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

Flux singularities in multiphase wavetrains and the Kadomtsev-Petviashvili equation with applications to stratified hydrodynamics

This paper illustrates how the singularity of the wave action flux causes the Kadomtsev-Petviashvili (KP) equation to arise naturally from the modulation of a two-phased wavetrain, causing the dispersion to emerge from the classical Whitham modulation theory. Interestingly, the coefficients of the resulting KP are shown to be related to the associated conservation of wave action for the original wavetrain, and so may be obtained prior to the modulation. This provides a universal form for the KP as a dispersive reduction from any Lagrangian with the appropriate wave action flux singularity. The theory is applied to the full water wave problem with two layers of stratification, illustrating how the KP equation arises from the modulation of a uniform flow state and how its coefficients may be extracted from the system

Flux singularities in multiphase wavetrains and the Kadomtsev‐Petviashvili equation with applications to stratified hydrodynamics

This paper illustrates how the singularity of the wave action flux causes the Kadomtsev-Petviashvili (KP) equation to arise naturally from the modulation of a two-phased wavetrain, causing the dispersion to emerge from the classical Whitham modulation theory. Interestingly, the coefficients of the resulting KP are shown to be related to the associated conservation of wave action for the original wavetrain, and therefore may be obtained prior to the modulation. This provides a universal form for the KP as a dispersive reduction from any Lagrangian with the appropriate wave action flux singularity. The theory is applied to the full water wave problem with two layers of stratification, illustrating how the KP equation arises from the modulation of a uniform flow state and how its coefficients may be extracted from the system

Modulational instability of two pairs of counter-propagating waves and energy exchange in two-component media

The dynamics of two pairs of counter-propagating waves in two-component media is considered within the framework of two generally nonintegrable coupled Sine-Gordon equations. We consider the dynamics of weakly nonlinear wave packets, and using an asymptotic multiple-scales expansion we obtain a suite of evolution equations to describe energy exchange between the two components of the system. Depending on the wave packet length-scale vis-a-vis the wave amplitude scale, these evolution equations are either four non-dispersive and nonlinearly coupled envelope equations, or four non-locally coupled nonlinear Schroedinger equations. We also consider a set of fully coupled nonlinear Schroedinger equations, even though this system contains small dispersive terms which are strictly beyond the leading order of the asymptotic multiple-scales expansion method. Using both the theoretical predictions following from these asymptotic models and numerical simulations of the original unapproximated equations, we investigate the stability of plane-wave solutions, and show that they may be modulationally unstable. These instabilities can then lead to the formation of localized structures, and to a modification of the energy exchange between the components. When the system is close to being integrable, the time-evolution is distinguished by a remarkable almost periodic sequence of energy exchange scenarios, with spatial patterns alternating between approximately uniform wavetrains and localized structures.Comment: 35 pages, 13 figure

Spatial Hamiltonian identities for nonlocally coupled systems

We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether's theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler-Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.Comment: 39 pages, 1 figur

Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves

Standing waves are a fundamental class of solutions of nonlinear wave equations with a spatial reflection symmetry, and they routinely arise in optical and oceanographic applications. At the linear level they are composed of two synchronized counterpropagating periodic traveling waves. At the nonlinear level, they can be defined abstractly by their symmetry properties. In this paper, general aspects of the modulational instability of standing waves are considered. This problem has difficulties that do not arise in the modulational instability of traveling waves. Here we propose a new geometric formulation for the linear stability problem, based on embedding the standing wave in a four-parameter family of nonlinear counterpropagating waves. Multisymplectic geometry is shown to encode the stability properties in an essential way. At the weakly nonlinear level we obtain the surprising result that standing waves are modulationally unstable only if the component traveling waves are modulation unstable. Systems of nonlinear wave equations will be used for illustration, but general aspects will be presented, applicable to a wide range of Hamiltonian PDEs, including water waves.</p

Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves

Standing waves are a fundamental class of solutions of nonlinear wave equations with a spatial reflection symmetry, and they routinely arise in optical and oceanographic applications. At the linear level they are composed of two synchronized counterpropagating periodic traveling waves. At the nonlinear level, they can be defined abstractly by their symmetry properties. In this paper, general aspects of the modulational instability of standing waves are considered. This problem has difficulties that do not arise in the modulational instability of traveling waves. Here we propose a new geometric formulation for the linear stability problem, based on embedding the standing wave in a four-parameter family of nonlinear counterpropagating waves. Multisymplectic geometry is shown to encode the stability properties in an essential way. At the weakly nonlinear level we obtain the surprising result that standing waves are modulationally unstable only if the component traveling waves are modulation unstable. Systems of nonlinear wave equations will be used for illustration, but general aspects will be presented, applicable to a wide range of Hamiltonian PDEs, including water waves.</p