On instability for the cubic nonlinear Schrodinger equation. Communicated by Prof. …

Global Well-posedness and Scattering for the Defocusing Cubic nonlinear Schrödinger equation in Four Dimensions Monica Vişan. For the Korteweg–de Vries (KdV) equation, there is a similar conserved energy for every s ≥ − 1. \\] In the first part of the paper, we analyze the one-parameter family of ground-state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. One of the simplest extensions of the cubic NLSE is the so-called cubic-quintic nonlinear Schrödinger (CQNLS) model, which in normalized units and 1D is (2) i q t (x, t) = − q xx (x, t) + G 1 | q (x, t) | 2 q (x, t) + G 2 | q (x, t) | 4 q (x, t) + V (x, t), where G 2 is the strength of quintic nonlinearity. We consider the cubic-quintic nonlinear Schrödinger equation: \\[ i\\partial_t u = -Δu - |u|^2u + |u|^4u. Exact Solutions > Nonlinear Partial Differential Equations > Second-Order Parabolic Partial Differential Equations > Schrodinger Equation with a Cubic Nonlinearity 1. i @w @t + @2w @x2 + k|w|2w = 0. The cubic nonlinear Schrödinger equation (CNLS) , , , is one of the most universal models that describe many physical nonlinear systems. We study the flow map … It arises as a simplified model for studying Bose–Einstein condensates [22, 27, 53], Kerr media in nonlinear optics [33, 63], and even freak waves in the ocean [21, 29]. We consider the Cauchy problem for the 1-dimensional periodic cubic nonlinear Schrödinger (NLS) equation with initial data below L 2. Google Scholar. Schrodinger (Schrodinger) equation with a cubic nonlinearity.¨ Here, w is a complex functions of real variables x and t; k is a real number, i2=−1. Search for other works by this author on: Oxford Academic. Correspondence to be sent to: visan@math.ucla.edu. Note. We prove that for each s > − 1 2 there exists a conserved energy which is equivalent to the H s-norm of the solution. 10.1016/j.crma.2007.03.006. Comptes ren-dus de l’Académie des sciences. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Monica Vişan Department of Mathematics, 520 Portola Plaza, UCLA, Los Angeles, CA 90095, USA. Série I, Mathématique, Elsevier, 2007, 344 (8), pp.483-486. Several monographs and numerous research papers have been devoted to studying the CNLS equation which possess special solution in the form of pulses which retain their shapes and velocities after interaction amongst themselves, see [3] , [4] , [9] , … The cubic nonlinearity is the most common nonlinearity in applications. hal-00127816 hal-00127816, version 1 - 29 Jan 2007 ON THE INSTABILITY FOR THE CUBIC NONLINEAR SCHRODINGER EQUATION¨ R´EMI CARLES Abstract. We consider the cubic nonlinear Schrödinger (NLS) equation as well as the modified Korteweg–de Vries (mKdV) equation in one space dimension.