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Register a new userOn asymptotic stability of N-solitons of the defocusing nonlinear Schrodinger equation
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We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic region of space time $|x| < 2t$ for large times and provide bounds for the error which decay as $t to infty$ for a general class of initial data whose difference from the non-vanishing background possesss a fixed number of finite moments and derivatives. Using properties of the scattering map for (GP) we derive as a corollary an asymptotic stability result for initial data which are sufficiently close to the N-dark soliton solutions of (GP).
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In this article, we prove the scattering for the quintic defocusing nonlinear Schrodinger equation on cylinder $mathbb{R} times mathbb{T}$ in $H^1$. We establish an abstract linear profile decomposition in $L^2_x h^alpha$, $0 < alpha le 1$, motivated by the linear profile decomposition of the mass-critical Schrodinger equation in $L^2(mathbb{R}^d )$, $dge 1$. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrodinger system, whose scattering can be proved by using the techniques in $1d$ mass critical NLS problem by B. Dodson, to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compactness/rigidity method. As a byproduct of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schrodinger system, we also prove the conjecture of the global well-posedness and scattering of the two-discrete-component quintic resonant nonlinear Schrodinger system made by Z. Hani and B. Pausader [Comm. Pure Appl. Math. 67 (2014)].
The Cauchy problem of the modified nonlinear Schr{o}dinger (mNLS) equation with the finite density type initial data is investigated via $overline{partial}$ steepest descent method. In the soliton region of space-time $x/tin(5,7)$, the long-time asymptotic behavior of the mNLS equation is derived for large times. Furthermore, for general initial data in a non-vanishing background, the soliton resolution conjecture for the mNLS equation is verified, which means that the asymptotic expansion of the solution can be characterized by finite number of soliton solutions as the time $t$ tends to infinity, and a residual error $mathcal {O}(t^{-3/4})$ is provided.
We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.
It has been conjectured that the defocusing nonlinear Schrodinger (NLS) equation on the half-line does not admit solitons. We give a proof of this conjecture.
We consider the Cauchy problem for the defocusing Schr$ddot{text{o}}$dinger (NLS) equation with finite density initial data begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, onumber &q(x,0)=q_0(x), quad lim_{x to pm infty}q_0(x)=pm 1. onumber end{align} Recently, for the space-time region $|x/(2t)|<1$ without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $bar{partial}$ generalization of the nonlinear steepest descent method. Their asymptotic result is the form begin{align} q(x,t)= T(infty)^{-2} q^{sol,N}(x,t) + mathcal{O}(t^{-1 }). end{align} However, for the space-time region $ |x/(2t)|>1$, there will be two stationary points appearing on the jump contour, the corresponding long-time asymptotics is still unknown. In this paper, for the region $|x/(2t)|>1, x/t=mathcal{O}(1)$, we found a different asymptotic expansion $$ q(x,t)= e^{-ialpha(infty)} left( q_{sol}(x,t;sigma_d^{(out)}) +t^{-1/2} h(x,t) right)+mathcal{O}left(t^{-3/4}right),$$ whose leading term is $N$-soliton solutions; the second $t^{-1/2}$ order term is soliton-soliton and soliton-radiation interactions; and the third term $mathcal{O}(t^{-3/4})$ is a residual error from a $overlinepartial$-equation. Additionally, the asymptotic stability property for the N-soliton solutions of the defocusing NLS equation sufficiently is obtained.
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