Search for: All records

Abstract The Riemann problem for the discrete conservation law is classified using Whitham modulation theory, a quasi‐continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogs of well‐known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite‐time blowup are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations. 

We prove the existence of a class of time-localized and space-periodic breathers (called q-gap breathers) in nonlinear lattices with time-periodic coe!cients. These q-gap breathers are the counterparts to the classical space-localized and time-periodic breathers found in space-periodic systems. Using normal form transformations, we establish rigorously the existence of such solutions with oscillating tails (in the time domain) that can be made arbitrarily small but finite. Due to the presence of the oscillating tails, these solutions are coined generalized q-gap breathers. Using a multiple-scale analysis, we also derive a tractable amplitude equation that describes the dynamics of breathers in the limit of small amplitude. In the presence of damping, we demonstrate the existence of transition fronts that connect the trivial state to the time-periodic ones. The analytical results are corroborated by systematic numerical simulations. 

The aim of this work is multifold. Firstly, it intends to present a complete, quantitative and self-contained description of the periodic traveling wave solutions and Whitham modulation equations for the Toda lattice, combining results from different previous works in the literature. Specifically, we connect the Whitham modulation equations and a detailed expression for the periodic traveling wave solutions of the Toda lattice. Along the way, some key details are filled in, such as the explicit expression of the characteristic speeds of the genus-one Toda–Whitham system. Secondly, we use these tools to obtain a detailed quantitative characterization of the dispersive shocks of the Toda system. Lastly, we validate the relevant analysis by performing a detailed comparison with direct numerical simulations. 

We provide the first experimental realization of wavenumber bandgap (𝑞−gap) breathers. Experiments are obtained in the setting of a time-periodic phononic lattice where the model and experiment exhibit good qualitative agreement. 𝑞−gap breathers are localized in time and periodic in space, and are the counterparts to the classical breathers found in space-periodic systems. We derive an exact condition for modulation instability that leads to the opening of wavenumber bandgaps in which the 𝑞−gap breathers can arise. The 𝑞−gap breathers become more narrow and larger in amplitude as the wavenumber goes further into the bandgap. In the presence of damping, these structures acquire a nonzero, oscillating tail. The controllable temporal localization that 𝑞−gap breathers make possible has potential applications in the creation of phononic frequency combs, energy harvesting or acoustic signal processing.