Multi-bump, self-similar, blow-up solutions of the Ginzburg–Landau equation

Abstract

For the Ginzburg-Landau equation (GL), we establish the existence and local uniqueness of two classes of multi-bump, self-similar, blowup solutions for all dimensions 2 < d < 4 (under certain conditions on the coefficients in the equation). In numerical simulation and via asymptotic analysis, one class of solutions was already found; the second class of multi-bump solutions is new.In the analysis, we treat the GL as a small perturbation of the cubic nonlinear Schrödinger equation (NLS). The existence result given here is a major extension of results established previously for the NLS, since for the NLS the construction only holds for d close to the critical dimension d = 2.The behaviour of the self-similar solutions is described by a nonlinear, nonautonomous ordinary differential equation (ODE). After linearisation, this ODE exhibits, hyperbolic behaviour near the origin and elliptic behaviour asymptotically. We call the region where the type of behaviour changes, the midrange. All of the bumps of the solutions we construct lie in the midrange.For the construction, we track a manifold of solutions of the ODE that satisfy the condition at the origin forward, and a manifold of solutions that satisfy the asymptotic conditions backward, to a common point in the midrange. Then, we show that these manifolds intersect transversally. We study the dynamics in the midrange by using geometric singular perturbation theory, adiabatic Melnikov theory, and the Exchange Lemma.

Finite-Time Blowup for a Complex Ginzburg--Landau Equation

“…when θ = 0 and |γ| is small. A result in the same spirit is obtained in [27] when the equation is "close" to the nonlinear Schrödinger equation iu t + ∆u + |u| α u = 0. The result in [31] was significantly extended in [14], where the authors give a rigorous justification of the numerical and formal arguments of [25,26].…”

Construction of a Blow-Up Solution for the Complex Ginzburg–Landau Equation in a Critical Case

“…Their argument is based on matching a numerical solution in an inner region with an analytical solution in an outer region. In the same direction we can also cite the work of Rottschäfer [Rot08] and [Rot13].…”

On the backward behavior of some dissipative evolution equations

“…Under this scenario, equation (3.3) is locally well-posed forward in time, and by referring to the energy identity (3.4), we see that the nonlinear term acts as a source forward in time, which can lead the solution to blow up in finite time [6] (see also [52,53] for other results concerning finite time blow-up of similar equations). On the other hand, for the CGLE defined in the whole space R d , the blow-up phenomenon has been carefully discussed in [13,55,62], etc. However, since α < 0, we see that the nonlinearity tends to damp the energy as the time goes backwards, thus the life span of solutions backward in time might be a subtle problem, which deserves a future study.…”