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Existence and asymptotics of nonlinear Helmholtz eigenfunctions | | Jesse Gell-Redman
; Andrew Hassell
; Jacob Shapiro
; Junyong Zhang
; | | Date: |
14 Aug 2019 | | Abstract: | We prove the existence and asymptotic expansion of a large class of solutions
to nonlinear Helmholtz equations of the form egin{equation*} (Delta -
lambda^2) u = N[u], end{equation*} where $Delta = -sum_j partial^2_j$ is
the Laplacian on $mathbb{R}^n$ with sign convention that it is positive as an
operator, $lambda$ is a positive real number, and $N[u]$ is a nonlinear
operator that is a sum of monomials of degree $geq p$ in $u$, $overline{u}$
and their derivatives of order up to two, for some $p geq 2$. Nonlinear
Helmholtz eigenfunctions with $N[u]= pm |u|^{p-1} u$ were first considered by
Guti’errez. Such equations are of interest in part because, for certain
nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution
equations, that is, solutions that are time-harmonic.
We show that, under the condition $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for
every $f in H^{k+2}(mathbb{S}^{n-1})$ of sufficiently small norm, there is a
nonlinear Helmholtz function taking the form egin{equation*} u(r, omega) =
r^{-(n-1)/2} Big( e^{-ilambda r} f(omega) + e^{+ilambda r} g(omega) +
O(r^{-epsilon}) Big), ext{ as } r o infty, quad epsilon > 0,
end{equation*} for some $g in H^{k}(mathbb{S}^{n-1})$. Moreover, we prove
the result in the general setting of asymptotically conic manifolds. | | Source: | arXiv, 1908.4890 | | Services: | Forum | Review | PDF | Favorites | |
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