Non-interacting gravity waves on the surface of a deep fluid

We study the interaction of gravity waves on the surface of an infinitely deep ideal fluid. Starting from Zakharov's variational formulation for water waves we derive an expansion of the Hamiltonian to an arbitrary order, in a manner that avoids a laborious series reversion associated with expressing the velocity potential in terms of its value at the free surface. The expansion kernels are shown to satisfy a recursion relation enabling us to draw some conclusions about higher-order wave-wave interaction amplitudes, without referring to the explicit forms of the individual lower-order kernels. In particular, we show that unidirectional waves propagating in a two-dimensional flow do not interact nonlinearly provided they fulfill the energy-momentum conservation law. Switching from the physical variables to the so-called normal variables we explain the vanishing of the amplitudes of fourth- and certain fifth-order non-generic resonant interactions reported earlier and outline a procedure for finding the one-dimensional wave vector configurations for which the higher order interaction amplitudes become zero on the resonant hypersurfaces.Comment: 13 page

Integrable Floquet dynamics

We discuss several classes of integrable Floquet systems, i.e. systems which do not exhibit chaotic behavior even under a time dependent perturbation. The first class is associated with finite-dimensional Lie groups and infinite-dimensional generalization thereof. The second class is related to the row transfer matrices of the 2D statistical mechanics models. The third class of models, called here "boost models", is constructed as a periodic interchange of two Hamiltonians - one is the integrable lattice model Hamiltonian, while the second is the boost operator. The latter for known cases coincides with the entanglement Hamiltonian and is closely related to the corner transfer matrix of the corresponding 2D statistical models. We present several explicit examples. As an interesting application of the boost models we discuss a possibility of generating periodically oscillating states with the period different from that of the driving field. In particular, one can realize an oscillating state by performing a static quench to a boost operator. We term this state a "Quantum Boost Clock". All analyzed setups can be readily realized experimentally, for example in cod atoms.Comment: 18 pages, 2 figures; revised version. Submission to SciPos

Semiclassical Quantisation of Finite-Gap Strings

We perform a first principle semiclassical quantisation of the general finite-gap solution to the equations of a string moving on R x S^3. The derivation is only formal as we do not regularise divergent sums over stability angles. Moreover, with regards to the AdS/CFT correspondence the result is incomplete as the fluctuations orthogonal to this subspace in AdS_5 x S^5 are not taken into account. Nevertheless, the calculation serves the purpose of understanding how the moduli of the algebraic curve gets quantised semiclassically, purely from the point of view of finite-gap integration and with no input from the gauge theory side. Our result is expressed in a very compact and simple formula which encodes the infinite sum over stability angles in a succinct way and reproduces exactly what one expects from knowledge of the dual gauge theory. Namely, at tree level the filling fractions of the algebraic curve get quantised in large integer multiples of hbar = 1/lambda^{1/2}. At 1-loop order the filling fractions receive Maslov index corrections of hbar/2 and all the singular points of the spectral curve become filled with small half-integer multiples of hbar. For the subsector in question this is in agreement with the previously obtained results for the semiclassical energy spectrum of the string using the method proposed in hep-th/0703191. Along the way we derive the complete hierarchy of commuting flows for the string in the R x S^3 subsector. Moreover, we also derive a very general and simple formula for the stability angles around a generic finite-gap solution. We also stress the issue of quantum operator orderings since this problem already crops up at 1-loop in the form of the subprincipal symbol.Comment: 53 pages, 22 figures; some significant typos corrected, references adde

Integrable approximation of regular regions with a nonlinear resonance chain

Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the non-integrable system. In the second step a sequence of canonical transformations is applied to the integrable approximation to match the shape of regular tori. We demonstrate the method for the generic standard map at various parameters.Comment: 10 pages, 8 figure