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Abstract We give a proof of local decay estimates for Schrödinger-type equations, which is based on the knowledge of Asymptotic Completeness. This approach extends to time dependent potential perturbations, as it does not rely on Resolvent Estimates or related methods. Global in time Strichartz estimates follow for quasi-periodic time-dependent potentials from our results. 

Abstract We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein–Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to continue the investigation of the occurrence of a novel modified scattering behavior of the solutions that involves a logarithmic slow-down of the decay rate along certain rays. This phenomenon is ultimately caused by the threshold resonance of the linear Klein–Gordon operator. It was previously uncovered for the special case of the zero potential in [51]. The Klein–Gordon model considered in this paper is motivated by the asymptotic stability problem for kink solutions arising in classical scalar field theories on the real line. 

In weak turbulence theory, the Kolmogorov–Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this paper, we construct a new class of time-dependent isotropic solutions to the decaying turbulence problems (whose solutions are energy conserved), with general initial conditions. These solutions exhibit the interesting property that the energy is cascaded from small wavenumbers to large wavenumbers. We can prove that starting with a regular initial condition whose energy at the infinity wave number |𝑝|=∞ is 0, as time evolves, the energy is gradually accumulated at {|𝑝|=∞}. Finally, all the energy of the system is concentrated at {|𝑝|=∞} and the energy function becomes a Dirac function at infinity 𝐸𝛿{|𝑝|=∞}, where E is the total energy. The existence of this class of solutions is, in some sense, the first complete rigorous mathematical proof based on the kinetic description for the energy cascade phenomenon for waves with quadratic nonlinearities. We only represent in this paper the analysis of the statistical description of acoustic waves (and equivalently capillary waves). However, our analysis works for other cases as well. 

Abstract In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N $$d$$ d -dimensional bosons for large N . The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form $$(N-1)^{-1}N^{d\beta }v(N^\beta \cdot )$$ ( N - 1 ) - 1 N d β v ( N β · ) for $$\beta \in [0,\frac{1}{4d})$$ β ∈ [ 0 , 1 4 d ) . We derive a sequence of N -body functions which approximate the true many-body dynamics in $$L^2({\mathbb {R}}^{dN})$$ L 2 ( R dN ) -norm to arbitrary precision in powers of $$N^{-1}$$ N - 1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution. 

We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread with time, and this fact answers the “weak turbulence” question for the nonlinear Schrödinger equation with random potentials. We also derive Ohm’s law for the porous medium equation.