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Register a new userLogarithmic Stability for Coefficients Inverse Problem of Coupled Schr{o}dinger Equations
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In this paper, we study an inverse coefficients problem for two coupled Schr{o}dinger equations with an observation of one component of the solution. The observation is done in a nonempty open subset of the domain where the equations hold. A logarithmic type stability result is obtained. The main method is based on the Carleman estimate for coupled Schr{o}dinger equations and coupled heatn equations, and the Fourier-Bros-Iagolnitzer transform.
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This paper investigates the identification of two coefficients in a coupled hyperbolic system with an observation on one component of the solution. Based on the the Carleman estimate for coupled wave equations a logarithmic type stability result is obtained by measurement data only in a suitably chosen subdomain under the assumption that the coefficients are given in a neighborhood of some subboundary.
This paper concerns inverse problems for strongly coupled Schrodinger equations. The purpose of this inverse problem is to retrieve a stationary potential in the strongly coupled Schrodinger equations from either boundary or internal measurements. Two stability results are derived from a new Carleman estimate for the strongly coupled Schrodinger equations.
We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{o}dinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on the other hand. We explain some connections between the two families of equations, and show how these connections may help having an insight in all cases. We insist on some specific aspects only, and refer to the cited articles for more details, and more complete statements. We try to give a general picture of the results, and present some heuristical arguments that can help the intuition, which are not necessarily found in the mentioned articles.
We analyze dynamical properties of the logarithmic Schr{o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.
We investigate the structure of nodal solutions for coupled nonlinear Schr{o}dinger equations in the repulsive coupling regime. Among other results, for the following coupled system of $N$ equations, we prove the existence of infinitely many nodal solutions which share the same componentwise-prescribed nodal numbers begin{equation}label{ab} left{ begin{array}{lr} -{Delta}u_{j}+lambda u_{j}=mu u^{3}_{j}+sum_{i eq j}beta u_{j}u_{i}^{2} ,,,,,,, in W , u_{j}in H_{0,r}^{1}(W), ,,,,,,,,j=1,dots,N, end{array} right. end{equation} where $W$ is a radial domain in $mathbb R^n$ for $nleq 3$, $lambda>0$, $mu>0$, and $beta <0$. More precisely, let $p$ be a prime factor of $N$ and write $N=pB$. Suppose $betaleq-frac{mu}{p-1}$. Then for any given non-negative integers $P_{1},P_{2},dots,P_{B}$, (ref{ab}) has infinitely many solutions $(u_{1},dots,u_{N})$ such that each of these solutions satisfies the same property: for $b=1,...,B$, $u_{pb-p+i}$ changes sign precisely $P_b$ times for $i=1,...,p$. The result reveals the complex nature of the solution structure in the repulsive coupling regime due to componentwise segregation of solutions. Our method is to combine a heat flow approach as deformation with a minimax construction of the symmetric mountain pass theorem using a $mathbb Z_p$ group action index. Our method is robust, also allowing to give the existence of one solution without assuming any symmetry of the coupling.
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