Degenerate Periodic Orbits and Homoclinic Torus Bifurcation

Bridges, Thomas J.; Donaldson, Neil M.

doi:10.1103/physrevlett.95.104301

Cited by 14 publications

(21 citation statements)

References 13 publications

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“…The formation of these waves is mathematically justified by the presence of a resonance in the dynamical system corresponding to the traveling waves [29]. The existence of generalised solitary waves can be deduced from the existence of the phase shift when the action of the stationary problem changes sign [6]. One can also use the existence of Smale's horseshoe dynamics on the zero energy set [7].…”

Section: Introductionmentioning

confidence: 99%

A plethora of generalised solitary gravity–capillary water waves

Clamond¹,

Dutykh

Durán

2015

J. Fluid Mech.

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The present study describes, first, an efficient algorithm for computing solutions in terms of capillary–gravity solitary waves of the irrotational Euler equations with a free surface and, second, provides numerical evidences of the existence of an infinite number of generalised solitary waves (solitary waves with undamped oscillatory wings). Using conformal mapping, the unknown fluid domain, which is to be determined, is mapped into a uniform strip of the complex plane. In the transformed domain, a Babenko-like equation is then derived and solved numerically.

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Section: Introductionmentioning

confidence: 99%

A plethora of generalised solitary gravity–capillary water waves

Clamond¹,

Dutykh

Durán

2015

J. Fluid Mech.

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“…See §3 of [9] for further elaboration of this example. The details of the theory justifying these results can be found in [8,9,7]. The purpose of this paper is to apply these ideas to the case of two-layer flows.…”

Section: Introductionmentioning

confidence: 93%

Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves

Bridges

Donaldson

2007

Physics of Fluids

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A geometric view of criticality for two-layer flows is presented. Uniform flows are classified by diagrams in the momentum-massflux space for fixed Bernoulli energy, and cuspoidal curves on these diagrams correspond to critical uniform flows. Restriction of these surfaces to critical flow leads to new sub-surfaces in energy-massflux space. While the connection between criticality and the generation of solitary waves is well known, we find that the nonlinear properties of these bifurcating solitary waves are also determined by the properties of the criticality surfaces. To be specific, the case of two layers with a rigid lid is considered, and application of the theory to other multi-layer flows is sketched.

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“…This system is universal in the sense that it arises near points of degeneracy for any two-parameter family of relative equilibria (Bridges & Donaldson 2005;Bridges 2006b). The sign s 1 is a property of the Jordan normal form structure and s 1 = −1 in this case.…”

Section: Critical Uniform Flows Degenerate Relative Equilibria and Smentioning

confidence: 99%

“…This relative equilibrium is degenerate precisely when the uniform flow is critical. There is a universal nonlinear normal form near degenerate relative equilibria (Bridges & Donaldson 2005;Bridges 2006b) and this normal form predicts the bifurcation of solitary waves.…”

Section: Critical Uniform Flows Degenerate Relative Equilibria and Smentioning

confidence: 99%

Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves

Bridges

Donaldson

2006

J. Fluid Mech.

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A generalization of criticality -called secondary criticality -is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary waves -from Stokes gravity waves in shallow water -is pervasive, even at low amplitude. The theory proceeds by generalizing concepts from hydraulics: three additional functionals are introduced which represent non-uniformity and extend the familiar mass flux, total head and flow force, the most important of which is the wave action flux. The theory works because the hydraulic quantities can be related to the governing equations in a precise way using the multi-symplectic Hamiltonian formulation of water waves. In this setting, uniform flows and Stokes waves coupled to a uniform flow are relative equilibria which have an attendant geometric theory using symmetry and conservation laws. A flow is then 'critical' if the relative equilibrium representation is degenerate. By characterizing successively non-uniform flows and unsteady flows as relative equilibria, a generalization of criticality is immediate. Recent results on the local nonlinear behaviour near a degenerate relative equilibrium are used to predict all the qualitative properties of the bifurcating dark solitary waves, including the phase shift. The theory of secondary criticality provides new insight into unsteady waves in shallow water as well. A new interpretation of the Benjamin-Feir instability from the viewpoint of hydraulics, and the connection with the creation of unsteady dark solitary waves, is given in Part 2.

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Degenerate Periodic Orbits and Homoclinic Torus Bifurcation

Cited by 14 publications

References 13 publications

A plethora of generalised solitary gravity–capillary water waves

A plethora of generalised solitary gravity–capillary water waves

Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves

Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves

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