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Register a new userOn a Schrodinger system arizing in nonlinear optics
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We study the nonlinear Schrodinger system [ begin{cases} displaystyle iu_t+Delta u-u+(frac{1}{9}|u|^2+2|w|^2)u+frac{1}{3}overline{u}^2w=0, idisplaystyle sigma w_t+Delta w-mu w+(9|w|^2+2|u|^2)w+frac{1}{9}u^3=0, end{cases} ] for $(x,t)in mathbb{R}^ntimesmathbb{R}$, $1leq nleq 3$ and $sigma,mu>0$. This system models the interaction between an optical beam and its third harmonic in a material with Kerr-type nonlinear response. We prove the existence of ground state solutions, analyse its stability, and establish local and global well-posedness results as well as several criteria for blow-up.
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We consider the quadratic Schrodinger system $$iu_t+Delta_{gamma_1}u+overline{u}v=0$$ $$2iv_t+Delta_{gamma_2}v-beta v+frac 12 u^2=0,$$ where $tinmathbf{R},,xin mathbf{R}^dtimes mathbf{R}$, in dimensions $1leq dleq 4$ and for $gamma_1,gamma_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.
We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y.Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows.
In this paper, we simplify the proof of M. Hamano in cite{Hamano2018}, scattering theory of the solution to eqref{NLS system}, by using the method from B. Dodson and J. Murphy in cite{Dodson2018}. Firstly, we establish a criterion to ensure the solution scatters in $ H^1(mathbb{R}^5) times H^1(mathbb{R}^5) $. In order to verify the correctness of the condition in scattering criterion, we must exclude the concentration of mass near the origin. The interaction Morawetz estimate and Galilean transform characterize a decay estimate, which implies that the mass of the system cannot be concentrated.
In this paper, we study the solutions to the energy-critical quadratic nonlinear Schrodinger system in ${dot H}^1times{dot H}^1$, where the sign of its potential energy can not be determined directly. If the initial data ${rm u}_0$ is radial or non-radial but satisfies the mass-resonance condition, and its energy is below that of the ground state, using the compactness/rigidity method, we give a complete classification of scattering versus blowing-up dichotomies depending on whether the kinetic energy of ${rm u}_0$ is below or above that of the ground state.
We study the existence of ground states for the coupled Schrodinger system begin{equation} left{begin{array}{lll} displaystyle -Delta u_i+lambda_i u_i= mu_i |u_i|^{2q-2}u_i+sum_{j eq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i u_iin H^1(mathbb{R}^n), quad i=1,ldots, d, end{array}right. end{equation} $ngeq 1$, for $lambda_i,mu_i >0$, $b_{ij}=b_{ji}>0$ (the so-called symmetric attractive case) and $1<q<n/(n-2)^+$. We prove the existence of a nonnegative ground state $(u_1^*,ldots,u_d^*)$ with $u_i^*$ radially decreasing. Moreover we show that, for $1<q<2$, such ground states are positive in all dimensions and for all values of the parameters.
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