arXiv subjects

Justin T. Cole

Publications and source records attributed to Justin T. Cole.

22 records · Page 2Linked to original sources

arXiv2018

On the Whitham system for the radial nonlinear Schrödinger equation

Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose-Einstein condensates, water waves and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one-dimensional NLS equation exhibits significant qualitative and quantitative differences.

nlin.PS↗

arXiv2018

Solitons, the Korteweg-de Vries equation with step boundary values and pseudo-embedded eigenvalues

The Korteweg-deVries (KdV) equation with step boundary conditions is considered, with an emphasis on soliton dynamics. When one or more initial solitons are of sufficient size they can propagate through the step; in this case the phase shift is calculated via the inverse scattering transform. On the other hand, when the amplitude is too small they become trapped. In the trapped case the transmission coefficient of the associated associated linear Schrödinger equation can become large at a point exponentially close to the continuous spectrum. This point is referred to as a {\it pseudo-embedded eigenvalue}. Employing the inverse problem it is shown that the continuous spectrum associated with a branch cut in the neighborhood of the pseudo-embedded eigenvalue plays the role of discrete spectra, which in turn leads to a trapped soliton in the KdV equation.

nlin.SI↗

arXiv2017

Tight-binding methods for general longitudinally driven photonic lattices -- edge states and solitons

A systematic approach for deriving tight-binding approximations in general longitudinally driven lattices is presented. As prototypes, honeycomb and staggered square lattices are considered. Time-reversal symmetry is broken by varying/rotating the waveguides, longitudinally, along the direction of propagation. Different sublattice rotation and structure are allowed. Linear Floquet bands are constructed for intricate sublattice rotation patterns such as counter rotation, phase offset rotation, as well as different lattice sizes and frequencies. An asymptotic analysis of the edge modes, valid in a rapid-spiraling regime, reveals linear and nonlinear envelopes which are governed by linear and nonlinear Schrodinger equations, respectively. Nonlinear states, referred to as topologically protected edge solitons are unidirectional edge modes. Direct numerical simulations for both the linear and nonlinear edge states agree with the asymptotic theory. Topologically protected modes are found; they possess unidirectionality and do not scatter at lattice defect boundaries.

nlin.PS↗

arXiv2017

Time-dependent spectral renormalization method

The spectral renormalization method was introduced by Ablowitz and Musslimani in 2005, [Opt. Lett. 30, pp. 2140-2142] as an effective way to numerically compute (time-independent) bound states for certain nonlinear boundary value problems. % of the nonlinear Schrödinger (NLS), Gross-Pitaevskii and water wave type equations to mention a few. In this paper, we extend those ideas to the time domain and introduce a time-dependent spectral renormalization method as a numerical means to simulate linear and nonlinear evolution equations. The essence of the method is to convert the underlying evolution equation from its partial or ordinary differential form (using Duhamel's principle) into an integral equation. The solution sought is then viewed as a fixed point in both space and time. The resulting integral equation is then numerically solved using a simple renormalized fixed-point iteration method. Convergence is achieved by introducing a time-dependent renormalization factor which is numerically computed from the physical properties of the governing evolution equation. The proposed method has the ability to incorporate physics into the simulations in the form of conservation laws or dissipation rates. This novel scheme is implemented on benchmark evolution equations: the classical nonlinear Schrödinger (NLS), integrable $PT$ symmetric nonlocal NLS and the viscous Burgers' equations, each of which being a prototypical example of a conservative and dissipative dynamical system. Numerical implementation and algorithm performance are also discussed.

nlin.PS↗