Nonlinear Theory for Coalescing Characteristics in Multiphase Whitham Modulation Theory

Abstract

The multiphase Whitham modulation equations with N phases have 2N characteristics which may be of hyperbolic or elliptic type. In this paper, a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly, a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation, that is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling wave solutions of coupled nonlinear Schrödinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.

Genuine Nonlinearity and its Connection to the Modified Korteweg - de Vries Equation in Phase Dynamics

“…As a result, starting from the WMEs one can systematically identify the most suitable dispersive long-wave model simply by assessing which of the relevant properties the characteristic c and linear dispersion relation σ satisfy. Combined with connections made in previous works [46,10] this essentially turns the phase modulation analysis into a flow chart-like process. This is visualised in figure 3, demonstrating the connections between each of these well-established equation and the conditions required on the respective linear quantity.…”

“…Appealing to solvability of this equation and manipulations almost identical to previous work [10] and leads to the vector system…”

“…Here, k and ω are the vectors containing each slow wavenumber and frequency respectively and A and B are now the vector-valued wave action and wave action flux associated with each phase of the solution. Emerging from this system are up to 2N characteristics c satisfying the zero determinant condition of a quadratic matrix pencil [10]:…”

“…Much like how the WMEs generalises to multiple phases, this approach and insight too extends naturally to waves with arbitrarily many phases However, it can be shown that the form of the dispersive dynamics may alter dependent on the nature of the characteristics. For example, when characteristics coalesce at the elliptic-hyperbolic transition point it is instead the dynamics of the two-way Boussinesq which become operational [9,10]. Such a dynamical change heralds both quantitative and qualitative differences in how the system evolves -new solutions may arise, how they bifurcate may be altered and stability properties of solution families can change.…”

Reappraisal of Whitham's 1967 theory for wave-mean flow interaction in shallow water

“…At coalescing characteristics the Whitham equations are no longer valid, but a remodulation with a slower time scale leads to a new modulation equation. It is shown in [6], that conservation of wave action, on the slower time scale, and after projection, leads to a modulation equation of the form…”