The Dirichlet Boundary Value Problem For Nonlinear Schr(?)dinger Equtions Of Nonautonomous

Schrodinger Equtions is a classics of partial differential equations.We achieve many scientific result for study of its quantity theoty.But when we looking for the quasi-periodic solutions of d-dimension partial differential equation(d> 2),we have to meet a small divison problems. The Hamiltonian systems and KAM theory is a very powerful tool to deal with the problems. The classical KAM theory is concerned with the existence of invariant tori (thus quasi-periodic solutions) for nearly integrable Hamiltonian systems. In order to obtain the quasi-periodic solutions of a partial differential equation, one may show the existence of the lower(finite) dimensional invariant tori for the infinite dimensional Hamiltonian systems defined by the partial differential equation.The paper research the Dirichlet boundary value problem for nonlinear schrodinger equtions of nonautonomous the followingwhere nonlinear function f is required to be real analytic and nondegenerate in some neighbourhood of the origin in C.Thus equation (1) is expressed the following formWe use the transformation u =εv(0 <ε<< 1) and desert the higher order term of v,the equation (2) is expressed the following formAfter that, To research the dirichlet boundary value problem for (1) change to looking for the quasi-periodic solution of problem (4).The frist, the equation (3) is replaced by the equation (5)Step 2, We gived the Hamiltonian function (6) of the equation (5) where AStep 3, We obtain the Birkhoff normal form for the Hamiltonian function (6) is expressed the following3.4.Lemma 3.4 For the Hamiltonian H =εG with nonlinearity ,there exists a real analytic symplectic change of coordinates F in a neighbourhood of the origin in l^a,p that for all real values of m takes it into its Birkhoff normal form up to order four. That iswhere m determinid coefficientsStep 4, We established the Cantor manifold theorem .For the existence ofεthe following assumptions are made.(1) Nondegeneracy.The normal form A + Q' is nondegeneracy in the sense thatwhere(2) Spectral Asymptotics.There exist d≥1,δ< d - 1,such thatwhere the dots stand for terms of order less than d in j. (3) Regularity.where A(l^a,p, l^a,p) denotes the class of all maps from some neighbourhood of the origin in l^a,p into l^a,p. which are real analytic in the real and imaginary parts of the complex coordinate q.Theorem3.2(Cantor manifold theorem) Suppose the HamiltonianH = A + Q' + R satisfies assumptions(1)-(3), andwith g > ,Δ= min(p - p, 1). Then there exist a Cantor manifoldεof real analytic,elliptic diophantine n-tori given by a Lipschitz continuous embedding ,where C has full density at the origin,and (?) is close to the inclusion map (?)₀Consequently,εis tangent to E at the origin .Finally,We obtain the important conclusion of the paper by using the Lemma 3.4 and Cantor manifold theoremTheorem3.1 Suppose the nonlinearity f is real analytic and nonde-gence,Then for all m∈M, n∈N, and all J = {j₁ <…< j_n} (?) N, there exist a Cantor manifoldε_J of real analytic,linearly stable, diophantine-n tori for the equation (1) given by a Lipschitz continuous embedding .which is a higher order perturbation of the inclusion map restricted to T_J[C].The Cantor set C has full density at the origin, whenceε_J has a tangent space at the origin equal to E_J. Moreover,ε_J is contained in the space of analytic functions on [0,Ï€].