Yi-Ping Ma

Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations

Abstract Dispersive shock waves (DSWs) in the Kadomtsev–Petviashvili (KP) equation and two dimensional Benjamin–Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time ( 2 + 1 ) dimensions to finding DSW solutions of ( 1 + 1 ) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg–de Vries (cKdV) and cylindrical Benjamin–Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the ( 2 + 1 ) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced ( 1 + 1 ) dimensional equations.

Depinning, front motion, and phase slips.

Pinning and depinning of fronts bounding spatially localized structures in the forced complex Ginzburg-Landau equation describing the 1:1 resonance is studied in one spatial dimension, focusing on regimes in which the structure grows via roll insertion instead of roll nucleation at either edge. The motion of the fronts is nonlocal but can be analyzed quantitatively near the depinning transition.

Conical wave propagation and diffraction in two-dimensional hexagonally packed granular lattices

Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wave packet and via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression, i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a nonoscillatory nature, resulting from the complex interplay among the discreteness, nonlinearity, and geometry of the packing. The transition between these two types of propagation is explored.

Homoclinic Snakes Bounded by a Saddle-Center Periodic Orbit

We describe a new variant of the so-called homoclinic snaking mechanism for the generation of infinitely many distinct localized patterns in spatially reversible partial differential equations on the real line. In standard snaking a branch of localized states undergoes infinitely many folds as the pattern grows in length by adding cells at either side. In the cases studied here the localized states have a defect or hump in the middle corresponding to an additional orbit homoclinic to the underlying spatially periodic orbit, and the folds accumulate on a parameter value where the periodic orbit undergoes a saddle-center transition. By analyzing an appropriate normal form in a spatial dynamics approach, it is shown that convergence of the folds is algebraic rather than exponential. Specifically the parameter value of the

Two-dimensional localized structures in harmonically forced oscillatory systems

n

Strong transmission and reflection of edge modes in bounded photonic graphene

th fold scales like

A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation

n^{-4}

EPOCH LIFETIMES IN THE DYNAMICS OF A COMPETING POPULATION

. The transition from this saddle-center mediated snaking to regular snaking is described by a codimension-two bifurcation that is also analyzed. The results ...

Defect-mediated snaking: A new growth mechanism for localized structures

Two-dimensional spatially localized structures in the complex Ginzburg–Landau equation with 1:1 resonance are studied near the simultaneous presence of a steady front between two spatially homogeneous equilibria and a supercritical Turing bifurcation on one of them. The bifurcation structures of steady circular fronts and localized target patterns are computed in the Turing-stable and Turing-unstable regimes. In particular, localized target patterns grow along the solution branch via ring insertion at the core in a process reminiscent of defect-mediated snaking in one spatial dimension. Stability of axisymmetric solutions on these branches with respect to axisymmetric and nonaxisymmetric perturbations is determined, and parameter regimes with stable axisymmetric oscillons are identified. Direct numerical simulations reveal novel depinning dynamics of localized target patterns in the radial direction, and of circular and planar localized hexagonal patterns in the fully two-dimensional system.

Linear and nonlinear traveling edge waves in optical honeycomb lattices

The propagation of linear and nonlinear edge modes in bounded photonic honeycomb lattices formed by an array of rapidly varying helical waveguides is studied. These edge modes are found to exhibit strong transmission (reflection) around sharp corners when the dispersion relation is topologically nontrivial (trivial). An asymptotic theory is developed that establishes the presence (absence) of typical edge states, including, in particular, armchair and zigzag edge states in the topologically nontrivial (trivial) case. In the presence of topological protection, nonlinear edge solitons can persist over very long distances.