Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory

El, Gennady; Grimshaw, Roger; Smyth, Noel F.

doi:10.1016/j.physd.2008.03.031

Cited by 41 publications

(61 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Ultimately, a finite solitary wavetrain emerges from this interaction process. We now use a modification of conduit DSW theory (Lowman & Hoefer 2013b) to determine the properties of this solitary wavetrain by applying the solitary wave resolution method originally developed in (El et al 2008).…”

Section: Conduit Equation Backgroundmentioning

confidence: 99%

“…Because of its ubiquity, we seek a deeper understanding of soliton fission that results from a broad initial condition, hereafter referred to as the box problem due to the initial profile's wide shape. A new method based on Whitham averaging theory (Whitham 1974) that does not require integrability was first proposed and applied to the Serre/Su-Gardner/Green-Naghdi equations for fully nonlinear shallow water waves in El et al (2008) and, partially, to the defocusing nonlinear Schrödinger equation with saturable nonlinearity in El et al (2007). The method draws upon principles first developed to describe DSWs that result from step initial data (El 2005).…”

Section: Introductionmentioning

confidence: 99%

“…In fact, in both of these experimental papers, solitary waves were generated by the fission of a large, initial disturbance. We apply modulation theory for solitary wave fission introduced in (El et al 2008) to the box problem for a PDE model of this fluid context known as the conduit equation Lowman & Hoefer (2013a) a t + (a 2 ) z − a 2 a −1 a t z z = 0.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solitary wave fission of a large disturbance in a viscous fluid conduit

et al. 2019

View full text Add to dashboard Cite

This paper presents a theoretical and experimental study of the long-standing fluid mechanics problem involving the temporal resolution of a large, localised initial disturbance into a sequence of solitary waves. This problem is of fundamental importance in a range of applications including tsunami and internal ocean wave modelling. This study is performed in the context of the viscous fluid conduit system-the driven, cylindrical, free interface between two miscible Stokes fluids with high viscosity contrast. Due to buoyancy induced nonlinear self-steepening balanced by stress induced interfacial dispersion, the disturbance evolves into a slowly modulated wavetrain and further, into a sequence of solitary waves. An extension of Whitham modulation theory, termed the solitary wave resolution method, is used to resolve the fission of an initial disturbance into solitary waves. The developed theory predicts the relationship between the initial disturbance's profile, the number of emergent solitary waves, and their amplitude distribution, quantifying an extension of the well-known soliton resolution conjecture from integrable systems to non-integrable systems that often provide a more accurate modelling of physical systems. The theoretical predictions for the fluid conduit system are confirmed both numerically and experimentally. The number of observed solitary waves is consistently within 1-2 waves of the prediction, and the amplitude distribution shows remarkable agreement. Universal properties of solitary wave fission in other fluid dynamics problems are identified.

show abstract

Section: Conduit Equation Backgroundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Solitary wave fission of a large disturbance in a viscous fluid conduit

et al. 2019

View full text Add to dashboard Cite

show abstract

“…A popular model here in the weakly nonlinear regime when the obstacle has a small amplitude is the forced Korteweg-de Vries (KdV) equation, see Akylas (1984), Cole (1985), Grimshaw and Smyth (1986), Lee et al (1989), Binder et al (2006), Grimshaw et al (2007) and the recent review by Grimshaw (2010). Various aspects of the extension to finite amplitudes in the long wave regime can be found in El et al (2006), El et al (2008), and El et al (2009).…”

Section: Introductionmentioning

confidence: 99%

Critical control in transcritical shallow-water flow over two obstacles

Grimshaw

Maleewong

2015

J. Fluid Mech.

View full text Add to dashboard Cite

The nonlinear shallow-water equations are often used to model flow over topography In this paper we use these equations both analytically and numerically to study flow over two widely separated localised obstacles, and compare the outcome with the corresponding flow over a single localised obstacle. Initially we assume uniform flow with constant water depth, which is then perturbed by the obstacles. The upstream flow can be characterised as subcritical, supercritical, and transcritical respectively. We review the well-known theory for flow over a single localised obstacle, where in the transcritical regime the flow is characterised by a local hydraulic flow over the obstacle, contained between an elevation shock propagating upstream and a depression shock propagating downstream. Classical shock closure conditions are used to determine these shocks. Then we show that the same approach can be used to describe the flow over two widely spaced localised obstacles. The flow development can be characterized by two stages. The first stage is the generation of upstream elevation shock and downstream depression shock from each obstacle alone, isolated from the other obstacle. The second stage is the interaction of two shocks between the two obstacles, followed by an adjustment to a hydraulic flow over both obstacles, with criticality being controlled by the higher of the two obstacles, and by the second obstacle when they have equal heights. This hydraulic flow is terminated by an elevation shock propagating upstream of the first obstacle and a depression shock propagating downstream of the second obstacle. A weakly nonlinear model for sufficiently small obstacles is developed to describe this second stage. The theoretical results are compared with fully nonlinear simulations obtained using a well-balanced finite volume method. The analytical results agree quite well with the nonlinear simulations for sufficiently small obstacles.

show abstract

“…Recently, Wang et al [24] have analyzed the water well resonance induced by preearthquake signals in a confined aquifer resorting to an analytical linearized approach. In the present paper, an exact analytical solution is proposed for the pressure waves propagating within a homogeneous confined coastal aquifer as a consequence of a train of generally asymmetric, triangular-like sea waves approximating the sequence of impulses to which can be reduced a train of solitons (e.g., El et al [25]), conventionally assuming an indefinite flow domain along the shoreline. …”

Section: Introductionmentioning

confidence: 99%

Analytical Solution for the Pressure Oscillations Caused by Trains of Solitary Waves within Confined Coastal Aquifers

Pannone

2012

ISRN Civil Engineering

View full text Add to dashboard Cite

An exact analytical solution is proposed for the pressure oscillations within deep coastal aquifers under the action of tidal level time-variations attributable to train of solitary waves originating off-shore. The purpose of the study is to relate the characteristics of the response of the system to amplitude, steepness, and asymmetry of the soliciting waves, in order to assess its vulnerability to events like violent seaquakes and consequent tsunamis. The time needed by the forcing perturbations, approximated by consecutive triangular impulses, to attain their maximum is assumed to be always smaller than the aquifer diffusive time, in order to evaluate the consequences of the sudden raise of water level along the shoreline, typical of those quite extreme phenomena.

show abstract

Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory

Cited by 41 publications

References 26 publications

Solitary wave fission of a large disturbance in a viscous fluid conduit

Solitary wave fission of a large disturbance in a viscous fluid conduit

Critical control in transcritical shallow-water flow over two obstacles

Analytical Solution for the Pressure Oscillations Caused by Trains of Solitary Waves within Confined Coastal Aquifers

Contact Info

Product

Resources

About