Inverse Random Source Scattering for the Helmholtz Equation with Attenuation

On some novel bright, dark and optical solitons to the cubic-quintic nonlinear non-paraxial pulse propagation model

“…The cubic-quintic nonlinear non-paraxial pulse propagation model accounts for the effects of backward scattering that are neglected in the more common nonlinear Schrödinger model. The Khater approach is employed to observe some new exact traveling wave solutions to the cubic-quintic nonlinear non-paraxial pulse propagation model (NPPP) equation (Li and Wang 2021;Marchner et al 2021).…”

Inverse source problems for the stochastic wave equations: far-field patterns

“…The regularity of random fields given in Definition 1 depends on the order −m. It has been studied in [20] and is stated in the following lemma. Now let us consider vector fields for n > 1.…”

“…Motivated by [13], a new model is developed for the random source, which is assumed to be a real-valued generalized microlocally isotropic Gaussian (GMIG) random field with its covariance operator being a classical pseudo-differential operator. It is shown that the principal symbol of the covariance operator can be uniquely determined by the amplitude of the near-field scattering data averaged over the frequency band, generated by a single realization of the random source, see [14,20] for acoustic waves, [14,15] for elastic waves, and [23] for biharmonic waves. The inverse random source problem for electromagnetic waves is considered in [21], where the source is modeled by a complex-valued centered GMIG random field whose real and imaginary parts are assumed to be independent and identically distributed, leading to the relation operator being zero.…”

An inverse source problem for the stochastic wave equation

“…Since the source F px, tq is a random field with low regularity, it is a distribution instead of a regular function. More precisely, it is shown in [25] that F px, ¤q W H¡1¡ ,p p0, T q for any ¡ 0 and p ¡ 1. Clearly, we can see that H ¡ 1 ¡ 0 as H p0, 1q, and the smaller the Hurst index H is, the lower the regularity of the source is.…”

“…Compared with the direct problems for stochastic wave equations driven by fBm, there are few work for the inverse source problems for the stochastic wave equations driven by fBm. Recently, [25] considered an inverse random source problem for the Helmholtz equation driven by a fractional Gaussian field. The approach was further extended to solve the inverse random source problem for time-harmonic Maxwell's equations driven by a centered complex-valued Gaussian vector field with correlated components [26].…”