Nonexistence of Subcritical Solitary Waves

Kozlov, Vladimir; Lokharu, Evgeniy; Wheeler, Miles H.

doi:10.1007/s00205-021-01659-y

Cited by 13 publications

(17 citation statements)

References 45 publications

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“…As it was recently proved in [13] that all solitary waves correspond to points on a part of lower boundary S = S − (r) (dashed curve in Figure 1). It is believed (but unproved) that the dashed curve ends with the point that determines the highest solitary wave.…”

Section: Introductionsupporting

confidence: 51%

Improved bound in the Benjamin and Lighthill conjecture

Lokharu

2020

Preprint

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The classical Benjamin and Lighthill conjecture about steady water waves states that the non-dimensional flow force constant of a solution is bounded by the corresponding constants of the supercritical and subcritical uniform streams respectively. These inequalities determine a parameter region that covers all steady motions. In fact not all points of the region determine a steady wave. In this note we prove a new and explicit lower bound for the flow force constant, which is asymptotically sharp in a certain sense. In particular, this recovers the well known inequality F < 2 for the Froude number, while significantly reducing the parameter region supporting steady waves.

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Section: Introductionsupporting

confidence: 51%

Improved bound in the Benjamin and Lighthill conjecture

Lokharu

2020

Preprint

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“…In particular, (nontrivial) solitary waves must be supercritical in that F > F cr . This fact has recently been proved for the case of homogeneous density by Kozlov, Lokharu, and Wheeler [KLW20]. A rigorous definition of F cr for the present system is given in Section 3.…”

Section: Introductionmentioning

confidence: 60%

Large-amplitude solitary waves in two-layer density stratified water

Sinambela¹

2020

Preprint

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We present a large-amplitude existence theory for two-dimensional solitary waves propagating through a two layer body of water. The domain of the fluid is bounded below by an impermeable flat ocean floor and above by a free boundary at constant pressure. For any piecewise smooth upstream density distribution and laminar background current, we construct a global curve of solutions. This curve bifurcates from the background current and, following along the curve, we find waves that are arbitrarily close to having horizontal stagnation points.The small-amplitude waves are constructed using a center manifold reduction technique. The large-amplitude theory is obtained through analytical global bifurcation together with refined qualitative properties of the waves.

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“…The cusp in the figure consists of two curves representing parallel flows. Subcritical flows on the upper curve give rise to bifurcations of small-amplitude Stokes waves, while the lower curve represents supercritical streams, supporting solitary waves; see [30]. It was recently proved in [35] that all steady waves with vorticity correspond to points inside the cuspidal region (the Benjamin and Lighthill conjecture).…”

Section: Statements Of Main Resultsmentioning

confidence: 91%

Global bifurcation and highest waves on water of finite depth

Kozlov,

Lokharu

2020

Preprint

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We consider the two-dimensional problem for steady water waves on finite depth with vorticity. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching either a solitary wave, the highest solitary wave or the highest Stokes wave. In contrast to previous studies we fix the Bernoulli constant and consider the wavelength as a bifurcation parameter, which guarantees that the limiting wave has a finite depth. In fact, this is the first rigorous proof of the existence of extreme Stokes waves on water of finite depth. Beside the existence of highest waves we provide a new result about the regularity of Stokes waves of arbitrary amplitude (including extreme waves).

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Nonexistence of Subcritical Solitary Waves

Cited by 13 publications

References 45 publications

Improved bound in the Benjamin and Lighthill conjecture

Improved bound in the Benjamin and Lighthill conjecture

Large-amplitude solitary waves in two-layer density stratified water

Global bifurcation and highest waves on water of finite depth

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