Extensions of the generalized Weierstrass representation to generic surfaces in 4-D Euclidean and pseudo-Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey-Stewartson hierarchy. Geometrically, these deformations are characterized by the invariance of an infinite set of functionals over surface. The Willmore functional (the total squared mean curvature) is the simplest of them. Various particular classes of surfaces and their integrable deformations are considered.

Inspired by the recent developments in the study of the thermodynamics of van der Waals fluids via the theory of nonlinear conservation laws and the description of phase transitions in terms of classical (dissipative) shock waves, we propose a novel approach to the construction of multi-parameter generalisations of the van der Waals model. The theory of integrable nonlinear conservation laws still represents the inspiring framework. Starting from a macroscopic approach, a four parameter family of integrable extended van der Waals models is indeed constructed in such a way that the equation of state is a solution to an integrable nonlinear conservation law linearisable by a Cole-Hopf transformation. This family is further specified by the request that, in regime of high temperature, far from the critical region, the extended model reproduces asymptotically the standard van der Waals equation of state. We provide a detailed comparison of our extended model with two notable empirical models such as Peng-Robinson and Soave's modification of the Redlich-Kwong equations of state. We show that our extended van der Waals equation of state is compatible with both empirical models for a suitable choice of the free parameters and can be viewed as a master interpolating equation. The present approach also suggests that further generalisations can be obtained by including the class of dispersive and viscous-dispersive nonlinear conservation laws and could lead to a new type of thermodynamic phase transitions associated to nonclassical and dispersive shock waves.

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We develop a theoretical frame for the study of classical and quantum gravitational waves based on the properties of a nonlinear ordinary differential equation for a function σ(η) of the conformal time η, called the auxiliary field equation. At the classical level, σ(η) can be expressed by means of two independent solutions of the "master equation" to which the perturbed Einstein equations for the gravitational waves can be reduced.At the quantum level, all the significant physical quantities can be formulated using Bogolubov transformations and the operator quadratic Hamiltonian corresponding to the classical version of a damped parametrically excited oscillator where the varying mass is replaced by the square cosmological scale factor a 2 (η). A quantum approach to the generation of gravitational waves is proposed on the grounds of the previous η−dependent Hamiltonian. An estimate in terms of σ(η) and a(η) of the destruction of quantum coherence due to the gravitational evolution and an exact expression for the phase of a gravitational wave corresponding to any value of η are also obtained. We conclude by discussing a few applications to quasi-de Sitter and standard de Sitter scenarios.

We consider the conformal A n Toda theory in AdS 2 . Due to the bulk full Virasoro symmetry, this system provides an instance of a non-gravitational AdS 2 /CFT 1 correspondence where the 1d boundary theory enjoys enhanced " 1 2 -Virasoro" symmetry. General boundary correlators are expected to be captured by the restriction of chiral correlators in a suitable WA n Virasoro extension. At next-to-leading order in weak coupling expansion they have been conjectured to match the subleading terms in the large central charge expansion of the dual WA n correlators. We explicitly test this conjecture on the boundary four point functions of the Toda scalar fields dual to WA n generators with next-to-minimal spin 3 and 4. Our analysis is valid in the generic rank case and extends previous results for specific rank-2 Toda theories. On the AdS side, the extension is straightforward and requires the computation of a finite set of tree Witten diagrams. This is due to simple rank dependence and selection rules of cubic and quartic couplings. On the boundary, we exploit crossing symmetry and specific meromorphic properties of the W-algebra correlators at large central charge. We present the required 4-point functions in closed form for any rank and verify the bulk-boundary correspondence in full details.