Large-Amplitude Solitary Water Waves with Vorticity

Wheeler, Miles H.

doi:10.1137/120891460

Cited by 36 publications

(107 citation statements)

References 40 publications

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“…We remark that results similar to Theorems 1.2 and 1.4 can be obtained by continuing from a nontrivial free wave (0, u, v, η) which is non-degenerate in the sense that the corresponding linearized operator is invertible. Such a wave is guaranteed to exist when the Froude number F is sufficiently close to 1 [10,12,31].…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…Before giving an outline, we compare the current paper to [31], in which we constructed large-amplitude solitary waves with R ≡ 0 and where F was allowed to vary. While in that paper the local problem near F = 1 was one of the main sources of difficulty, in the current paper the local problem (with F > 1 fixed) is relatively straightforward.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…Hur [13] used these exponential asymptotics together with a moving planes argument to show symmetry and monotonicity under certain assumptions. In [31] we removed one of these assumptions, and used the resulting monotonicity in our continuation argument in an essential way. However, with a non-constant surface pressure, the free surface may not decay exponentially (see Appendix A.1 for an example).…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

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Solitary Water Waves of Large Amplitude Generated by Surface Pressure

Wheeler

2015

Arch Rational Mech Anal

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We consider exact nonlinear solitary water waves on a shear flow with an arbitrary distribution of vorticity. Ignoring surface tension, we impose a non-constant pressure on the free surface. Starting from a uniform shear flow with a flat free surface and a supercritical wave speed, we vary the surface pressure and use a continuation argument to construct a global connected set of symmetric solitary waves. This set includes waves of depression whose profiles increase monotonically from a central trough where the surface pressure is at its lowest, as well as waves of elevation whose profiles decrease monotonically from a central crest where the surface pressure is at its highest. There may also be two waves in this connected set with identical surface pressure, only one of which is a wave of depression.

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

Section: Statement Of the Main Resultsmentioning

confidence: 99%

Section: Statement Of the Main Resultsmentioning

confidence: 99%

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Solitary Water Waves of Large Amplitude Generated by Surface Pressure

Wheeler

2015

Arch Rational Mech Anal

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“…In the case of zero vorticity, the rigorous existence theory goes back to the constructions in [4,33] of small-amplitude waves and it includes singular bifurcation result in [2] of large-amplitude waves. Recently, these results have been extended to include an arbitrary distribution of vorticity in [35,42,80,81].…”

Section: Traveling Waves and The Bifurcation Conditionmentioning

confidence: 99%

Pressure Transfer Functions for Interfacial Fluids Problems

Chen¹,

Hur²,

Walsh³

2016

J. Math. Fluid Mech.

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We make a consistent derivation, from the governing equations, of the pressure transfer function in the smallamplitude Stokes wave regime and the hydrostatic approximation in the small-amplitude solitary water wave regime, in the presence of a background shear flow. The results agree with the well-known formulae in the zero vorticity case, but they incorporate the effects of vorticity through solutions to the Rayleigh equation. We extend the results to permit continuous density stratification and to internal waves between two constant-density fluids. Several examples are discussed.Mathematics Subject Classification. 35R35, 76B15, 76B47, 76B70.

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“…On the other hand, the argument in [16] is based on an implicit function theorem of Nash-Moser type. Largeamplitude solitary waves were constructed in [34]. …”

Section: Unidirectional Waves With Vorticitymentioning

confidence: 99%

Small-amplitude steady water waves with vorticity

Lokharu¹

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The problem of describing two-dimensional traveling water waves is considered. The water region is of finite depth and the interface between the region and the air is given by the graph of a function. We assume the flow to be incompressible and neglect the effects of surface tension. However we assume the flow to be rotational so that the vorticity distribution is a given function depending on the values of the stream function of the flow. The presence of vorticity increases the complexity of the problem and also leads to a wider class of solutions.First we study unidirectional waves with vorticity and verify the Benjamin-Lighthill conjecture for flows whose Bernoulli constant is close to the critical one. For this purpose it is shown that every wave, whose slope is bounded by a fixed constant, is either a Stokes or a solitary wave. It is proved that the whole set of these waves is uniquely parametrised (up to translation) by the flow force which varies between its values for the supercritical and subcritical shear flows of constant depth. We also study largeamplitude unidirectional waves for which we prove bounds for the free-surface profile and for Bernoulli's constant.Second, we consider small-amplitude waves with counter currents. Such flows admit layers, where the fluid flows in different directions. In this case we prove that the initial nonlinear free-boundary problem can be reduced to a finite-dimensional Hamiltonian system with a stable equilibrium point corresponding to a uniform stream. As an application of this result, we prove the existence of non-symmetric wave profiles. Furthermore, using a different method, we prove the existence of periodic waves with an arbitrary number of crests per period.i ii Populärvetenskaplig sammanfattningI denna avhandling betraktar vi problemet att beskriva tvådimensionella vandrande vattenvågor. Vattenregionen antas ha ändligt djup och ytan mellan området och luften beskrivs av grafen till en funktion. Vi antar att flödet är inkompressibelt and bortser från ytspänningseffekter. Vidare antar vi att flödet är inte är rotationsfritt, så att vorticiteten beskrivs av en given funktion som beror på värden av strömmningsfunktionen för flödet. Existensen av vorticitet ökar komplexiteten hos problemet och leder också till en större lösningsklass.Först studerar vi enkelriktade vågor med vorticitet och verifierar Benjamin-Lighthillkonjekturen för flöden vars Bernoullikonstant ligger nära den kritiska nivån. Av denna anledning visas att varje våg vars lutning är begränsad av en fix konstant är antingen en Stokesvåg eller en solitär våg. Hela mängden av dessa vågor kan på ett entydigt sätt parametriseras (upp till translationer) av flödeskraften som varierar mellan värden för superkritiska och subkritiska skjuvflöden på konstant djup. Vi studerar också enkelriktade vågor med hög amplitud där vi visar begränsningar för den fria ytprofilen och för Bernoullikonstanten.Sedan betraktar vi lågamplitudsvågor med motströmmar. Sådana flöden tillåter lager där vätskan flödar i olika...

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Large-Amplitude Solitary Water Waves with Vorticity

Cited by 36 publications

References 40 publications

Solitary Water Waves of Large Amplitude Generated by Surface Pressure

Solitary Water Waves of Large Amplitude Generated by Surface Pressure

Pressure Transfer Functions for Interfacial Fluids Problems

Small-amplitude steady water waves with vorticity

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