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Standing Waves on an Infinitely Deep Perfect Fluid Under Gravity

- G. Iooss, P. Plotnikov, J. Toland
- Mathematics
- 11 June 2005

The existence of two-dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established. When formulated as a second-order equation for a real-valued function… Expand

Nash-Moser Theory for Standing Water Waves

- P. Plotnikov, J. Toland
- Mathematics
- 1 August 2001

Abstract We consider a perfect fluid in periodic motion between parallel vertical walls, above a horizontal bottom and beneath a free boundary at constant atmospheric pressure. Gravity acts… Expand

Modelling nonlinear hydroelastic waves

- P. Plotnikov, J. Toland
- Physics, Medicine
- Philosophical Transactions of the Royal Society A…
- 28 July 2011

This paper uses the special Cosserat theory of hyperelastic shells satisfying Kirchoff’s hypothesis and irrotational flow theory to model the interaction between a heavy thin elastic sheet and an… Expand

Convexity of Stokes Waves of Extreme Form

- P. Plotnikov, J. Toland
- Mathematics
- 1 March 2004

Existence is established of a piecewise-convex, periodic, planar curve S below which is defined a harmonic function which simultaneously satisfies prescribed Dirichlet and Neumann boundary conditions… Expand

NONUNIQUENESS OF SOLUTIONS OF THE PROBLEM OF SOLITARY WAVES AND BIFURCATION OF CRITICAL POINTS OF SMOOTH FUNCTIONALS

- P. Plotnikov
- Mathematics
- 30 April 1992

The problem of solitary waves on the surface of an ideal fluid is considered. By means of a variational principle it is shown that for an infinite set of values of the Froude number this problem has… Expand

Small Divisor Problem in the Theory of Three-dimensional Water Gravity Waves

- G. Iooss, P. Plotnikov
- Mathematics, Physics
- 23 January 2006

We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity g and resulting from the nonlinear interaction of two simply periodic… Expand

Boundary value problems for transport equations

- P. Plotnikov, J. Sokolowski
- Physics
- 2012

The first order scalar differential equation \( {\rm \mathbf{u}} . \nabla \varphi + b\varphi = f ,\) which is called the transport equation, is one of the basic equations of mathematical physics. It… Expand

A Proof of the Stokes Conjecture in the Theory of Surface Waves

- P. Plotnikov
- Mathematics
- 1 February 2002

This article gives a proof of the famous Stokes conjecture that a gravity wave of greatest height on water has a corner with contained angle 2π/3 at its singular point.

Inhomogeneous boundary value problems for compressible Navier–Stokes and transport equations☆

- P. Plotnikov, E. V. Ruban, J. Sokolowski
- Mathematics
- 1 August 2009

Abstract In the paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. The well-posedness for inhomogeneous boundary value… Expand

Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis

- P. Plotnikov, E. V. Ruban, J. Sokolowski
- Mathematics, Computer Science
- SIAM J. Math. Anal.
- 22 October 2008

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