Traveling Gravity Water Waves with Critical Layers

Aasen, Ailo; Varholm, Kristoffer

doi:10.1007/s00021-017-0316-7

Cited by 16 publications

(41 citation statements)

References 35 publications

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“…They also exist in the presence of vorticity [25], even without capillarity [9,14]. In that case, one may even construct arbitrary large kernels [1,10], and corresponding multi-dimensional solution sets [23].…”

Section: Introductionmentioning

confidence: 99%

On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

et al. 2019

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We study the bifurcation of periodic travelling waves of the capillary-gravity Whitham equation. This is a nonlinear pseudodifferential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillarygravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov-Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and therefore positive and convex; the threshold value for this to happen turns out to be T = 4 π 2 , not the critical Bond number 1 3 ), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions; and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in L 2 (S) (but not in L ∞ ) towards one of the two constant solution curves. 4 π 2 , we are able in Theorem 2.6 to show that the kernel is completely monotone, a delicate structural property shared by the kernel for the linear dispersion in the pure gravity case (not its inverse). Moreover, we

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Section: Introductionmentioning

confidence: 99%

On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

et al. 2019

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“…We will be interested in the analysis of small-amplitude waves over streams with counter-currents. A similar setup appeared in a several recent papers including [28] and [22] on the existence of Stokes waves with critical layers, [8] and [1] concerning the existence of bimodal waves, and [9], [23] constructing trimodal and N-modal waves respectively. In all mentioned papers the authors prove existence using a local bifurcation argument of Crandall and Rabinowitz (see [7]) and its generalizations to the case of multi-dimensional kernels.…”

Section: Introductionmentioning

confidence: 85%

“…Using this ansatz in (2.6)-(2.8), we find after taking the limit → 0 the following equations for (Φ (1) , η (1) ):…”

Section: Dispersion Equationmentioning

confidence: 99%

Small-amplitude steady water waves with critical layers: Non-symmetric waves

Kozlov

Lokharu

2019

Journal of Differential Equations

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The problem for two-dimensional steady water waves with vorticity is considered. Using methods of spatial dynamics, we reduce the problem to a finite dimensional Hamiltonian system. The reduced system describes all smallamplitude solutions of the problem and, as an application, we give a proof of the existence of non-symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one.

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“…The first approximation for such waves is given by a combination of two cos-functions with different periods. In [1] the authors, using different bifurcation parameters, construct bimodal waves that are different from those obtained in [11] and [12]. Later, in 2015, Ehrnström and Wahlén [13] established existence of trimodal waves.…”

Section: Steady Waves Over Streams With Counter-currentsmentioning

confidence: 99%

“…In the papers [11], [1], [13] the authors construct symmetric small-amplitude and periodic waves for which the linear approximation is given by a combinations of two or three co-sinus functions with different wavelengths. In [11] they state a natural question: are there waves for which the linear approximation is given by any number of basic modes?…”

Section: N-modal Wavesmentioning

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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Traveling Gravity Water Waves with Critical Layers

Cited by 16 publications

References 35 publications

On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

Small-amplitude steady water waves with critical layers: Non-symmetric waves

Small-amplitude steady water waves with vorticity

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