This paper is concerned with the nonlinear Schrödinger equation iut = uxx − mu − f (|u|2)u, (1) on the finite x-interval [0, π] with Dirichlet boundary conditions u(t, 0) = 0 = u(t, π), − ∞ < t < ∞.

The nonlinear Schr6dinger (NLS) equation i ~ t + V2 ~ + al~lS ~ = 0 is a canonical and universal equation which is of major importance in continuum echanics, plasma physics and optics. This paper argues that much of the observed solution behavior in the critical case sd = 4, where d is dimension

parabolic equation on bounded domain with critical Sobolev exponent Takashi SUZUKI (Osaka Univ.) The semilinear parabolic equation ut −∆u = |u|p−1 u in Ω × (0, T), u|∂Ω = 0, u|t=0 = u0(x) (1) was introduced pure mathematically, but its profiles of the solution, particularly blowup in finite and in

The system of two coupled nonlinear Schrödinger equations has wide applications in physics. In the past, the main attention has been their solitary waves. Here we turn our attention to their periodic wave solutions. In this paper, the stability of the periodic solutions is studied analytically

in the nonlinear Schr6dinger equation

in R 3 is known to exhibit finite time blowup for data of negative energy. It also admits the special static solutions φ(x, a) = (3a) 1 4 (1 + a|x | 2) − 1 2 for all a> 0 which are linearly unstable. We view these functions as a curve in the energy space ˙ H 1 × L 2. We show that in a small neighborhood of itself, which lies on a stable hyper-surface of radial data, this curve acts as a one-dimensional attractor.

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Abstract. We study an adiabatic invariant for the time-dependent Schrfdinger equation which gives the transition probability across a gap from time t' to time t. When the hamiltonian depends analytically on time, and t' = -oo, t = + oo we give sufficient conditions so that this adiabatic

Abstract -A perturbation theory for determining the stability characteristics of spatial optical solitons with a 2D transverse profile in a transparent medium with a weak saturation of nonlinear refractive index is developed. For Kerr nonlinearity, a new solution of linearized equations for weak

nonlinear Schr6dinger (VNLS) hierarchies are contained within the KP hierarchy in this way. These hierarchies approximate the KP hierarchy in the limit of large N, and this permits the equations of the KP hierarchy to be approximated by nonlinear equations in 1 + 1 dimensions.