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The purpose of this paper is to face up the statistical mechanics of dense spin glasses using the well-known Ising case as a prelude for testing the methodologies we develop and then focusing on the Gaussian case as the main subject of our investigation. We tackle the problem of solving for the quenched statistical pressures of these models both at the replica symmetric level and under the first step of replica symmetry breaking by relying upon two techniques: the former is an adaptation of the celebrated Guerra's interpolation (closer to probability theory in its spirit) and the latter is an adaptation of the transport partial differential equation (closer to mathematical physics in spirit). We recover, in both assumptions, the same expression for quenched statistical pressure and self-consistency equation found with other techniques, including the well-known replica trick technique. [ABSTRACT FROM AUTHOR]

This paper surveys recent progress in the analysis of nonlinear partial differential equations using Anderson localization and semi-algebraic sets method. We discuss the application of these tools from linear analysis to nonlinear equations such as the nonlinear Schrödinger equations, the nonlinear Klein–Gordon equations (nonlinear wave equations), and the nonlinear random Schrödinger equations on the lattice. We also review the related linear time-dependent problems. [ABSTRACT FROM AUTHOR]

The complexification of the independent variables of nonlinear integrable evolution partial differential equations (PDEs) in two space dimensions, like the celebrated Kadomtsev-Petviashvili and Davey-Stewartson (DS) equations, yields nonlinear integrable equations in genuine 4 + 2, namely, in four real space dimensions (x1, x2, y1, y2) and two real time dimensions (t1, t2), as opposed to two complex space dimensions and one complex time dimension. The associated initial value problem for such equations, namely, the problem where the dependent variables are specified for all space variables at t1 = t2 = 0, can be solved via a non-local d-bar formalism. Here, the details of this formalism for the 4 + 2 DS system are presented. Furthermore, the linearised version of the 3 + 1 reduction of the 4 + 2 DS system is discussed. The construction of the nonlinear 3 + 1 reduction remains open, in spite of the fact that multi-soliton solutions for the 3 + 1 DS system already exist. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlocal parabolic equation associated with initial and Dirichlet boundary conditions. First, we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells. Then we obtain a threshold of global existence and blow up for solutions with critical initial energy. Furthermore, for those solutions that satisfy J(u0) ≤ d and I(u0) ≠ 0, we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow-up solutions increases exponentially. [ABSTRACT FROM AUTHOR]

Recently, there has been a significant development of the abstract theory of Friedrichs systems in Hilbert spaces [Ern, A., Guermond, J.-L., and Caplain, G., Commun. Partial Differ. Equations 32, 317-341 (2007) and Antonić, N. and Burazin, K., Commun. Partial Differ. Equations 35, 1690-1715 (2010)] and its applications to specific problems in mathematical physics. However, these applications were essentially restricted to real systems. We check that the already developed theory of abstract Friedrichs systems can be adjusted to the complex setting, with some necessary modifications, which allows for applications to partial differential equations with complex coeffi- cients. We also provide examples where the involved Hilbert space is not the space of square integrable functions, as it was the case in previous studies, but rather its closed subspace or the space Hs(Rd;Cr), for real s. This setting appears to be suitable for particular systems of partial differential equations, such as the Dirac system, the Dirac-Klein-Gordon system, the Dirac-Maxwell system, and the time-harmonic Maxwell system, which are all addressed in the paper. Moreover, for the time-harmonic Maxwell system, we also applied a suitable version of the two-field theory with partial coercivity assumption which is developed in this paper. [ABSTRACT FROM AUTHOR]