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We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schrodinger equation in a bounded domain of $mathbb R^n$, $ngeq 2$, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be Holder stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.
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We are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrodinger equation with unknown source and potential terms. The well-posedness of the direct scattering problem is first established. Three uniqueness results are then obtained for the corresponding inverse problems in determining the variance of the source, the potential and the expectation of the source, respectively, by the associated far-field measurements. First, a single realization of the passive scattering measurement can uniquely recover the variance of the source without the a priori knowledge of the other unknowns. Second, if active scattering measurement can be further obtained, a single realization can uniquely recover the potential function without knowing the source. Finally, both the potential and the first two statistic moments of the random source can be uniquely recovered with full measurement data. The major novelty of our study is that on the one hand, both the random source and the potential are unknown, and on the other hand, both passive and active scattering measurements are used for the recovery in different scenarios.
In this paper, we establish the existence of ground state solutions for a fractional Schrodinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrodinger equations. Our results explain the effect of each term of the Schrodinger equation : The fractional power, the power of the nonlinearity and the harmonic potential.
In this paper, we show the scattering of the solution for the focusing inhomogenous nonlinear Schrodinger equation with a potential begin{align*} ipartial_t u+Delta u- Vu=-|x|^{-b}|u|^{p-1}u end{align*} in the energy space $H^1(mathbb R^3)$. We prove a scattering criterion, and then we use it together with Morawetz estimate to show the scattering theory.
In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrodinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(mathbb{R})$, begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2sigma q^2(x,t)bar q(-x,t)=0,quadsigma=pm1, q(x,0)&=q_0(x). end{align*} We show that the solution can be represented by the solution of a Riemann-Hilbert problem (RH problem), and assuming no discrete spectrum, we majorly apply $barpartial$-steepest cescent descent method on analyzing the long-time asymptotic behavior of it.
We consider the Cauchy problem for the nonlinear wave equation $u_{tt} - Delta_x u +q(t, x) u + u^3 = 0$ with smooth potential $q(t, x) geq 0$ having compact support with respect to $x$. The linear equation without the nonlinear term $u^3$ and potential periodic in $t$ may have solutions with exponentially increasing as $ t to infty$ norm $H^1({mathbb R}^3_x)$. In [2] it was established that adding the nonlinear term $u^3$ the $H^1({mathbb R}^3_x)$ norm of the solution is polynomially bounded for every choice of $q$. In this paper we show that $H^k({mathbb R}^3_x)$ norm of this global solution is also polynomially bounded. To prove this we apply a different argument based on the analysis of a sequence ${Y_k(ntau_k)}_{n = 0}^{infty}$ with suitably defined energy norm $Y_k(t)$ and $0 < tau_k <1.$
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