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Register a new userScattering theory For Quadratic Nonlinear Schrodinger System in dimension six
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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.
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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.
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.
We study the Schrodinger-Debye system over $mathbb{R}^d$ iu_t+frac 12Delta u=uv,quad mu v_t+v=lambda |u|^2 and establish the global existence and scattering of small solutions for initial data in several function spaces in dimensions $d=2,3,4$. Moreover, in dimension $d=1$, we prove a Hayashi-Naumkin modified scattering result.
We prove sharp $L^infty$ decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schrodinger equation with small initial data chosen from a weighted Sobolev space. Specifically, we work with an averaged version of the dispersion-managed NLS in the strong dispersion management regime. The proof adapts techniques from Hayashi-Naumkin and Kato-Pusateri, which established small-data modified scattering for the standard $1d$ cubic NLS.
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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