Numerical investigations of waves interactions from forced Korteweg de Vries equations

Soliton generated by the Korteweg de Vries (KdV) equation forms a group of solitons ladder. During full interaction of multi-soliton solutions, three types of peaks were obtained, namely single, flat and double peak. Soliton generated by the forced Korteweg de Vries (fKdV) equation forms uniform solitons trains with equal amplitude. Various aspects of solitons interactions of the fKdV equation for free surface flow over uneven bottom topography have been investigated. Fluid flowing over uneven bottom topography can support wave propagation that generates upstream and downstream nonlinear wavetrains. Such forced nonlinear solitary waves occur naturally in the shallow water near the coastal region. The fKdV equation models the above phenomena in many cases, such as in the transcritical, weakly nonlinear and weakly dispersive region. Numerical method which involves the pseudo-spectral method is used to solve the fKdV equation as it is difficult to obtain the solution analytically, due to the presence of the forcing term and the broken symmetry. A group of uniform solitons having the same amplitude and speed will not collide when the bump size and bump speed are constant. A wave profile with time-dependent transcritical velocity was investigated with a variation of Froude number. As the Froude number changes, two sets of solitary waves travelling upstream were discovered. A set of these solitary waves have nearly uniform amplitude, while another set comprises of solitary waves with variable amplitude, which forms a pairwise and two pairwise interactions pattern in the transcritical region. In the case of multiple bumps, upstream-advancing nonlinear solitary waves which may be generated continuously and interact with each other when the distance between bumps, width and height of bumps were varied. Interesting interaction patterns of the collision between uniform solitons will provide a better understanding of the forcing caused by multiple bumps on water flow at the uneven bottom topography of a shallow water in a rectangular channel

Time dependent hydraulic falls and trapped waves over submerged obstructions

We consider the classical problem of a free surface flowing past one or more disturbances in a channel. The fluid is assumed to be inviscid and incompressible, and the flow, irrotational. Both the effects of gravity and surface tension are considered. The stability of critical flow steady solutions, which have subcritical flow upstream of the disturbance and supercritical flow downstream, is investigated. We compute the initial steady solution using boundary integral equation techniques based on Cauchy integral formula and advance the solution forward in time using a mixed Euler-Lagrange method along with Adams-Bashforth-Moulton scheme. Both gravity and gravity-capillary critical flow solutions are found to be stable. The stability of solutions with a train of waves trapped between two disturbances is also investigated in the pure gravity and gravity-capillary cases

Free surface flows over submerged obstructions

Steady and unsteady two-dimensional free surface flows subjected to one or multiple disturbances are considered. Flow configurations involving either a single fluid or two layers of fluid of different but constant densities, are examined. Both the effects of gravity and surface tension are included. Fully nonlinear boundary integral equation techniques based on Cauchy’s integral formula are used to derive integro-differential equations to model the problem. The integro-differential equations are discretised and solved iteratively using Newton’s method. Both forced solitary waves and critical flow solutions, where the flow upstream is subcritical and downstream is supercritical, are obtained. The behaviour of the forced wave is determined by the Froude and Bond numbers and the orientation of the disturbance. When a second disturbance is placed upstream in the pure gravity critical case, trapped waves have been found between the disturbances. However, when surface tension is included, trapped waves are shown only to exist for very small values of the Bond number. Instead, it is shown that the disturbance must be placed downstream in the gravity-capillary case to see trapped waves. The stability of these critical hydraulic fall solutions is examined. It is shown that the hydraulic fall is stable, but the trapped wave solutions are only stable in the pure gravity case. Critical, flexural-gravity flows, where a thin sheet of ice rests on top of the fluid are then considered. The flows in the flexural-gravity and gravity-capillary cases are shown to be similar. These similarities are investigated, and the physical significance of both configurations, examined. When two fluids are considered, the situation is more complex. The rigid lid approximation is assumed, and four types of critical flow are investigated. Trapped wave solutions are found to exist in some cases, depending on the Froude number in the lower layer

Dispersive shock waves and modulation theory

There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs

Nonautonomous analysis of steady Korteweg-de Vries waves under nonlocalised forcing

Abstract not availableSanjeeva Balasuriya, Benjamin J. Binde

Nonlinear dynamics of three-dimensional solitary waves

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 105-108).In problems of dispersive wave propagation governed by two distinct restoring-force mechanisms, the phase speed of linear sinusoidal wavetrains may feature a minimum, cmin, at non-zero wavenumber, kmin. Examples include waves on the surface of a liquid in the presence of both gravity and surface tension, flexural waves on a floating ice sheet, in which case capillarity is replaced by the flexural rigidity of the ice, and internal gravity waves in layered flows in the presence of interfacial tension. The focus here is on deep-water gravity-capillary waves, where cmin = 23 cm/s with corresponding wavelength Amin = 27r/kmin = 1.71 cm. In this instance, ignoring viscous dissipation, cmin is known to be the bifurcation point of two-dimensional (plane) and three-dimensional (fully localized) solitary waves, often referred to as "lumps"; these are nonlinear disturbances that propagate at speeds below cmin without change of shape owing to a perfect balance between the opposing effects of wave dispersion and nonlinear steepening. Moreover, Cmin is a critical forcing speed, as the linear inviscid response to external forcing moving at Cmin grows unbounded in time, and nonlinear effects as well as viscous dissipation are expected to play important parts near this resonance. In the present thesis, various aspects of the dynamics of gravity-capillary lumps are investigated theoretically. Specifically, it is shown that steep gravity-capillary lumps of depression can propagate stably and they are prominent nonlinear features of the forced response near resonant conditions, in agreement with companion experiment for the generation of gravity-capillary lumps on deep water. These findings are relevant to the generation of ripples by wind and to the wave drag associated with the motion of small bodies on a free surface.by Yeunwoo Cho.Ph.D

Symmetry and Complexity

Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry

NIAC Phase I Final Report: On-Orbit, Collision-Free Mapping of Small Orbital Debris

Sub-centimeter orbital debris is currently undetectable using ground-based radar and optical methods. However, the pits in Space Shuttle windows produced by paint chips (e.g. the 3.8mm diameter pit produced by a 0.2mm paint chip on STS-7) demonstrate that small debris can cause serious damage to spacecraft. Recent analytical, computational and experimental work has shown that charged objects moving quickly through a plasma will cause the formation of solitons in the plasma density. Due to their exposure to the solar wind plasma environment, even the smallest space debris will be charged. Depending on the debris size, charge and velocity, the plasma signature of the solitons may be detected by simple instrumentation on spacecraft. We will describe the amplitude and velocity of solitons that may be produced by mm-cm scale orbital debris in LEO. We will discuss the feasibility of mapping sub-cm orbital debris using a fleet of CubeSats equipped with Langmuir probes. The time and fleet size required to map the debris will also be described. Plasma soliton detection would be the first collision-free method of mapping the small debris population