Daniel J. Ratliff

Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg-de Vries equation

Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.

Phase dynamics of periodic waves leading to the Kadomtsev–Petviashvili equation in 3+1 dimensions

The Kadomstev–Petviashvili (KP) equation is a well-known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper, it is shown that the KP equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a universal form, with the coefficients determined by the components of the conservation of wave action. The theory is derived for a general class of partial differential equations generated by a Lagrangian using phase modulation. The theory extends to any space dimension and time, but the emphasis in the paper is on the case of 3+1. Motivated by light bullets and quantum vortex dynamics, the theory is illustrated by showing how defocusing NLS in 3+1 bifurcates to KP in 3+1 at criticality. The generalization to N>3 is also discussed.

Double criticality and the two-way Boussinesq equation in stratified shallow water hydrodynamics

Double criticality and its nonlinear implications are considered for stratified N–layer shallow water flows with N = 1,  2,  3. Double criticality arises when the linearization of the steady problem about a uniform flow has a double zero eigenvalue. We find that there are two types of double criticality: non-semisimple (one eigenvector and one generalized eigenvector) and semi-simple (two independent eigenvectors). Using a multiple scales argument, dictated by the type of singularity, it is shown that the weakly nonlinear problem near double criticality is governed by a two-way Boussinesq equation (non-semisimple case) and a coupled Korteweg-de Vries equation (semisimple case). Parameter values and reduced equations are constructed for the examples of two-layer and three-layer stratified shallow water hydrodynamics.

On the Elliptic-Hyperbolic Transition in Whitham Modulation Theory

The dispersionless Whitham modulation equations in one space dimension and time are generically hyperbolic or elliptic and break down at the transition, which is a curve in the frequency-wavenumber plane. In this paper, the modulation theory is reformulated with a slow phase and different scalings resulting in a phase modulation equation near the singular curves which is a geometric form of the two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multiperiodic, quasi-periodic, and multipulse localized solutions. This theory shows that the elliptic-hyperbolic transition is a rich source of complex behavior in nonlinear wave fields. There are several examples of these transition curves in the literature to which the theory applies. For illustration the theory is applied to the complex nonlinear Klein--Gordon equation which has two singular curves in the manifold of periodic traveling waves.

Vanishing characteristic speeds and critical dispersive points in nonlinear interfacial wave problems

Criticality plays a central role in the study of reductions and stability of hydrodynamical systems. At critical points, it is often the case that nonlinear reductions with dispersion arise to govern solution behavior. By considering when such models become bidirectional and lose their initial dispersive properties, it will be shown that higher order dispersive models may be supported in hydrodynamical systems. Precisely, this equation is a two-way Boussinesq equation with sixth order dispersion. The case of two layered shallow water is considered to illustrate this, and it is reasoned why such an environment is natural for such a system to emerge. Further, it is demonstrated that the regions in the parameter space for nontrivial flow, which admit this reduction, are vast and in fact form a continuum. The reduced model is then numerically simulated to illustrate how the two-way and higher dispersive properties suggest more exotic families of solitary wave solutions can emerge in stratified flows.

Nonlinear modulation near the Lighthill instability threshold in 2+1 Whitham theory

The dispersionless Whitham modulation equations in 2+1 (two space dimensions and time) are reviewed and the instabilities identified. The modulation theory is then reformulated, near the Lighthill instability threshold, with a slow phase, moving frame and different scalings. The resulting nonlinear phase modulation equation near the Lighthill surfaces is a geometric form of the 2+1 two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multi-periodic, quasi-periodic and multi-pulse localized solutions. For illustration the theory is applied to a complex nonlinear 2+1 Klein–Gordon equation which has two Lighthill surfaces in the manifold of periodic travelling waves. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.