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Register a new userThe defocusing nonlinear Schrodinger equation with $t$-periodic data: New exact solutions
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Jonatan Lenells
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We consider solutions of the defocusing nonlinear Schrodinger (NLS) equation on the half-line whose Dirichlet and Neumann boundary values become periodic for sufficiently large $t$. We prove a theorem which, modulo certain assumptions, characterizes the pairs of periodic functions which can arise as Dirichlet and Neumann values for large $t$ in this way. The theorem also provides a constructive way of determining explicit solutions with the given periodic boundary values. Hence our approach leads to a class of new exact solutions of the defocusing NLS equation on the half-line.
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We consider the nonlinear Schrodinger equation on the half-line with a given Dirichlet (Neumann) boundary datum which for large $t$ tends to the periodic function $g_0^b(t)$ ($g_1^b(t)$). Assuming that the unknown Neumann (Dirichlet) boundary value tends for large $t$ to a periodic function $g_1^b(t)$ ($g_0^b(t)$), we derive an easily verifiable condition that the functions $g_0^b(t)$ and $g_1^b(t)$ must satisfy. Furthermore, we introduce two different methods, one based on the formulation of a Riemann-Hilbert problem, and one based on a perturbative approach, for constructing $g_1^b(t)$ ($g_0^b(t)$) in terms of $g_0^b(t)$ ($g_1^b(t)$).
We consider the nonlinear Schrodinger equation on the half-line with a given Dirichlet boundary datum which for large $t$ tends to a periodic function. We assume that this function is sufficiently small, namely that it can be expressed in the form $alpha g_0^b(t)$, where $alpha$ is a small constant. Assuming that the Neumann boundary value tends for large $t$ to the periodic function $g_1^b(t)$, we show that $g_1^b(t)$ can be expressed in terms of a perturbation series in $alpha$ which can be constructed explicitly to any desired order. As an illustration, we compute $g_1^b(t)$ to order $alpha^8$ for the particular case that $g_0^b(t)$ is the sum of two exponentials. We also show that there exist particular functions $g_0^b(t)$ for which the above series can be summed up, and therefore for these functions $g_1^b(t)$ can be obtained in closed form. The simplest such function is $exp(iomega t)$, where $omega$ is a real constant.
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.
The unified transform method (UTM) provides a novel approach to the analysis of initial-boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann-Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrodinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial-boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial-boundary value problem on a finite interval with $x$-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle), belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg-de Vries (KdV) and modified KdV equations.
An explicit lifespan estimate is presented for the derivative Schrodinger equations with periodic boundary condition.
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