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Register a new userRecovery of a cubic non-linearity in the wave equation in the weakly non-linear regime
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We study the inverse problem of recovery a compactly supported non-linearity in the semilinear wave equation $u_{tt}-Delta u+ alpha(x) |u|^2u=0$, in two and three dimensions. We probe the medium with complex-valued harmonic waves of wavelength $h$ and amplitude $h^{-1/2}$, then they propagate in the weakly non-linear regime; and measure the transmitted wave when it exits the support of $alpha$. We show that one can extract the Radon transform of $alpha$ from the phase shift of such waves, and then one can recover $alpha$. We also show that one can probe the medium with real-valued harmonic waves and obtain uniqueness for the linearized problem.
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We study the inverse problem of recovery a non-linearity $f(x,u)$, which is compactly supported in $x$, in the semilinear wave equation $u_{tt}-Delta u+ f(x,u)=0$. We probe the medium with either complex or real-valued harmonic waves of wavelength $sim h$ and amplitude $sim 1$. They propagate in a regime where the non-linearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits $text{supp}_x f$. We show that one can recover $f(x,u)$ when it is an odd function of $u$, and we can recover $alpha(x)$ when $f(x,u)=alpha(x)u^{2m}$. This is done in an explicit way as $hto0$.
The aim of this paper is to study, in dimensions 2 and 3, the pure-power non-linear Schrodinger equation with an external uniform magnetic field included. In particular, we derive a general criteria on the initial data and the power of the non-linearity so that the corresponding solution blows up in finite time, and we show that the time for blow up to occur decreases as the strength of the magnetic field increases. In addition, we also discuss some observations about Strichartz estimates in 2 dimensions for the Mehler kernel, as well as similar blow-up results for the non-linear Pauli equation.
We study the problem of unique recovery of a non-smooth one-form $mathcal A$ and a scalar function $q$ from the Dirichlet to Neumann map, $Lambda_{mathcal A,q}$, of a hyperbolic equation on a Riemannian manifold $(M,g)$. We prove uniqueness of the one-form $mathcal A$ up to the natural gauge, under weak regularity conditions on $mathcal A,q$ and under the assumption that $(M,g)$ is simple. Under an additional regularity assumption, we also derive uniqueness of the scalar function $q$. The proof is based on the geometric optic construction and inversion of the light ray transform extended as a Fourier Integral Operator to non-smooth parameters and functions.
We consider the focusing cubic half-wave equation on the real line $$i partial_t u + |D| u = |u|^2 u, widehat{|D|u}(xi)=|xi|hat{u}(xi), (t,x)in Bbb R_+times Bbb R.$$ We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small $L^2$-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its $H^1$-norm on a finite time interval, followed by (ii) a saturation regime in which the $H^1$-norm remains stationary large forever in time.
We investigate the weakly non-linear evolution of cosmic gravitational clustering in phase space by looking at the Zeldovich solution in the discrete wavelet transform (DWT) representation. We show that if the initial perturbations are Gaussian, the relation between the evolved DWT mode and the initial perturbations in the weakly non-linear regime is quasi-local. That is, the evolved density perturbations are mainly determined by the initial perturbations localized in the same spatial range. Furthermore, we show that the evolved mode is monotonically related to the initial perturbed mode. Thus large (small) perturbed modes statistically correspond to the large (small) initial perturbed modes. We test this prediction by using QSO Ly$alpha$ absorption samples. The results show that the weakly non-linear features for both the transmitted flux and identified forest lines are quasi-localized. The locality and monotonic properties provide a solid basis for a DWT scale-by-scale Gaussianization reconstruction algorithm proposed by Feng & Fang (Feng & Fang, 2000) for data in the weakly non-linear regime. With the Zeldovich solution, we find also that the major non-Gaussianity caused by the weakly non-linear evolution is local scale-scale correlations. Therefore, to have a precise recovery of the initial Gaussian mass field, it is essential to remove the scale-scale correlations.
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