On the theory of long waves

Lavrentʹev, Mikhail Alekseevich

doi:10.1007/bf00854079

Cited by 23 publications

(26 citation statements)

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“…The Euler equations along with the model system (1.1) admit traveling wave solutions, i.e. waves that propagate without change in shape or speed [3,5,22]. Solitary waves form a special class of traveling wave solutions of these systems.…”

Section: Introductionmentioning

confidence: 99%

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

2017

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Abstract. We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes a high-order finite-element method with smooth, periodic splines in space and explicit Runge-Kutta methods in time. Other forms of solutions such as cnoidal waves and dispersive shock waves are also considered. The differences between solutions of the Serre equations and the Euler equations are also studied.

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Section: Introductionmentioning

confidence: 99%

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

2017

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“…Lavrentiev [24] constructed a solitary wave as the limit of a sequence of Stokes waves of increasing period, while Friedrichs and Hyers [16] gave an existence proof based upon a series expansion and Beale [3] used a Nash-Moser implicit-function theorem. A global branch of large-amplitude solutions was obtained by Amick and Toland [1,2] using a formulation of the problem as an integral equation, and Plotnikov [26] used a variational formulation of the problem to demonstrate the nonuniqueness of large-amplitude solitary waves.…”

Section: Introductionmentioning

confidence: 99%

Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity

Groves

Wahlén

2008

Physica D: Nonlinear Phenomena

102

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“…In finite depth, Lavrantiev (1946) proved the existence of a solitary gravity wave as the limit of a periodic wave, when the wavelength tends to infinity. He also proved the non-existence of a solitary wave of depression.…”

Section: Discussionmentioning

confidence: 99%

Cnoidal-type surface waves in deep water

Clamond

2003

J. Fluid Mech.

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Two-dimensional potential flows due to progressive surface waves in deep water are considered. For periodic waves, only gravity is included in the dynamic boundary condition, but both gravity and surface tension are taken into account for solitary waves. The validity of the steady first-order cnoidal wave approximation, i.e. the periodic solution of KdV, is extended to infinite depth by renormalizations. This renormalized cnoidal wave (RCW) solution is expressed as a Fourier-Padé approximation. It is analytically simpler and more accurate than fifth-order Stokes approximations. It is also capable of describing the recently discovered sharp-crested wave. A sharp-crested wave is obtained when the fluid velocity at the crest is larger than the phase speed. When the wavelength is infinite, RCW yields an algebraic solitary wave. Depending on the surface tension, the solitary wave involves one or two interfaces: a wave of depression; a wave of depression with a pocket of air; a wave of elevation with a pocket of air. Solitary waves are found for all values of the surface tension. However, this does not necessarily mean that these waves are solutions of the exact equations. Moreover, RCW approximate solitary waves always present a dipole singularity. It is also shown that a cnoidal wave in deep water can be rewritten as a periodic distribution of dipoles, each dipole representing an algebraic solitary wave. This provides a new paradigm for descriptions of water wave phenomena.

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On the theory of long waves

Cited by 23 publications

References 0 publications

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity

Cnoidal-type surface waves in deep water

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