On the Whitham modulation equations for the Toda lattice and the quantitative characterization of its dispersive shocks
arXiv:2312.10755 · doi:10.1016/j.physd.2024.134315
Abstract
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.
arXiv comment23 pages, 5 figures, 58 references
References in corpus (24)
- Dispersive shock waves and modulation theory
- Dispersive, superfluid-like shock waves in nonlinear optics
- On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
- Universal nature of the nonlinear stage of modulational instability
- Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
- Shock Wave Structure in a Strongly Nonlinear Granular Lattice with Viscous Dissipation
- Dispersive dam-break flow of a photon fluid
- Observation of two-wave structure in strongly nonlinear dissipative granular chains
- Shocks in non-loaded bead chains with impurities
- Observation of shock waves in a large Bose-Einstein condensate
- Solitonic dispersive hydrodynamics: theory and observation
- On the Whitham equations for the defocusing nonlinear Schrodinger equation with step initial data
- Observation of dispersive shock waves, solitons, and their interactions in viscous fluid conduits
- Hydrodynamic Optical Soliton Tunneling
- Solitary and shock waves in discrete double power-law materials
- Riemann problems and dispersive shocks in self-focusing media
- Characterizing Traveling Wave Collisions in Granular Chains Starting from Integrable Limits: the case of the KdV and the Toda Lattice
- Finite genus solutions for the Ablowitz-Ladik hierarchy
- Demonstration of dispersive rarefaction shocks in hollow elliptical cylinder chains
- Soliton trapping, transmission and wake in modulationally unstable media
- Oblique interactions between solitons and mean flows in the Kadomtsev-Petviashvili equation
- Observation of ultra-slow shock waves in a tunable magnetic lattice
- Nonlinear interactions between solitons and dispersive shocks in focusing media
- Dynamical chaos in the integrable Toda chain induced by time discretization