Asymptotic Analysis

We study the Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic boundary data for large t > 0 . We derive an expression for the Dirichlet to Neumann map to all orders in the perturbative expansion of a small ϵ > 0 in the case of the asymptotically periodic boundary data. More precisely, we show that if the unknown Neumann boundary data are asymptotically periodic for large t in the sense that u x ( 0 , t ) and u x x ( 0 , t ) tend to periodic functions g ˜ 1 ( t ) and g ˜ 2 ( t ) for large t, respectively, then the periodic functions g ˜ 1 ( t ) and g ˜ 2 ( t ) can be characterized in terms of the given asymptotically periodic Dirichlet boundary datum u ( 0 , t ) . Moreover, we determine effectively the Fourier coefficients of the functions g ˜ 1 ( t ) and g ˜ 2 ( t ) by solving a certain recursive algebraic equations.

We study the resonances of 2 × 2 systems of one dimensional Schrödinger operators which are related to the mathematical theory of molecular predissociation. We determine the precise positions of the resonances with real parts below the energy where bonding and anti-bonding potentials intersect transversally. In particular, we find that imaginary parts (widths) of the resonances are exponentially small and that the indices are determined by Agmon distances for the minimum of two potentials.

We consider a family of linear viscoelastic shells with thickness 2 ε (where ε is a small parameter), clamped along a portion of their lateral face, all having the same middle surface S. We formulate the three-dimensional mechanical problem in curvilinear coordinates and provide existence and uniqueness of (weak) solution of the corresponding three-dimensional variational problem.

We are interested in studying the limit behavior of both the three-dimensional problems and their solutions when ε tends to zero. To do that, we use asymptotic analysis methods. First, we formulate the variational problem in a fixed domain independent of ε. Then we assume an asymptotic expansion of the scaled displacements field, u ( ε ) = ( u i ( ε ) ) , and we characterize the zeroth order term as the solution of a two-dimensional scaled limit problem. Moreover, we find that, depending on the order of the applied forces, the limit of the field u ( ε ) is the solution of one of the two sets of two-dimensional variational equations derived, which can be described as viscoelastic membrane shell and viscoelastic flexural shell problems. In both cases, we find a model which presents a long-term memory that takes into account the deformations at previous times. We finally comment on the existence and uniqueness of solution for the two-dimensional variational problems found and announce convergence results.

In this paper we study the validity of a Gausson (soliton) dynamics of the logarithmic Schrödinger equation in presence of a smooth external potential.