Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves

Abstract

A new type of wave-mean flow interaction is identified and studied in which a smallamplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, largescale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg-de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket-mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wave packet by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Maiden et al. (2018) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.

Dynamic soliton–mean flow interaction with non-convex flux

“…In contrast, due to scale separation, the interaction between a soliton and a large-scale mean flow with amplitude O(1) is primarily a one-way nonlinear process -the mean flow exhibits a small phase shift due to soliton interaction -that can be viewed as 'soliton steering' by the dynamically evolving mean flow. This distinct type of dynamic wave-mean interaction is also realised for linearised shallow-water wave packets propagating over large-scale nonlinear dispersive mean flows (Congy et al 2019). However, non-convexity and positive dispersion supports an inherently two-way process: the kink imparts a polarity reversal to DSWs and RWs while the mean changes the amplitude and speed of the kink.…”

“…Soliton-mean flow interaction in the focusing NLS equation was investigated in Biondini & Lottes (2019). A similar problem involving the interaction of linear wavepackets with shallow-water wave mean flows modelled by the KdV equation was studied using an analogous modulation theory framework in Congy, El & Hoefer (2019). Aside from the focusing NLS case, for which mean flow evolution is described by an elliptic system of equations, and the present work, the models previously investigated in the context of soliton-mean flow interaction were limited to dispersive conservation laws with hyperbolic, convex flux.…”

Nematic Dispersive Shock Waves from Nonlocal to Local

“…The issue of internal resonance and its relation to the existence of the PDSW and RDSW regimes merits further study. In this regard, the recent work [55] on the interaction of linear wavepackets and DSWs could be relevant.…”

Soliton gas in integrable dispersive hydrodynamics

“…Indeed, the full modulation system contains D equations for λ j (X, T ), and the wave conservation equations are always consistent with, but not equivalent to the full system (see [47] for the KdV spectral modulation theory). However, in the harmonic and soliton limits corresponding to the collapsed spectral gaps or bands the dimension D of the vector λ decreases, enabling the necessary closure for the system of wave conservation laws in (3.9) under the additional constraint of constant background (see [82], [21] for the relevant theory of the dynamic wave-mean flow interaction showing how the effects of nonconstant background can be included). In particular, in the harmonic limit system (3.9) transforms into the kinematic system (3.3).…”

“…The branchcuts of R(z) will be specified tailed description of finite-gap solutions of NLS equation can be found in [32][33][34]. We t the finite-gap theory provides a natural r the construction of random solutions to tion by assuming a uniform distribution of ase vector θ (0) ∈ T n [21,35].…”