Atanas Stefanov

Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein–Gordon equations

We show decay estimates for the propagator of the discrete Schrodinger and Klein–Gordon equations in the form . This implies a corresponding (restricted) set of Strichartz estimates. Applications of the latter include the existence of excitation thresholds for certain regimes of the parameters and the decay of small initial data for relevant lp norms. The analytical decay estimates are corroborated with numerical results.

On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension

Based on the recent work [Komech et al., “Dispersive estimates for 1D discrete Schrodinger and Klein-Gordon equations,” Appl. Anal. 85, 1487 (2006)] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrodinger operator, Hϕ=(−Δ+V)ϕ=−(ϕn+1+ϕn−1−2ϕn)+Vnϕn. We show that under appropriate decay conditions on the general potential (and a nonresonance condition at the spectral edges), the spectrum of H consists of finitely many eigenvalues of finite multiplicities and the essential (absolutely continuous) spectrum, while the resolvent satisfies the limiting absorption principle and the Puiseux expansions near the edges. These properties imply the dispersive estimates ‖eitHPa.c.(H)‖lσ2→l−σ2≲t−3/2 for any fixed σ>52 and any t>0, where Pa.c.(H) denotes the spectral projection to the absolutely continuous spectrum of H. In addition, based on the scattering theory for the discrete Jost solutions and the previous results by Stefanov and Kevrekidis [“Asymptotic behaviour of small ...

Asymptotic stability of small gap solitons in nonlinear Dirac equations

We prove dispersive decay estimates for the one-dimensional Dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear Dirac equations with quintic and higher-order nonlinear terms.

On the Existence of Solitary Traveling Waves for Generalized Hertzian Chains

We consider the question of existence of “bell-shaped” (i.e., nonincreasing for x>0 and nondecreasing for x<0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p=3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego (Proc. Am. Math. Soc. 133:1763, 2005) formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke–Wattis result (Commun. Math. Phys. 161:391, 1994) by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model.

ASYMPTOTIC STABILITY OF SMALL BOUND STATES IN THE DISCRETE NONLINEAR SCHRODINGER EQUATION

Asymptotic stability of small bound states in one dimension is proved in the frame- work of a discrete nonlinear Schrodinger equation with septic and higher power-law nonlinearities and an external potential supporting a simple isolated eigenvalue. The analysis relies on the dis- persive decay estimates from Pelinovsky and Stefanov (J. Math. Phys., 49 (2008), 113501) and the arguments of Mizumachi (J. Math. Kyoto Univ., 48 (2008), pp. 471-497) for a continuous nonlinear Schrodinger equation in one dimension. Numerical simulations suggest that the actual decay rate of perturbations near the asymptotically stable bound states is higher than the one used in the analysis.

Traveling waves and their tails in locally resonant granular systems

In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as mass-in-mass systems. We use three distinct approaches to identify relevant traveling waves. The first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier tranformed variant of the problem, or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of resonances, which will be discussed in detail. All three approaches can be utilized in either the displacement or the strain formulation. Typical resulting computations in finite domains result in the solitary waves bearing symmetric non-vanishing tails at both ends of the computational domain. Importantly, however, a countably infinite set of anti-resonance conditions is identified for which solutions with genuinely rapidly decaying tails arise.

Calderón–Zygmund Operators on Mixed Lebesgue Spaces and Applications to Null Forms

The boundedness of Calderon–Zygmund operators is proved in the scale of the mixed Lebesgue spaces. As a consequence, the boundedness of the bilinear null forms

Traveling waves for monomer chains with precompression

Q_{i j} (u,v) \,{=}\,\p_i u\p_j v \,{-}\, \p_j u\p_i v

Nonlinearity management in higher dimensions

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Traveling Waves for the Mass in Mass Model of Granular Chains

Q_0(u,v)\,{=}\,u_t v_t \,{-}\,\nabla_x u\,{\cdot}\, \nabla_x v