Renormalization of isoenergetically degenerate Hamiltonian flows, and instability of solitons in shear hydrodynamic flows
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Part I of this Thesis presents a study of the renormalization group transformation acting on an appropriate space of Hamiltonian functions in two angle and two action variables. In particular, we study the existence of real invariant tori, on which the flow is conjugate to a rotation with the rotation number equal to the golden mean (ω-tori). We demonstrate that the stable manifold of the renormalization operator at the “simple” fixed point contains isoenergetically degenerate Hamiltonians possessing shearless ω-tori. We also show that one-parameter families of Hamiltonians transverse to the stable manifold undergo a bifurcation: for a certain range of the parameter values the members of these families posses two distinct ω-tori, the members of such families lying on the stable manifold posses one shearless ω-torus, while the members corresponding to other parameter values do not posses any. We also present some numerical evidence for universality associated with the breakup of shearless invariant tori, and compute the relevant critical renormalization and scaling eigenvalues. Part II of the Thesis presents a stability analysis of plane solitonsin hydrodynamic shear flows obeying a (2+1) analogue of the Benjamin–Ono equation. The instability region and the short-wave instability threshold for plane solitons are found numerically. We also determine the dependence of the growth rate on the propagation angle in the longwave limit and demonstrate the existence of a critical angle which separates two types of behaviour of the growth rate.