Scintillation in the White-Noise Paraxial Regime

In this paper the white-noise paraxial wave model is considered. This model describes for instance the propagation of laser beams in the atmosphere in some typical scaling regimes. The closed-form equations for the second- and fourth-order moments of the field are solved in two particular situations. The first situation corresponds to a random medium with a transverse correlation radius smaller than the beam radius. This is the spot-dancing regime: the beam shape spreads out as in a homogeneous medium and its center is randomly shifted according to a Gaussian process whose variance grows like the third power of the propagation distance. The second situation corresponds to a plane-wave initial condition, a small amplitude for the medium fluctuations, and a large propagation distance. This is the scintillation regime: the normalized variance of the intensity converges to one exponentially with the propagation distance, corresponding to strong intensity fluctuations and in agreement with the conjecture that the statistics of the field becomes complex Gaussian.     

Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case

We prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method.     

Spectral analysis of nonselfadjoint Schr(o)dinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schr(o)dinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator,and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schr(o)dinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schr(o)dinger boundary value problem are given.