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Register a new userOn the Mountain-pass algorithm for the quasi-linear Schrodinger equation
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We discuss the application of the Mountain Pass algorithm to the so-called quasi-linear Schrodinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical algorithm is not directly applicable.
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The numerical solution of a linear Schrodinger equation in the semiclassical regime is very well understood in a torus $mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in $mathbb{R}^d$, a setting more suitable for many applications. In this paper we extend the theory of splitting methods to this end. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schrodinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schrodinger operator.
For non-homotopic maps $u,vin C^{infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $gamma_p(u,v)$ for which there exist paths $u_tin W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $|du_t|_{L^p}^pleq gamma_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows that $gamma_p(u,v)<infty$ for $pin [1,k)$, and using their construction, one can obtain an estimate of the form $gamma_p(u,v)leq frac{C(u,v)}{k-p}$. When $M$ and $N$ are oriented, and $u$ and $v$ induce different maps on real cohomology in degree $k-1$, we show that the growth $gamma_p(u,v)sim frac{1}{k-p}$ as $pto k$ is sharp, and obtain a lower bound for the coefficient $liminf_{pto k}(k-p)gamma_p(u,v)$ in terms of the min-max masses of certain non-contractible loops in the space of codimension-$k$ integral cycles in $M$. In the process, we establish lower bounds for a related smaller quantity $gamma_p^*(u,v)leqgamma_p(u,v)$, for which there exist critical points $u_pin W^{1,p}(M,N)$ of the $p$-energy functional satisfying $gamma_p^*(u,v)leq |du_p|_{L^p}^pleq gamma_p(u,v).$
Numerical methods that approximate the solution of the Vlasov-Poisson equation by a low-rank representation have been considered recently. These methods can be extremely effective from a computational point of view, but contrary to most Eulerian Vlasov solvers, they do not conserve mass and momentum, neither globally nor in respecting the corresponding local conservation laws. This can be a significant limitation for intermediate and long time integration. In this paper we propose a numerical algorithm that overcomes some of these difficulties and demonstrate its utility by presenting numerical simulations.
For the solution $q(t)=(q_n(t))_{ninmathbb Z}$ to one-dimensional discrete Schrodinger equation $${rm i}dot{q}_n=-(q_{n+1}+q_{n-1})+ V(theta+nomega) q_n, quad ninmathbb Z,$$ with $omegainmathbb R^d$ Diophantine, and $V$ a small real-analytic function on $mathbb T^d$, we consider the growth rate of the diffusion norm $|q(t)|_{D}:=left(sum_{n}n^2|q_n(t)|^2right)^{frac12}$ for any non-zero $q(0)$ with $|q(0)|_{D}<infty$. We prove that $|q(t)|_{D}$ grows {it linearly} with the time $t$ for any $thetainmathbb T^d$ if $V$ is sufficiently small.
We derive asymptotic formulas for the solution of the derivative nonlinear Schrodinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem.
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