Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations

A two-dimensional (2D) generalization of the stabilized Kuramoto - Sivashinsky (KS) system is presented. It is based on the Kadomtsev-Petviashvili (KP) equation including dissipation of the generic (Newell -- Whitehead -- Segel, NWS) type and gain. The system directly applies to the description of gravity-capillary waves on the surface of a liquid layer flowing down an inclined plane, with a surfactant diffusing along the layer's surface. Actually, the model is quite general, offering a simple way to stabilize nonlinear waves in media combining the weakly-2D dispersion of the KP type with gain and NWS dissipation. Parallel to this, another model is introduced, whose dissipative terms are isotropic, rather than of the NWS type. Both models include an additional linear equation of the advection-diffusion type, linearly coupled to the main KP-NWS equation. The extra equation provides for stability of the zero background in the system, opening a way to the existence of stable localized pulses. The consideration is focused on the case when the dispersive part of the system of the KP-I type, admitting the existence of 2D localized pulses. Treating the dissipation and gain as small perturbations and making use of the balance equation for the field momentum, we find that the equilibrium between the gain and losses may select two 2D solitons, from their continuous family existing in the conservative counterpart of the model (the latter family is found in an exact analytical form). The selected soliton with the larger amplitude is expected to be stable. Direct simulations completely corroborate the analytical predictions.Comment: a latex text file and 16 eps files with figures; Physical Review E, in pres

Experimental Investigations of Capillary Effects on Nonlinear Free-Surface Waves

This thesis presents the results of three experiments on various aspects of the effects of surface tension on nonlinear free-surface waves. The first two experiments focus on capillary effects on the breaking of short-wavelength gravity waves, a problem of interest in areas of physical oceanography and remote sensing. The third experiment is concerned with the bifurcation of solitary capillary-gravity waves, a problem that is relevant in the study of nonlinear, dispersive wave systems. In the first set of experiments, streamwise profile measurements were made of spilling breakers at the point of incipient breaking. Both wind-waves and mechanically generated waves were investigated in this study, with gravity wavelengths in the range of 10--120 cm. Although it has been previously argued that the crest shape is dependent only on the surface tension, the results reported herein are to the contrary as several geometrical parameters used to describe the crest change significantly with the wavelength. However, the non-dimensional crest shape is self-similar, with two-shape parameters that depend on a measure of the local wave slope. This self-similarity persists over the entire range of wavelengths and breaker conditions measured, indicating a universal behavior in the near-crest dynamics that is independent of the method used to generate the wave. The measured wave slope is found to be related to the wave growth rate and phase-speed prior to breaking, a result that contributes towards the development of a breaking criterion for unsteady capillary-gravity waves. The second set of experiments examines the cross-stream surface structure in the turbulent breaking zone generated by short-wavelength breakers. Waves in this study were generated using a mechanical wedge and ranged in wavelength from 80--120 cm. To isolate the effects of surface tension on the flow, the important experimental parameters were adjusted to produce Froude-scaled, dispersively-focused wave packets. The results show the development of ``quasi''-2D streamwise ripples along with smaller cross-stream ripples that grow as breaking develops and can become comparable in amplitude to the streamwise ripples for larger breakers. It is found that the amplitude of the cross-stream surface ripples scale as \bar{&lambda}^3, where \bar{&lambda} is the average wavelength of the wave packet. The cross-stream ripple activity appears to be highest in the ``troughs'' of the larger streamwise ripples, with the appearance of persistent ``scar''-like features. Based on these observations, a simple model for the coupling between the vorticity and capillary structure in the breaking zone is conjectured. The third set of experiments focuses on the generation of capillary-gravity waves by a pressure source moving near the minimum phase speed cmin. Near this minimum, nonlinear capillary-gravity solitary waves, or ``lumps'', have been shown to exist theoretically. We identify an abrupt transition to a wave-like state that features a localized solitary wave that trails the pressure forcing. This trailing wave is steady, fully localized in 3D, elongated in the cross-stream relative to the streamwise direction, and has a one-to-one relationship between height and phase speed. All of these characterisitics are commensurate with the freely propogating ``lumps'' computed by previous authors, and a quantitative comparison between these previous numerical calculations and the current experiments is presented. At speeds closer to cmin, a new time-dependent state is observed that can qualitatively be described by the shedding of solitary depressions from the tips of a ``V''-shaped pattern. These results are discussed in conjunction with a new theoretical model for these waves that employs nonlinear and viscous effects, both of which are crucial in capturing the salient features of the surface response. While discussed in the context of water waves, these results have applicaiton to other wave systems where nonlinear and dispersive effects are important

Dynamics of Nonlinear Gravity-Capillary Waves in Deep Water Near Resonance

The minimum phase speed of linear gravity-capillary waves in deep water ($c_\mathrm{min}$) is known to be the bifurcation point of three-dimensional solitary waves (``lumps"). In the present thesis, various aspects of unsteady gravity-capillary lumps are investigated in the context of three sets of experiments. In the first set, cinematic shadowgraph and refraction-based techniques are utilized to measure the temporal evolution of the free surface deformation pattern downstream of a surface pressure source as it moves along a towing tank, while numerical simulations using a model equation are used to extend the experimental results. The focus of this study is on exploring the characteristics of the observed periodic shedding of lump-like depressions for towing speeds close to $c_\mathrm{min}$. From the experiments, it is found that the speed-amplitude characteristics and the shape of the depressions are nearly the same as those of the freely propagating gravity-capillary lumps of inviscid potential theory. The periodic behavior is found to be analogous to the periodic generation of two-dimensional solitary waves in shallow water by a source moving at trans-critical speeds of pure gravity waves. In the second set of experiments, the effect of viscous dissipation on freely propagating lumps is examined. A steady forced lump is first generated by applying appropriate forcing and towing speed. The forcing is then removed suddenly and the change in shape and speed of the lump is measured as it propagates freely under the action of viscosity. It is found that the localized structure of the lump is maintained during the decay and the first measurement of the decay rate of gravity-capillary lumps is reported. In the third set of experiments, the interactions of state III lumps generated by two pressure sources moving in parallel straight lines are investigated. The sources are adjusted to produce nearly identical periodic responses. The first lump generated by each source, collides with the lump from the other source in the center-plane of the two sources. It was observed that a steep depression is formed during the collision but breaks up soon after and radiates energy away in the form of small-amplitude radial waves. After the collision, a quasi-steady pattern is formed with several rows of localized depressions that are similar to lumps but exhibit periodic oscillations in depth

Self-focusing dynamics of patches of ripples

The dynamics of focussing of extended patches of nonlinear capillary gravity waves within the primitive fluid dynamic equations is presented. It is found that, when the envelope has certain properties, the patch focusses initially in accordance to predictions from nonlinear Schrodinger equation, and focussing can concentrate energy to the vicinity of a point or a curve on the fluid surface. After initial focussing, other effects dominate and the patch breaks up into a complex set of localised structures lumps and breathers - plus dispersive radiation. We perform simulations both in the inviscid regime and for small viscosities. Lastly we discuss throughout the similarities and differences between the dynamics of ripple patches and self-focussing light beams. (C) 2016 The Authors. Published by Elsevier B.V

Structure and dynamics of solitary waves in fluid media

This research deals with the study of nonlinear solitary waves in fluid media. The equations which model surface and internal waves in fluids have been studied and used in this research. The approach to study the structure and dynamics of internal solitary waves in near-critical situations is the traditional theoretical and numerical study of nonlinear wave processes based on the methods of dynamical systems. The synergetic approach has been exploited, which presumes a combination of theoretical and numerical methods. All numerical calculations were performed with the desktop personal computer. Traditional and novel methods of mathematical physics were actively used, including Fourier analysis technique, inverse scattering method, Hirota method, phase-plane analysis, analysis of integral invariants, finite-difference method, Petviashvili and Yang–Lakoba numerical iterative techniques for the numerical solution of Partial Differential Equation. A new model equation, dubbed the Gardner–Kawahara equation, has been suggested to describe wave phenomena in the near-critical situations, when the nonlinear and dispersive coefficients become anomalously small. Such near-critical situations were not studied so far, therefore this study is very topical and innovative. Results obtained will shed a light on the structure of solitary waves in near-critical situation, which can occur in two-layer fluid with strong surface tension between the layers. A family of solitary waves was constructed numerically for the derived Gardner–Kawahara equation; their structure has been investigated analytically and numerically. The problem of modulation stability of quasi-monochromatic wave-trains propagating in a media has also being studied. The Nonlinear Schrödinger equation (NLSE) has been derived from the unidirectional Gardner–Ostrovsky equation and a general Shrira equation which describes both surface and internal long waves in a rotating fluid. It was demonstrated that earlier obtained results (Grimshaw & Helfrich, 2008; 2012; Whitfield & Johnson, 2015a; 2015b) on modulational stability/instability are correct within the limited range of wavenumbers where the Ostrovsky equation is applicable. In the meantime, results obtained in this Thesis and published in the paper (Nikitenkova et al., 2015) are applicable in the wider range of wavenumbers up to k = 0. It was shown that surface and internal oceanic waves are stable with respect to selfmodulation at small wavenumbers when k → 0 in contrast to what was mistakenly obtained in (Shrira, 1981). In Chapter 4 new exact solutions of the Kadomtsev-Petviashvili equation with a positive dispersion are obtained in the form of obliquely propagating skew lumps. Specific features of such lumps were studied in details. In particular, the integral characteristics of single lumps (mass, momentum components and energy) have been calculated and presented in terms of lump velocity. It was shown that exact stationary multi-lump solutions can be constructed for this equation. As the example, the exact bilump solution is presented in the explicit form and illustrated graphically. The relevance of skew lumps to the real physical systems is discussed

Dynamics of three-dimensional gravity-capillary solitary waves in deep water

A model equation for gravity-capillary waves in deep water is proposed. This model is a quadratic approximation of the deep water potential flow equations and has wavepacket-type solitary wave solutions. The model equation supports line solitary waves which are spatially localized in the direction of propagation and constant in the transverse direction, and lump solitary waves which are spatially localized in both directions. Branches of both line and lump solitary waves are computed via a numerical continuation method. The stability of each type of wave is examined. The transverse instability of line solitary waves is predicted by a similar instability of line solitary waves in the nonlinear Schrödinger equation. The spectral stability of lumps is predicted using the waves' speed energy relation. The role of wave collapse in the stability of these waves is also examined. Numerical time evolution is used to confirm stability predictions and observe dynamics, including instabilities and solitary wave collisions

Hydroelastic solitary waves in deep water

The problem of waves propagating on the surface of a two-dimensional ideal fluid of infinite depth bounded above by an elastic sheet is studied with asymptotic and numerical methods. We use a nonlinear elastic model that has been used to describe the dynamics of ice sheets. Particular attention is paid to forced and unforced dynamics of waves having near-minimum phase speed. For the unforced problem, we find that wavepacket solitary waves bifurcate from nonlinear periodic waves of minimum speed. When the problem is forced by a moving load, we find that, for small-amplitude forcing, steady responses are possible at all subcritical speeds, but for larger loads there is a transcritical range of forcing speeds for which there are no steady solutions. In unsteady computations, we find that if the problem is forced at a speed in this range, very large unsteady responses are obtained, and that when the forcing is released, a solitary wave is generated. These solitary waves appear stable, and can coexist within a sea of small-amplitude waves