Solution of the Dirichlet boundary value problem for the Sine-Gordon equation

The sine-Gordon equation in light cone coordinates is solved when Dirichlet conditions on the L-shape boundaries of the strip [0,T]X[0,infinity) are prescribed in a class of functions that vanish (mod 2 pi) for large x at initial time. The method is based on the inverse spectral transform (IST) for the Schroedinger spectral problem on the semi-line solved as a Hilbert boundary value problem. Contrarily to what occurs when using the Zakharov-Shabat eigenvalue problem, the spectral transform is regular and in particular the discrete spectrum contains a finite number of eigenvalues (and no accumulation point).Comment: LaTex file, to appear in Physics Letters

Exactly solvable model of the 2D electrical double layer

We consider equilibrium statistical mechanics of a simplified model for the ideal conductor electrode in an interface contact with a classical semi-infinite electrolyte, modeled by the two-dimensional Coulomb gas of pointlike $\pm$ unit charges in the stability-against-collapse regime of reduced inverse temperatures $0\le \beta<2$. If there is a potential difference between the bulk interior of the electrolyte and the grounded interface, the electrolyte region close to the interface (known as the electrical double layer) carries some nonzero surface charge density. The model is mappable onto an integrable semi-infinite sine-Gordon theory with Dirichlet boundary conditions. The exact form-factor and boundary state information gained from the mapping provide asymptotic forms of the charge and number density profiles of electrolyte particles at large distances from the interface. The result for the asymptotic behavior of the induced electric potential, related to the charge density via the Poisson equation, confirms the validity of the concept of renormalized charge and the corresponding saturation hypothesis. It is documented on the non-perturbative result for the asymptotic density profile at a strictly nonzero $\beta$ that the Debye-H\"uckel $\beta\to 0$ limit is a delicate issue.Comment: 14 page

Pokrovsky-Talapov Model at finite temperature: a renormalization-group analysis

We calculate the finite-temperature shift of the critical wavevector $Q_{c}$ of the Pokrovsky-Talapov model using a renormalization-group analysis. Separating the Hamiltonian into a part that is renormalized and one that is not, we obtain the flow equations for the stiffness and an arbitrary potential. We then specialize to the case of a cosine potential, and compare our results to well-known results for the sine-Gordon model, to which our model reduces in the limit of vanishing driving wavevector Q=0. Our results may be applied to describe the commensurate-incommensurate phase transition in several physical systems and allow for a more realistic comparison with experiments, which are always carried out at a finite temperature

Structure of the broken phase of the sine-Gordon model using functional renormalization

We study in this paper the sine-Gordon model using functional Renormalization Group (fRG) at Local Potential Approximation (LPA) using different RG schemes. In $d=2$, using Wegner-Houghton RG we demonstrate that the location of the phase boundary is entirely driven by the relative position to the Coleman fixed point even for strongly coupled bare theories. We show the existence of a set of IR fixed points in the broken phase that are reached independently of the bare coupling. The bad convergence of the Fourier series in the broken phase is discussed and we demonstrate that these fixed-points can be found only using a global resolution of the effective potential. We then introduce the methodology for the use of Average action method where the regulator breaks periodicity and show that it provides the same conclusions for various regulators. The behavior of the model is then discussed in $d\ne 2$ and the absence of the previous fixed points is interpreted.Comment: 43 pages, 32 figures, accepted versio