The Unified Method: I Non-Linearizable Problems on the Half-Line

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions.Comment: 39 page

The European Court of Human Rights and minority religions: messages generated and messages received

This contribution introduces a collection of studies focused on engagements of religious minorities with the European Court of Human Rights (ECtHR). Setting out first the global importance of the ECtHR as a standard setter in the protection of the rights of religious minorities, the text goes on to introduce the ten contributions that together make up the present special issue on the European Court of Human Rights and Religious Minorities. Beyond briefly summar- ising the contexts of the special issue, this contribution indicates that the first part of the special issue entails critical assessments of some of the Court’s case law dealing with religious minority claims (exploring on their clarity and consistency – or lack thereof – and controversiality), and that the second part offers insight into the grassroots level impact of the Court’s case law on religious minority claims. It explains how each of these contributions deepens our understanding of the ECtHR in its approach to and impact on religious minorities. And it introduces the fact that, rather uniquely, this collection of texts offers a rare vantage point on the ‘circle of life’ of the Court’s case law on religious minorities

Integrable nonlinear equations on a circle

The concept of integrable boundary value problems for soliton equations on $\mathbb{R}$ and $\mathbb{R}_+$ is extended to bounded regions enclosed by smooth curves. Classes of integrable boundary conditions on a circle for the Toda lattice and its reductions are found.Comment: 23 page

Linearizability of the Perturbed Burgers Equation

We show in this letter that the perturbed Burgers equation $u_t = 2uu_x + u_{xx} + \epsilon ( 3 \alpha_1 u^2 u_x + 3\alpha_2 uu_{xx} + 3\alpha_3 u_x^2 + \alpha_4 u_{xxx} )$ is equivalent, through a near-identity transformation and up to order \epsilon, to a linearizable equation if the condition $3\alpha_1 - 3\alpha_3 - 3/2 \alpha_2 + 3/2 \alpha_4 = 0$ is satisfied. In the case this condition is not fulfilled, a normal form for the equation under consideration is given. Then, to illustrate our results, we make a linearizability analysis of the equations governing the dynamics of a one-dimensional gas.Comment: 10 pages, RevTeX, no figure

Ablowitz-Ladik system with discrete potential. I. Extended resolvent

Ablowitz-Ladik linear system with range of potential equal to {0,1} is considered. The extended resolvent operator of this system is constructed and the singularities of this operator are analyzed in detail.Comment: To be published in Theor. Math. Phy

The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation

We establish the existence of a real solution y(x,T) with no poles on the real line of the following fourth order analogue of the Painleve I equation, x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure

Dispersive wave runup on non-uniform shores

Historically the finite volume methods have been developed for the numerical integration of conservation laws. In this study we present some recent results on the application of such schemes to dispersive PDEs. Namely, we solve numerically a representative of Boussinesq type equations in view of important applications to the coastal hydrodynamics. Numerical results of the runup of a moderate wave onto a non-uniform beach are presented along with great lines of the employed numerical method (see D. Dutykh et al. (2011) for more details).Comment: 8 pages, 6 figures, 18 references. This preprint is submitted to FVCA6 conference proceedings. Other author papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Asymptotics for a special solution to the second member of the Painleve I hierarchy

We study the asymptotic behavior of a special smooth solution y(x,t) to the second member of the Painleve I hierarchy. This solution arises in random matrix theory and in the study of Hamiltonian perturbations of hyperbolic equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is known and relatively simple, but it turns out to be more subtle when x and t tend to infinity simultaneously. We distinguish a region of algebraic asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain rigorous asymptotics in both regions. We also discuss two critical transitional asymptotic regimes.Comment: 19 page

Universality for orthogonal and symplectic Laguerre-type ensembles

We give a proof of the Universality Conjecture for orthogonal (beta=1) and symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels K_{n,beta}, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (beta=2) Laguerre-type ensembles have been proved by the fourth author in [23]. The varying weight case at the hard spectral edge was analyzed in [13] for beta=2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in [7], [8] analogous results for Hermite-type ensembles. As in [7], [8] we use the version of the orthogonal polynomial method presented in [25], [22] to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from [23].Comment: 75 page