Stability of Solitary Waves for the nonlinear Schrödinger Equation

Abstract

Consider the one-dimensional focusing nonlinear Schrödinger equation with subcritical/critical exponent. <br /> This thesis examines a question derived from the so-called soliton resolution conjecture. The NLS admits regular solutions called solitons. The soliton resolution conjecture claims that every global solution of the NLS will eventually resolve into a sum of soliton-like solutions and a radiation component which disperses like a linear solution. <br /> We consider the related question of 'asymptotic stability'. For initial data close to a soliton, does the solution resolve into a soliton-like solution and radiation? Specifically, we examine the linearisation of the NLS around the soliton. Let L denote the Hamiltonian of the resulting linear equation. <br /> We show the following in this thesis. Firstly, we fully characterise the spectrum of L. Apart from several well-known eigenvalues in 0, iL admits a resonance in ±1 for p = 3, a symmetrical pair of eigenvalues ±E in (-1, 1){0} for 3 &lt; p &lt; 5, as well as two additional generalised eigenvalues in 0 for p = 5. Secondly, based on the above characterisation of the spectrum of L, we show the existence of a wave operator for 3 &lt; p &lt; 5, mapping L onto the free Schrödinger operator. This is accomplished by constructing a distorted Fourier transform mapping L onto a multiplication operator. Thirdly, we show that the wave operator acts as a bounded operator from L^q to L^q for every 1 = q = 8. As a consequence, for 3 &lt; p &lt; 5, the linearised equation allows for the same dispersive estimates as the free equation. Lastly, for 3 &lt; p &lt; 5, we show a local smoothing estimate for the linearised equation. Due to the absence of resonances, this local smoothing estimate allows for significantly stronger local decay than the case of the free equation.