Reviews in Mathematical Physics

This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.

We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but also includes direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.

In this paper, we discuss an initial boundary value problem for the stochastic viscoelastic wave equation involving the nonlinear damping term |u_t|^q-2u_t and a source term of the type |u|^p-2u. We firstly establish the local existence and uniqueness of solution by the Galerkin approximation method and an elementary measure-theoretic argument. Moreover, we also show that the solution is global for q ≥ p. Secondly, by the technique of [Existence of a solution of the wave equation with nonlinear damping and source term, J. Differential Equations 109 (1994) 295–308] with a modification in the energy functional, we prove that the local solution of the stochastic equations will blow up with positive probability or explosive in energy sense for p > q. This result extends earlier ones obtained by Liang and Gao [Explosive solutions of stochastic viscoelastic wave equations with damping, Rev. Math. Phys. 23(8) (2011) 883–902] in which only linear damping is considered. Furthermore, upon comparing our stochastic equations with their deterministic counterparts, we find that our results indicates that the presence of noise might affect the occurrence of blow-up.

The multiconfiguration Dirac–Fock (MCDF) model uses a linear combination of Slater determinants to approximate the electronic N-body wave function of a relativistic molecular system, resulting in a coupled system of nonlinear eigenvalue equations, the MCDF equations. In this paper, we prove the existence of solutions of these equations in the weakly relativistic regime. First, using a new variational principle as well as the results of Lewin on the multiconfiguration non-relativistic model, and Esteban and Séré on the single-configuration relativistic model, we prove the existence of critical points for the associated energy functional, under the constraint that the occupation numbers are not too small. Then, this constraint can be removed in the weakly relativistic regime, and we obtain non-constrained critical points, i.e. solutions of the multiconfiguration Dirac–Fock equations.