Criticality in shallow water hydrodynamics,

 degenerate relative equilibria and

  the bifurcation of solitary waves

 Department of Mathematics
 University of Surrey


Secondary criticality of water waves - steady flow
  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.
  T. J. Bridges & N. M. Donaldson. Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves, J. Fluid Mech. 565 381-417 (2009)
JFM website  

Degenerate periodic orbits and homoclinic torus bifurcation
  A one-parameter family of periodic orbits with frequency ω and energy E of an autonomous Hamiltonian system is degenerate when E'(ω)=0. In this paper, new features of the nonlinear bifurcation near this degeneracy are identified. A new normal form is found where the coefficient of the nonlinear term is determined by the curvature of the energy-frequency map. An important property of the bifurcating "homoclinic torus" is the homoclinic angle and a new asymptotic formula for it is derived. The theory is constructive, and so is useful for physical applications and in numerics.
  T. J. Bridges & N. M. Donaldson. Degenerate periodic orbits and homoclinic bifurcation, Phys. Rev. Lett. 96 104301 (2005)   PRL website    

Krein signature in the elliptic-hyperbolic transition of periodic orbits
  In the linearization about periodic orbits, Krein signature is a symplectic invariant of Floquet multipliers on the unit circle. Of interest here is the role of Krein signature when two Floquet multipliers meet at +1 and undergo a elliptic to hyperbolic transition. It is shown, using the normal form at the transition, that the symplectic invariant at the transition point is determined by the Krein signature of the colliding elliptic Floquet multipliers.
  T. J. Bridges. Krein signature in the elliptic-hyperbolic transition of periodic orbits, Internal Report (2011)    
  R. S. MacKay. Three topics in Hamiltonian dynamics, Dynamical Systems and Chaos, Vol 2 (Hachioji) 34-43, World Scientific Publishers (1984)

Criticality for two-layer flows and internal solitary waves
  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 specif ic, the case of two layers with a rigid lid is considered, and application of the theory to other multi-layer flows is sketched.
  T. J. Bridges & N. M. Donaldson. Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves. Phys. Fluids 19 072111 (2007)    PoF website    
Degenerate relative equilibria and homoclinic bifurcation
  A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension (2n+2) with an n-dimensional abelian symmetry group. Examples for n=1,2,3 are presented to illustrate application of the theory.
  T. J. Bridges. Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation, J. Diff. Eqns. 244 1629-1674 (2008)     JDE website    
  T. J. Bridges. The intrinsic second derivative, Internal report (2011)

Secondary criticality of water waves - unsteady flow
  The theory for criticality presented in Part 1 is extended to the unsteady problem, and a new formulation of the Benjamin-Feir instability for Stokes waves in finite depth coupled to a mean flow, which takes the criticality matrix as an organizing centre, is presented. The generation of unsteady dark solitary waves at points of stability changes and their connection with the steady dark solitary waves of Part 1 are also discussed.
  T. J. Bridges & N. M. Donaldson. Secondary criticality of water waves. Part 2. Unsteadiness and the Benjamin-Feir instability from the viewpoint of hydraulics, J. Fluid Mech. 565 419-439 (2006)     JFM website  

Criticality manifolds for two-layer flow with a free surface
  The role of criticality manifolds is explored both for the classification of all uniform flows and for the bifurcation of solitary waves, in the context of two fluid layers of differing density with an upper free surface. While the weakly nonlinear bifurcation of solitary waves in this context is well known, it is shown herein that the critical nonlinear behaviour of the bifurcating solitary waves and generalized solitary waves is determined by the geometry of the criticality manifolds. By parametrizing all uniform flows, new physical results are obtained on the implication of a velocity difference between the two layers on the bifurcating solitary waves.
  T. J. Bridges & N. M. Donaldson. Criticality manifolds and their role in the generation of solitary waves for two-layer flow with a free surface, European J. Mech. B/Fluids 28 117-126 (2009)   EJMBF website    

Homoclinic bifurcation of invariant tori
  A family of smooth invariant tori of a Hamiltonian system can be parameterized by the values of the actions or the frequencies. These parameterizations are related by the action-frequency map. The purpose of this paper is to show that when the action-frequency map is degenerate, it signals a homoclinic bifurcation. Remarkably, the nonlinear properties of this homoclinic bifurcation to invariant tori are determined by the curvature of the action-frequency map. A homoclinic angle is also generated which is analogous to a Hannay-Berry phase shift. The theory is constructive and so can usefully be combined with computation. Some implications for quantization, and the generation of solitary waves are also discussed.
  T. J. Bridges. Degeneracy of the action-frequency map: A mechanism for homoclinic bifurcation of invariant tori, Phys. Rev. E 79 088803 (2009)   PRE website    

Generating steady dark solitary waves
  Various classes of steady and unsteady dark solitary waves (DSWs) are known to exist in modulation equations for water waves in finite depth. However, there is a class of steady DSWS of the full water-wave problem which are missed by the classical modulation equations such as the Hasimoto-Ono, Benney-Roskes, and Davey-Stewartson. These steady DSWs, recently discovered by Bridges and Donaldson, are pervasive in finite depth, arise through secondary criticality of Stokes gravity waves, and are synchronized with the Stokes wave. In this paper, the role of DSWs in modulation equations for water waves is reappraised. The intrinsic unsteady nature of existing modulation equations filters out some interesting solutions. On the other hand, the geometry of DSWs in modulation equations is very similar to the full water wave problem and these geometrical properties are developed. A model equation is proposed which illustrates the general nature of the emergence of steady DSWs due to wave-generated mean flow coupled to a periodic wave. Although the existing modulation equations are intrinsically unsteady, it is shown that there are also important shortcomings when one wants to use them for stability analysis of DSWs.
  T. J. Bridges. Steady dark solitary waves emerging from wave-generated meanflow: The role of modulation equations, Chaos 15 037113 (2005)   Chaos website    

  Department of Mathematics     University of Surrey