Related papers: Inverse scattering transform for the Kundu-Eckhaus…
In this paper we present a hybrid approach to numerically solve two-dimensional electromagnetic inverse scattering problems, whereby the unknown scatterer is hosted by a possibly inhomogeneous background. The approach is `hybrid' in that it…
We study the validity of the Neumann or Born series approach in solving the Helmholtz equation and coefficient identification in related inverse scattering problems. Precisely, we derive a sufficient and necessary condition under which the…
We consider the problem \begin{equation}(1)\;\;\; \begin{cases} S_k(D^2u)= \lambda |x|^{\sigma} (1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B, \end{cases} \end{equation} where $B$ denotes the unit ball in…
In this work, we construct the Born and inverse Born approximation and series to recover two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An…
The inverse scattering problem is applied to 2-dimensional partial differential equations called soliton equations such as the KdV equation and so on. It is also used to integrate the Einstein equations with axial symmetry. These inverse…
We present the solution of the weak noise theory (WNT) for the Kardar-Parisi-Zhang equation in one dimension at short time for flat initial condition (IC). The non-linear hydrodynamic equations of the WNT are solved analytically through a…
In this work, inverse scattering transform for the sixth-order nonlinear Schr\"{o}dinger equation with both zero and nonzero boundary conditions at infinity is given, respectively. For the case of zero nonzero boundary conditions, in terms…
We delve into the inverse scattering transform of the real-valued vector modified Korteweg--de Vries equation, emphasizing the challenges posed by $N$ pairs of higher-order poles in the transmission coefficient and the enhanced spectral…
We study a stationary scattering problem related to the nonlinear Helmholtz equation $-\Delta u - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $\varphi \in L^\infty(\mathbb{R}^N)$,…
We develop the inverse scattering transform method for the Novikov equation $u_t-u_{txx}+4u^2u_x=3u u_xu_{xx}+u^2u_{xxx}$ considered on the line $x\in(-\infty,\infty)$ in the case of non-zero constant background. The approach is based on…
In this paper, we consider the scattering theory of the radial solution to focusing energy-subcritical Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$ using the method from \cite{Dodson2016}. The…
We start with a Riemann-Hilbert problem (RHP) related to a BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s +1)\otimes O(2s))$, $s\geq 1$. We consider two Riemann-Hilbert problems: the first formulated on the real axis $\mathbb{R}$ in the…
We study the inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of…
Inverse scattering problem is discussed for the Maxwell's equations. A reduction of the Maxwell's system to a new Fredholm second-kind integral equation with a {\it scalar weakly singular kernel} is given for electromagnetic (EM) wave…
This is a review of the main ideas of the inverse scattering method (ISM) for solving nonlinear evolution equations (NLEE), known as soliton equations. As a basic tool we use the fundamental analytic solutions (FAS) of the Lax operator L.…
We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then…
In this paper, we study the inverse scattering problem for energy-dependent Schr\"{o}dinger equations on the half-line with energy-dependent boundary conditions at the origin. Under certain positivity and very mild regularity assumptions,…
We present a detailed analysis of the solution of the focusing nonlinear Schr\"odinger equation with initial condition $\psi(x,0)=N {\rm sech}(x)$ in the limit $N\to\infty$. We begin by presenting new and more accurate numerical…
In this manuscript we set up the direct and inverse scattering problems for step-like traveling-wave solutions of the nonlinear Schr\"odinger equation. Specifically, we consider initial data $u(x,0)$ satisfying $u(x,0)\to u_0^\ell(x)$ as…
The initial inverse problem of finding solutions and their initial values ($t = 0$) appearing in a general class of fractional reaction-diffusion equations from the knowledge of solutions at the final time ($t = T$). Our work focuses on the…