x,t,u,u<sub>x</sub>,u<sub>xx</sub>)+b(x,t,u,u<sub>x</sub>),(x,t)∈(-1,1)×(0,T],u(±1,t)=0,u(x,0)=■(x),where a is strongly nonlinear with respect to u<sub>xx</sub>and ■ is not necessarily small.We also dealwith nonuniform case.展开更多
Using the methods of dynamical systems for the (n+ 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions axe obtained. Fo...Using the methods of dynamical systems for the (n+ 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions axe obtained. For the double sine-Gordon equation, the exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.展开更多
The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odin...The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.展开更多
EQ rot 1 nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is...EQ rot 1 nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2 ) one order higher than its interpolation error O(h), the superclose results of order O(h2 ) in broken H1 -norm are obtained. At the same time, the global superconvergence in broken H1 -norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4 ) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQ rot 1 element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.展开更多
文摘Nonlinear Schrodinger equation (NSE) arises in many physical problems. It is a very important equation. A lot of works studied the wellposed, the existence of solution of NSE etc. And there are many works studied the numerical methods for it. Recently, since the development of infinite dimensional dynamic system the dynamical behavior of NSE has been investigated. The paper [1] studied the long time wellposedness, the existence of universal attractor and the estimate of Lyapunov exponent for NSE with weakly damped. At the same time it was need to study the large time new computational methods and to discuss its convergence error estimate, the existence of approximate attractors etc. In this pape we study the NSE with weakly damped (1.1). We assume,where 0【λ【2 is a constant. If we wish to construct the higher accuracy computational scheme, it will be difficult that staigh from the equation (1.1). Therefore we start with (1. 4) and use fully discrete Fourier spectral method with time difference to
基金This study was supported by the National Natural Science Foundation of China (Grant No.50479047) and partly by the National Science Fund for Distinguished Young Scholars of China (Estuarine and Coastal Science, Grant No.40225014)
文摘This paper aims to propose an improved numerical model for wave breaking in the nearshore region based on the fully nonlinear form of Boussinesq equations. The model uses the κ equation turbulence scheme to determine the eddy viscosity in the Boussinesq equations. To calculate the turbulence production term in the equation, a new formula is derived based on the concept of surface roller. By use of this formula, the turbulence production in the one-equation turbulence scheme is directly related to the difference between the water particle velocity and the wave celerity. The model is verified by Hansen and Svendsen’s experimental data (1979) in terms of wave height and setup and setdown. The comparison between the model and experimental results of wave height and setup and setdown shows satisfactory agreement. The modeled turbulence energy decreases as waves attenuate in the surf zone. The modeled production term peaks at the breaking point and decreases as waves propagate shoreward. It is also suggested that both convection and diffusion play their important roles in the transport of turbulence energy immediately after wave breaking. When waves approach to the shoreline, the production and dissipation of turbulence energy are almost balanced. By use of the slot technique for the simulation of the movable shoreline boundary, wave runup in the swash zone is well simulated by the present model.
文摘In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is used to study the asymptotic behaviour of Kuramoto-Sivashinsky equation and to construct the bifurcation diagrams.
基金financed by the Alexander von Humboldt Foundationcontinued in March 2009 at the Mathematisches Forschungsinstitut Oberwolfach in the "Research in Pairs"program
文摘We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions.When these conditions are violated,there can be blow up of the gradient in the interior or on the boundary of the domain.In particular we derive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before.Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense,where detachment can occur.Another consequence is this:if interior gradient blow up occurs,Perron-type solutions can in general become discontinuous,so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.
文摘In this paper, coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations with variable coefficients are studied, which can be used to describe the interaction among the modes in nonlinear optics and Bose-Einstein condensation. Some novel bright-dark solitons and dark-dark solitons are obtained by modified Sine-Gordon equation method. Moreover, some figures are provided to illustrate how the soliton solutions propagation is determined by the different values of the variable group velocity dispersion terms, which can be used to model various phenomena.
基金This article contributes to the project"Systematic multi-scale modeling and analysis for geophysical flow"of the Collaborative Research Center TRR 181"Energy Transfers in Atmosphere and Ocean"funded by the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)under project number 274762653.
文摘In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is elliptic,for example,in the class of convex functions.We show that the notion of Monge-Ampere measures and Aleksandrov generalized solutions extends to this equation,subject to a weaker notion of convexity which we call bi-planar convexity.While the equation is also elliptic in the class of bi-planar convex functions,the contrary is not necessarily true.This is a substantial difference compared to the classical Monge-Ampere equation where ellipticity and convexity coincide.We provide explicit counter-examples:classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced.We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.
基金Knowledge Innovation Programs of the Chinese Academy of Sciences under contract Nos KZCX2-YW-201 and KZCX1-YW-12Natural Science Fund supported by the Educational Department of Inner Mongolia under contract Nos NJzy080005,and NJ09011A Grant from Science Fund for Young Scholars of Inner Mongolia University under contract NoND0801
文摘Interfacial waves propagating along the interface between a three-dimensional two-fluid system with a rigid upper boundary and an uneven bottom are considered. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. A set of higher-order Boussinesq-type equations in terms of the depth-averaged velocities accounting for stronger nonlinearity are derived. When the small parameter measuring frequency dispersion keeping up to lower-order and full nonlinearity are considered, the equations include the Choi and Camassa's results (1999). The enhanced equations in terms of the depth-averaged velocities are obtained by applying the enhancement technique introduced by Madsen et al. (1991) and Schaffer and Madsen (1995a). It is noted that the equations derived from the present study include, as special cases, those obtained by Madsen and Schaffer (1998). By comparison with the dispersion relation of the linear Stokes waves, we found that the dispersion relation is more improved than Choi and Camassa's (1999) results, and the applicable scope of water depth is deeper.
文摘In this paper, we extend the results established by Caffarelli, Nirenberg and Spruck for elliptic case to parabolic case and study the classical solvability for fully nonlinear and non-uniformly parabolic equation-Dtuf(K(u))=ψ (x,t) subject to the initial-boundary condition u=(x, t). Here K(u) denotes the principal curvatures of the graph (x. u(x)).
基金The authors would like to thank the Deanship of Scientific Research,Majmaah University,Saudi Arabia,for funding this work under project No.R-1441-26.
文摘In this study,the sine-Gordon equation method is modified to deal with variable-coefficient systems containing imaginary parts,such as nonlinear Schrödinger systems.These are of considerable importance in many fields of research,including ocean engineering and optics.As an example,we apply the modified method to variable-coefficient coupled nonlinear Schrödinger equations and Davey-Stewartson system with variable coefficients,treating them as one-dimensional and two-dimensional systems,respectively.As a result of this application,novel solitary wave solutions are obtained for both cases.Moreover,some figures are provided to illustrate how the solitary wave propagation is determined by the different values of the variable group velocity dispersion terms,which can be used to model various phenomena.
基金Supported in part by the National Natural Science Foundation of China under Grant Nos.11671369,11271340the Natural Science Foundation of the Education Department of Henan Province under Grant Nos.14A110009,16A110022
文摘The lowest order H^1-Galerkin mixed finite element method(for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order RaviartThomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h^2)/O(h^2+ τ~2) in H^1-norm and H(div; ?)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively,which improve the results in the previous literature.
文摘Interior regularity results for viscosity solutions of fully nonlinear uniformly parabolicequations under the Dini condition, which improve and generalize a result due to Kovats, are obtainedby the use of the approximation lemma.
基金Project supported by the National Natural Science Foundation of China.
文摘This paper is devoted to the investigation of C<sup>1,α</sup> regularity of viscosity solutions for aclass of fully nonlinear partial differential equations. The Dirichlet problem of elliptic equa-tions and the first boundary value problem of parabolic equations are treated in this paper.
基金Supported by the Open Office of Mathematica Institute,Academia Sinica.
文摘We discuss the existence of global classical solution for the uniformly parabolicequation■ut=a(x,t,u,u<sub>x</sub>,u<sub>xx</sub>)+b(x,t,u,u<sub>x</sub>),(x,t)∈(-1,1)×(0,T],u(±1,t)=0,u(x,0)=■(x),where a is strongly nonlinear with respect to u<sub>xx</sub>and ■ is not necessarily small.We also dealwith nonuniform case.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 11671179)the Natural Science Foundation of Yunnan Province (Grant No. 2005A0092M).
文摘Using the methods of dynamical systems for the (n+ 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions axe obtained. For the double sine-Gordon equation, the exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.
文摘The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.
基金Supported by the National Natural Science Foundation of China (Nos. 10971203 11101381)+3 种基金Tianyuan Mathe-matics Foundation of National Natural Science Foundation of China (No. 11026154)Natural Science Foundation of Henan Province (No. 112300410026)Natural Science Foundation of the Education Department of Henan Province (Nos. 2011A110020 12A110021)
文摘EQ rot 1 nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2 ) one order higher than its interpolation error O(h), the superclose results of order O(h2 ) in broken H1 -norm are obtained. At the same time, the global superconvergence in broken H1 -norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4 ) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQ rot 1 element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.