Hamiltonian Aspects of Three-Layer Stratified Fluids

The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Viríssimo and Milewski are also discussed.

Modular Korteweg - de Vries equation: Riemann, cnoidal and solitary waves

<p>The review paper by Oleg Rudenko [1] suggests several examples of elastic systems with so-called modular nonlinearities. In this study we consider the modular Korteweg - de Vries (KdV) equation in the form u_t + 6 u u_x + u_{xxx} = 0. This equation is not integrable by means of the Inverse Scattering Transform in the general case, but sign-defined functions which never change the sign satisfy the integrable KdV equation, and hence possess an exact solution. Firstly, we consider the dispersionless limit of the modular KdV equation and analyze the evolution of a simple nonlinear wave (Riemann wave) and its Fourier transform including the asymptotics when the wave tends to break [2]. Then, we study the structure of travelling waves. If the waves propagate on a pedestal and do not cross the zero level u = 0, they coincide with the well-known travelling wave solutions of the classic KdV equation in the form of cnoidal and solitary waves. If the pedestal is zero, the structure of sign-varying travelling waves is expressed through Jacobi elliptic functions. The interaction of solitary waves of different polarities is studied numerically using an implicit pseudo-spectral method. The simulation has revealed the inelastic character of the collision; in the course of the interaction the solitons can alter their amplitudes (the small soliton decreases and the large one grows) and emit small-amplitude waves. The inelastic effects are most pronounced when the solitons’ amplitudes are close. When their amplitudes differ significantly, the maximum wave height which is attained during the absorb-emit interaction tends to the sum of the heights of the solitons with the polarity inherited from the large soliton, as predicted in the frameworks of different long-wave integrable models in [3, 4]. As a result of the collision the solitons may experience non-classic phase shifts as they both jump back.</p><p>[1] O.V. Rudenko. Physics – Uspekhi, Vol. 56(7), 683-690 (2013).</p><p>[2] E. Tobisch, and E. Pelinovsky. Appl. Math. Lett., Vol. 97, 1-5 (2019).</p><p>[3] A.V. Slunyaev, and E.N. Pelinovsky. Phys. Rev. Lett., Vol. 117, 214501 (2016).</p><p>[4] A. Slunyaev. Stud. Appl. Math., Vol. 142, 385-413 (2019).</p>

Shock wave occurrence and soliton propagation in polariton condensates

We study the effects of the gain and the loss of polaritons on the wave propagation in polariton condensates. This system is described by a modified Gross–Pitaevskii equation. In the case of small damping, we use the reductive perturbation method to transform this equation; we get a modified Burgers equation in the dispersionless limit and a damped Korteweg – de Vries equation in a more general case. We demonstrate that the shock wave occurrence depends on the gain and the loss of polaritons in the dispersionless polariton condensate. The resolution of the damped Korteweg – de Vries equation shows that the soliton behaves like a damped wave in the case of a constant damping. Based on an asymptotic solution, the survival time and the distance traveled by this soliton are evaluated. We solve the damped Korteweg – de Vries equation and the modified Gross–Pitaevskii numerically to validate the analytical calculations and discuss especially the soliton propagation in the system.