Do peaked solitary water waves indeed exist?

Many models of shallow water waves admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa-Holm equation. Besides, it is proved that Kelvin's theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, the phase speed of the peaked solitary waves has nothing to do with wave height. In addition, the kinetic energy of the peaked solitary waves either increases or almost keeps the same from free surface to bottom. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.Comment: 53 pages, 13 figures, 7 tables. Accepted by Communications in Nonlinear Science and Numerical Simulatio

Traveling surface waves of moderate amplitude in shallow water

We study traveling wave solutions of an equation for surface waves of moderate amplitude arising as a shallow water approximation of the Euler equations for inviscid, incompressible and homogenous fluids. We obtain solitary waves of elevation and depression, including a family of solitary waves with compact support, where the amplitude may increase or decrease with respect to the wave speed. Our approach is based on techniques from dynamical systems and relies on a reformulation of the evolution equation as an autonomous Hamiltonian system which facilitates an explicit expression for bounded orbits in the phase plane to establish existence of the corresponding periodic and solitary traveling wave solutions

Interactions of large amplitude solitary waves in viscous fluid conduits

The free interface separating an exterior, viscous fluid from an intrusive conduit of buoyant, less viscous fluid is known to support strongly nonlinear solitary waves due to a balance between viscosity-induced dispersion and buoyancy-induced nonlinearity. The overtaking, pairwise interaction of weakly nonlinear solitary waves has been classified theoretically for the Korteweg-de Vries equation and experimentally in the context of shallow water waves, but a theoretical and experimental classification of strongly nonlinear solitary wave interactions is lacking. The interactions of large amplitude solitary waves in viscous fluid conduits, a model physical system for the study of one-dimensional, truly dissipationless, dispersive nonlinear waves, are classified. Using a combined numerical and experimental approach, three classes of nonlinear interaction behavior are identified: purely bimodal, purely unimodal, and a mixed type. The magnitude of the dispersive radiation due to solitary wave interactions is quantified numerically and observed to be beyond the sensitivity of our experiments, suggesting that conduit solitary waves behave as "physical solitons." Experimental data are shown to be in excellent agreement with numerical simulations of the reduced model. Experimental movies are available with the online version of the paper.Comment: 13 pages, 4 figure

Transverse instability of concentric soliton waves

Should it be a pebble hitting water surface or an explosion taking place underwater, concentric surface waves inevitably propagate. Except for possibly early times of the impact, finite amplitude concentric water waves emerge from a balance between dispersion or nonlinearity resulting in solitary waves. While stability of plane solitary waves on deep and shallow water has been extensively studied, there are no analogous analyses for concentric solitary waves. On shallow water, the equation governing soliton formation -- the nearly concentric Korteweg-de Vries -- has been deduced before without surface tension, so we extend the derivation onto the surface tension case. On deep water, the envelope equation is traditionally thought to be the nonlinear Schr\"{o}dinger type originally derived in the Cartesian coordinates. However, with a systematic derivation in cylindrical coordinates suitable for studying concentric waves we demonstrate that the appropriate envelope equation must be amended with an inverse-square potential, thus leading to a Gross-Pitaevskii equation instead. Properties of both models for deep and shallow water cases are studied in detail, including conservation laws and the base states corresponding to axisymmetric solitary waves. Stability analyses of the latter lead to singular eigenvalue problems, which dictate the use of analytical tools. We identify the conditions resulting in the transverse instability of the concentric solitons revealing crucial differences from their plane counterparts. Of particular interest here are the effects of surface tension and cylindrical geometry on the occurrence of transverse instability