Smallamplitude, traveling, space periodic solutions –called Stokes waves– of the gravitywater waves equations with infinite depth are linearly unstable with respect to longwaveperturbations, as predicted by Benjamin and Feir in 1967. We completely describe thebehavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave,as the Floquet exponent is turned on. We prove in particular the conjecture that a pair ofnonpurely imaginary eigenvalues depicts a closed figure eight, parameterized by the Floquetexponent, in full agreement with numerical simulations. Our new spectral approach to theBenjaminFeir instability phenomenon uses Kato’s theory of similarity transformation toreduce the problem to determine the eigenvalues of a 4×4 complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory we blockdiagonalize such matrix into a pair of 2 × 2 Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.
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Full description of BenjaminFeir instability of Stokes waves in deep water
Research Group:
Paolo Ventura
Schedule:
Thursday, September 23, 2021  16:00 to 17:00
Location:
A133
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Hybrid: in presence and online
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Upcoming events

Simone Carano
The relaxed area with respect to the strict convergence in BV
Friday, October 22, 2021  14:00 to 15:30

Qingtang Su
Nonlinear Modulational Instability of the Stokes Waves in 2d Full Water Waves
Tuesday, October 26, 2021  17:00 to 18:00
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