Existence and Asymptotics of Nonlinear Helmholtz Eigenfunctions

Abstract

We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form (\Delta -\lambda 2 )u = N [u], where \Delta = -\sum j \partial 2 j is the Laplacian on \BbbR n , \lambda is a positive real number, and N [u] is a nonlinear operator depending polynomially on u and its derivatives of order up to order two. Nonlinear Helmholtz eigenfunctions with N [u] = \pm | u| p - 1 u were first considered by Guti\' errez [Math. Ann., 328 (2004), pp. 1--25]. We show that for suitable nonlinearities and for every f \in H k+4 (\BbbS n - 1 ) of sufficiently small norm, there is a nonlinear Helmholtz function taking the form u(r, \omega ) = r - (n - 1)/2 (e - i\lambda r f (\omega ) + e +i\lambda r b(\omega ) + O(r - \epsilon )), as r \rightar \infty , \epsilon > 0, for some b \in H k (\BbbS n - 1 ). Moreover, we prove the result in the general setting of asymptotically conic manifolds. The proof uses an elaboration of anisotropic Sobolev spaces defined by Vasy [A minicourse on microlocal analysis for wave propagation, in Asymptotic Analysis in General Relativity, London Math. Soc. Lecture Note Ser. 443, Cambridge University Press, Cambridge, 2018, pp. 219--374], between which the Helmholtz operator \Delta -\lambda 2 acts invertibly. These spaces have a variable spatial weight \sansl \pm , varying in phase space and distinguishing between the two ``radial sets"" corresponding to incoming oscillations, e - i\lambda r , and outgoing oscillations, e +i\lambda r . Our spaces have, in addition, module regularity with respect to two different ``test modules"" and have algebra (or pointwise multiplication) properties that allow us to treat nonlinearities N [u] of the form specified above.

Propagation of singularities and Fredholm analysis of the time-dependent Schrödinger equation

“…Then u = Q − u + Q + u gives a decomposition such that u ± ∈ X s,l ± . One of the main goals of this work is to set up potential applications to the nonlinear Schrödinger equation along the lines of [7,8] for the nonlinear Helmholtz equation, which in turn was inspired by works [12], [6] on nonlinear wave equations. As such, we include module regularity estimates which typically arise in microlocal approaches to nonlinear analysis.…”

“…[24]). Our approach is, to the best of our knowledge, essentially different to any previous method for treating the time-dependent Schrödinger equation, although inspired by previous Fredholm treatments of non-elliptic problems for the wave equation [27], [1], [12], and the Helmholtz equation [8]. The first example of Fredholm theory used to treat a non-elliptic problem appears to be Faure and Sjöstrand's treatment of Anosov flows in [4].…”

“…We expect that our method will be advantageous for analyzing the large-time asymptotics of solutions. In the case of defocusing NLS, a multiplication result for module regularity spaces, along the lines of [8] in the Helmholtz case, will lead directly to a small-data result. Large data results should be achievable by combining our techniques with a priori estimates on solutions provided by Strichartz or Morawetz estimates.…”

“…2 this was denoted σ base,m,l (A) in [8] Remark 2.1. The symbols in Definition 2.1 can also be characterized by a regularity condition when they are thought of as functions on T * par R n+1 , namely they are conormal to the boundary with the appropriate weights.…”

The Feynman problem for the Klein–Gordon equation

“…The Gell-Redman-Haber-Vasy method also allows for solving a non-linear Feynman problem [GHV16]. Closely related techniques were recently used by Hassell-Gell-Redman-Schapiro-Zhang to prove the existence of standing waves for the non-linear Helmholtz equation [GHSZ19], and it is expected that the Feynman problem for the Klein-Gordon equation could be solved by a mixture of the techniques in the two works [GHV16,GHSZ19].…”