Related papers: Pathological set of initial data for scaling-super…
In this paper we consider Schr\"odinger equations with sublinear dispersion relation on the one-dimensional torus $\T := \R /(2 \pi \Z)$. More precisely, we deal with equations of the form $\partial_t u = \ii {\cal V}(\omega t)[u]$ where…
We establish rates of convergence of solutions to scaling (or similarity) profiles in a coagulation type system modelling submonolayer deposition. We prove that, although all memory of the initial condition is lost in the similarity limit,…
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for…
A method is developed for solving quasilinear convection diffusion problems starting on a coarse mesh where the data and solution-dependent coefficients are unresolved, the problem is unstable and approximation properties do not hold. The…
In this paper we develop the formalism for the stochastic approach to inflation at all order in slow-roll parameters. This is done by including the momentum and Hamiltonian constraints into the stochastic equations. We then specialise to…
The Schr\"odingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schr\"odinger-type equations with unitary evolution. It does so via the so-called warped phase…
Using the Fredholm theory of the linear time-dependent Schr\"odinger equation set up in our previous article arXiv:2201.03140, we solve the final-state problem for the nonlinear Schr\"odinger problem $$ (D_t + \Delta + V) u = N[u], \quad…
A statistical model of self-organization in a generic class of one-dimensional nonlinear Schrodinger (NLS) equations on a bounded interval is developed. The main prediction of this model is that the statistically preferred state for such…
This is the second part of a two-paper series studying the nonlinear Schr\"odinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schr\"odinger equation.…
We investigate the existence and local uniqueness of normalized $k$-peak solutions for the fractional Schr\"odinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points.…
In this paper we obtain a stabilization result for the Schr\"odinger equation under generic assumptions on the potential. Then we consider the Schr\"odinger equation with a potential which has a random time-dependent amplitude. We show that…
We consider the energy supercritical defocusing nonlinear Schr\"odinger equation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal…
We consider the initial-value problem for the one-dimensional nonlinear Schr\"odinger equation in the presence of an attractive delta potential. We show that for sufficiently small initial data, the corresponding global solution decomposes…
We consider the existence of normalized solutions to nonlinear Schr\"odinger equations on noncompact metric graphs in the $L^2$ supercritical regime. For sufficiently small prescribed mass ($L^2$ norm), we prove existence of positive…
In this paper, we integrate neural networks and Gaussian wave packets to numerically solve the Schr\"odinger equation with a smooth potential near the semi-classical limit. Our focus is not only on accurately obtaining solutions when the…
We consider the Cauchy problem for quadratic derivative fractional nonlinear Schr\"odinger equations on $\mathbb{R}$ or $\mathbb{T}$. We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed…
We study the final state problem for the nonlinear Schr\"{o}dinger equation with a critical long-range nonlinearity and a long-range linear potential. Given a prescribed asymptotic profile which is different from the free evolution, we…
Fitted finite element methods are constructed for a singularly perturbed convection-diffusion problem in two space dimensions. Exponential splines as basis functions are combined with Shishkin meshes to obtain a stable parameter-uniform…
Having discovered a dimension mistake in two key formulas of the Classical Nucleation Theory (CNT) but wishing to remain in the style of this theory, we propose to approach nucleation on the basis of the Zeldovich unsteady rate formula,…
We construct finite energy blow-up solutions for the radial self-dual Chern-Simons-Schr\"odinger equation with a continuum of blow-up rates. Our result stands in stark contrast to the rigidity of blow-up of $H^{3}$ solutions proved by the…