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We introduce and systematically investigate the generation of dispersive shock waves, which arise naturally in physical settings such as optical waveguide arrays and superfluids confined within optical lattices. The underlying physically…
A perturbation theory for the Nonlinear Schroedinger Equation (NLSE) in 1D on a lattice was developed. The small parameter is the strength of the nonlinearity. For this purpose secular terms were removed and a probabilistic bound on small…
The long-time asymptotic behavior of solutions to the focusing nonlinear Schr\"odinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the…
We discuss the dynamics of a Bose-Einstein condensate initially loaded into every third site of an optical lattice using a description based upon the discrete nonlinear Schrodinger equation. An analytic solution is developed for the case of…
We analyze the physics of Bose-Einstein condensates confined in 2D quasi-periodic optical lattices, which offer an intermediate situation between ordered and disordered systems. First, we analyze the time-of-flight interference pattern that…
The Nonlinear Schroedinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a…
The interaction of nonlinear Schr\"{o}dinger (NLS) solitons with potential wells with variable shapes is investigated numerically. For fixed initial velocities below the threshold for transmission, the outcome pattern as a function of the…
We show that the phenomenon of modulational instability in arrays of Bose-Einstein condensates confined to optical lattices gives rise to coherent spatial structures of localized excitations. These excitations represent thin disks in 1D,…
We present an analytical model of integrable turbulence in the focusing nonlinear Schr\"odinger (fNLS) equation, generated by a one-parameter family of finite-band elliptic potentials in the semiclassical limit. We show that the spectrum of…
We study the cubic-quartic nonlinear Schr\"odinger equation (NLS) in two and three spatial dimension. This equation arises in the mean-field description of Bose-Einstein condensates with Lee-Huang-Yang correction. We first prove global…
We study theoretically the interference patterns produced by the overlap of an array of Bose-Einstein condensates that have no phase coherence among them. We show that density-density correlations at different quasimomenta, which play an…
The paper studies asymptotics of moving gap solitons in nonlinear periodic structures of finite contrast ("deep grating") within the one dimensional periodic nonlinear Schr\"odinger equation (PNLS). Periodic structures described by a finite…
Bose-Einstein condensates initially arranged in a long chain freely expand and interfere. If the initial phases of the condensates are identical, the initial density distribution is restored periodically during the expansion, giving rise to…
Time double-slit interference experiments have been achieved and presented as complementary to spatial double-slit interference experiments, providing a further confirmation of the wave-particle duality. Numerical solutions of the free…
On examples of Bose-Einstein condensates embedded in two-dimensional optical lattices we show that in nonlinear periodic systems modulational instability and inter-band tunneling are intrinsically related phenomena. By direct numerical…
The phenomenon of intrinsic localization in discrete nonlinear extended systems, i.e. the (generic) existence of discrete breathers, is shown to be not restricted to periodic solutions but it also extends to more complex (chaotic) dynamical…
We study a system of inhomogeneous nonlinear Schr\"odinger equations that emerge in optical media with a $\chi^{(2)}$ nonlinearity. This nonlinearity, whose local strength is subject to a cusp-shaped spatial modulation, $\chi^{(2)}\sim…
We show that the width of the wave-packet of a class of generalized nonlinear Schrodinger equations (NLSE) trapped in an arbitrary time-dependent harmonic well in any dimensions is universally determined by the same Hill's equation. This…
We prove for a class of nonlinear Schr\"odinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and…
This work focuses on the study of time-periodic solutions, including breathers, in a nonlinear lattice consisting of elements whose contacts alternate between strain-hardening and strain-softening. The existence, stability, and bifurcation…