In this article we consider the(complex)Ginzburg-Landau equation,we discretize in time using the implicit Euler scheme,and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prov...In this article we consider the(complex)Ginzburg-Landau equation,we discretize in time using the implicit Euler scheme,and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.展开更多
By establishing concept an transient solutions of general nonlinear systems converging to its equilibrium set, long-time behavior of solutions for cellular neural network systems is studied. A stability condition in g...By establishing concept an transient solutions of general nonlinear systems converging to its equilibrium set, long-time behavior of solutions for cellular neural network systems is studied. A stability condition in generalized sense is obtained. This result reported has an important guide to concrete neural network designs.展开更多
The object of this paper is to establish the relation between stability and convergence of the numerical methods for the evolution equation u(t) - Au - f(u) = g(t) on Banach space V, and to prove the long-time error e...The object of this paper is to establish the relation between stability and convergence of the numerical methods for the evolution equation u(t) - Au - f(u) = g(t) on Banach space V, and to prove the long-time error estimates for the approximation solutions. At first, we give the definition of long-time stability, and then prove the fact that stability and compatibility imply the uniform convergence on the infinite time region. Thus, we establish a general frame in order to prove the long-time convergence. This frame includes finite element methods and finite difference methods of the evolution equations, especially the semilinear parabolic and hyperbolic partial differential equations. As applications of these results we prove the estimates obtained by Larsson [5] and Sanz-serna and Stuart [6].展开更多
In this paper,the authors apply■steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data iqt+qxx+1(lq|^(2)q)_(x)=0,q(x,0)=q0(x),...In this paper,the authors apply■steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data iqt+qxx+1(lq|^(2)q)_(x)=0,q(x,0)=q0(x),where lim x→±∞ qo(x)=g0(x)=q±and|q±|=1.Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert problem.They compute the long time asymptotic expansion of the solution in differeit space-time regions.For the regionζ=x/t with|ζ+2|<1,the long time asymptotic is given by q(x,t)=T(∞)^(-2)q^(r)Λ(x,t)+O(t^(-3/4)),in which the leading term is N(I)solitons,the second term is a residual error from a■equation.For the regionζ+2|>1,the long time asymptotic is given by q(x,t)=t(∞)^(-2)q^(r)Λ(x,t)-t^(-1/2)if11+O(t^(-3/4)) in which the leading term is N(I)solitons,the second t^(-1/2)order term is soliton-radiation interactions and the third term is a residual error from a■equation.These results are verification of the soliton resolution conjectuore for the derivative Schrödinger equation.In their case of finite density type initial data,the phase functionθ(z)is more complicated that in finite mass initial data.Moreover,two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis,respectively.展开更多
文摘In this article we consider the(complex)Ginzburg-Landau equation,we discretize in time using the implicit Euler scheme,and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.
文摘By establishing concept an transient solutions of general nonlinear systems converging to its equilibrium set, long-time behavior of solutions for cellular neural network systems is studied. A stability condition in generalized sense is obtained. This result reported has an important guide to concrete neural network designs.
文摘The object of this paper is to establish the relation between stability and convergence of the numerical methods for the evolution equation u(t) - Au - f(u) = g(t) on Banach space V, and to prove the long-time error estimates for the approximation solutions. At first, we give the definition of long-time stability, and then prove the fact that stability and compatibility imply the uniform convergence on the infinite time region. Thus, we establish a general frame in order to prove the long-time convergence. This frame includes finite element methods and finite difference methods of the evolution equations, especially the semilinear parabolic and hyperbolic partial differential equations. As applications of these results we prove the estimates obtained by Larsson [5] and Sanz-serna and Stuart [6].
基金supported by the National Natural Science Foundation of China(Nos.51879045,1202624,118013233,11671095)。
文摘In this paper,the authors apply■steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data iqt+qxx+1(lq|^(2)q)_(x)=0,q(x,0)=q0(x),where lim x→±∞ qo(x)=g0(x)=q±and|q±|=1.Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert problem.They compute the long time asymptotic expansion of the solution in differeit space-time regions.For the regionζ=x/t with|ζ+2|<1,the long time asymptotic is given by q(x,t)=T(∞)^(-2)q^(r)Λ(x,t)+O(t^(-3/4)),in which the leading term is N(I)solitons,the second term is a residual error from a■equation.For the regionζ+2|>1,the long time asymptotic is given by q(x,t)=t(∞)^(-2)q^(r)Λ(x,t)-t^(-1/2)if11+O(t^(-3/4)) in which the leading term is N(I)solitons,the second t^(-1/2)order term is soliton-radiation interactions and the third term is a residual error from a■equation.These results are verification of the soliton resolution conjectuore for the derivative Schrödinger equation.In their case of finite density type initial data,the phase functionθ(z)is more complicated that in finite mass initial data.Moreover,two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis,respectively.
