Andre Neveu

A recent extension of a variationally optimized perturbation, combined with renormalization group properties in a straightforward way, can provide approximations to nonperturbative quantities such as the chiral symmetry breaking order parameters typically. We apply this to evaluate, up to third order in this modified perturbation, the ratio F$\pi${}/$\Lambda${}MS\ifmmode\bar\else\textasciimacron\fi{}, where F$\pi${} is the pion decay constant and $\Lambda${}MS\ifmmode\bar\else\textasciimacron\fi{} the basic QCD scale in the MS\ifmmode\bar\else\textasciimacron\fi{} scheme. Using experimental F$\pi${} input value we obtain $\Lambda${}MS\ifmmode\bar\else\textasciimacron\fi{}nf=2$\simeq${}255-15+40\,\,MeV, where quoted errors are estimates of theoretical uncertainties of the method. This compares reasonably well with some recent lattice simulation results. We briefly discuss prospects (and obstacles) for extrapolation to $\alpha${}S($\mu${}) at perturbative $\mu${} values.

We introduce an extension of a variationally optimized perturbation method, by combining it with renormalization group properties in a straightforward (perturbative) form. This leads to a very transparent and efficient procedure, with a clear improvement of the nonperturbative results with respect to previous similar variational approaches. This is illustrated here by deriving optimized results for the mass gap of the O(2N) Gross-Neveu model, compared with the exactly known results for arbitrary N. At large N, the exact result is reproduced already at the very first order of the modified perturbation using this procedure. For arbitrary values of N, using the original perturbative information only known at two-loop order, we obtain a controllable percent accuracy or less, for any N value, as compared with the exactly known result for the mass gap from the thermodynamical Bethe Ansatz. The procedure is very general and can be extended straightforwardly to any renormalizable Lagrangian model, being systematically improvable provided that a knowledge of enough perturbative orders of the relevant quantities is available.

The dynamics of a nonlinear and dispersive long surface capillary-gravity wave model equation is studied analytically in its short-wave limit. We exhibit its Lax pair and some non-trivial conserved quantities. Through a change of functions, an unexpected connection between this classical surface water-wave model and the sine-Gordon (or sinh-Gordon) equation is established. Numerical and analytical studies show that in spite of integrability their solutions can develop singularities and multivaluedness in finite time. These features can be traced to the fact that the surface tension term in the energy involves second-order derivatives. It would be interesting to see in an experiment whether such singularities actually appear, for which surface tension would be specifically responsible.

A new nonlinear equation governing asymptotic dynamics of short surface waves is derived by using a short-wave perturbative expansion in an appropriate reduction of the Euler equations. This reduction corresponds to a Green-Naghdi-type equation with a cinematic discontinuity in the surface. The physical system under consideration is an ideal fluid (inviscid, incompressible and without surface tension) in which takes place a steady surface motion. An ideal surface wind on a lake which produces surface flow is a physical environment conducive to the above-mentioned phenomenon. The equation obtained admits peakon solutions with amplitude, velocity and width in interrelation.