Rigorous numerics for NLS: Bound states, spectra, and controllability

Abstract

In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schrödinger equation (NLS); specifically, to determining bound-state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof [6] of the local exact controllability of NLS.

Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

“…Proof. Given the three numerical approximations at λ ∈ {0.1446, 0.2346, 1.0846}, for each of the corresponding numerical approximations, the MATLAB script script proof theorem 1.m computes the coefficients Y , Z 0 , Z 1 and Z 2 given respectively by (20), (21), (27) and (28) and it verifies with INTLAB (interval arithmetic in MATLAB) the existence of an interval I = (r min , r max ) such that for each r ∈ I, p(r) < 0, with p(r) the radii polynomial as defined in (18). By Lemma 3.5, there exists a uniquex ∈ B r (x) such that f (x) = 0, with f given component-wise in (10).…”

“…Combining the bounds Y , Z 0 , Z 1 and Z 2 given respectively by (20), (21), (27) and (28), we have explicitly constructed the radii polynomial p(r) as defined in (18).…”

“…In an effort to overcome these difficulties, the strengths of functional analysis, topology, algebraic topology (Conley index theory), numerical analysis, nonlinear analysis and scientific computing have recently been combined, giving rise to novel computer-assisted approaches to study infinite dimensional nonlinear problems. A growing literature on computational methods (functional analytic and topological) is providing mathematically rigorous proofs of existence of special bounded solutions for PDEs [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21], DDEs [22,23,24] and infinite dimensional maps [25,26,27]. Besides these recent successes in dynamical systems, ill-posed equations and problems with indefinite tails (e.g.…”

Automatic differentiation for Fourier series and the radii polynomial approach

“…The radii polynomial approach has been applied to a host of problems in differential equations/dynamical systems theory including the study of initial value problems [8], equilibrium solutions of partial differential equations [2,9,10], periodic solutions of ordinary, delay and partial differential equations [1,11,12,13], stable/unstable invariant manifolds for differential equations [14], solutions of boundary value problems [15], eigenvalue/eigenfunction problems [16], connecting orbit problems for differential equations [17,18], and standing wave patterns [19].…”

Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

“…For example equilibria and periodic orbits of ordinary, delay, partial differential equations (PDEs) as well as systems of PDEs are considered in [10,11,12,13,14,15,16,17]. The same approach is applied in order to validate transverse connecting orbits for ordinary differential equations in [18,19,20], to study symmetric pulses, kinks and radially symmetric solutions in reaction diffusion equations [21,22], to solve initial and boundary value problems for ordinary differential equations in a mathematically rigorous way [19,23], and to validate series expansions for the Floquet normal form for linear differential equations with periodic coefficients. This leads to methods for validated computations of the linear stable and unstable bundles of periodic orbits in differential equations [24].…”