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Register a new userScattering of the three-dimensional cubic nonlinear Schrodinger equation with partial harmonic potentials
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In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrodinger equation (NLS) with partial harmonic potential begin{equation*}tag{NLS} ipartial_t u + left(Delta_{mathbb{R}^3 }-x^2 right) u = |u|^2 u, quad u|_{t=0} = u_0. end{equation*} Our main result shows that the solution $u$ scatters for any given initial data $u_0$ with finite mass and energy. The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson cite{D3,D1,D2} in his study of scattering for the mass-critical nonlinear Schrodinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle applies.
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In this article, we prove the scattering for the quintic defocusing nonlinear Schrodinger equation on cylinder $mathbb{R} times mathbb{T}$ in $H^1$. We establish an abstract linear profile decomposition in $L^2_x h^alpha$, $0 < alpha le 1$, motivated by the linear profile decomposition of the mass-critical Schrodinger equation in $L^2(mathbb{R}^d )$, $dge 1$. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrodinger system, whose scattering can be proved by using the techniques in $1d$ mass critical NLS problem by B. Dodson, to approximate the nonlinear profile, we can prove scattering in $H^1$ by using the concentration-compactness/rigidity method. As a byproduct of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schrodinger system, we also prove the conjecture of the global well-posedness and scattering of the two-discrete-component quintic resonant nonlinear Schrodinger system made by Z. Hani and B. Pausader [Comm. Pure Appl. Math. 67 (2014)].
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic region of space time $|x| < 2t$ for large times and provide bounds for the error which decay as $t to infty$ for a general class of initial data whose difference from the non-vanishing background possesss a fixed number of finite moments and derivatives. Using properties of the scattering map for (GP) we derive as a corollary an asymptotic stability result for initial data which are sufficiently close to the N-dark soliton solutions of (GP).
New exact analytical bound-state solutions of the radial Dirac equation in 3+1 dimensions for two sets of couplings and radial potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of equations that can be transformed into polynomial equations. Several analytical results found in the literature, including the Dirac oscillator, are obtained as particular cases of this unified approach.
We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.
The three-dimensional Schrodinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharges, and the corresponding wave functions are connected by intertwining relations. All models are partially integrable by construction: each of them has at least one second order symmetry operator.
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