Multimode solutions of first-order elliptic quasilinear systems obtained from Riemann invariants

Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links between two techniques, that of the symmetry reduction method and of the generalized method of characteristics. A variant of the conditional symmetry method for constructing this type of solution is proposed. A specific feature of that approach is an algebraic-geometric point of view, which allows the introduction of specific first-order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method for solving elliptic homogeneous systems of PDEs. A further generalization of the Riemann invariants method to the case of inhomogeneous systems, based on the introduction of specific rotation matrices, enables us to weaken the integrability condition. It allows us to establish a connection between the structure of the set of integral elements and the possibility of constructing specific classes of simple mode solutions. These theoretical considerations are illustrated by the examples of an ideal plastic flow in its elliptic region and a system describing a nonlinear interaction of waves and particles. Several new classes of solutions are obtained in explicit form, including the general integral for the latter system of equations

Nonclassical Symmetry Reduction and Riemann Wave Solutions

AbstractIn this paper we employ the “nonclassical symmetry method” in order to obtain Riemann multiple wave solutions of a system of first-order quasilinear differential equations. We show how to construct a Lie module of vector fields which are symmetries of the system supplemented by certain first-order differential constraints. We demonstrate the usefulness of our approach on several examples

Description of surfaces associated with CPN−1CP^{N-1} sigma models on Minkowski space

The objective of this paper is to construct and investigate smooth orientable surfaces in $R^{N^2-1}$ by analytical methods. The structural equations of surfaces in connection with $CP^{N-1}$ sigma models on Minkowski space are studied in detail. This is carried out using moving frames adapted to surfaces immersed in the $su(N)$ algebra. The first and second fundamental forms of this surface as well as the relations between them as expressed in the Gauss-Weingarten and Gauss-Codazzi-Ricci equations are found. The Gaussian curvature, the mean curvature vector and the Willmore functional expressed in terms of a solution of $CP^{N-1}$ sigma model are obtained. An example of a surface associated with the $CP^1$ model is included as an illustration of the theoretical results.Comment: 19 pages, 1 figure; shorter version, some typos and minor mistakes correcte

On certain classes of solutions of the Weierstrass-Enneper system inducing constant mean curvature surfaces

Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing certain classes of solutions to this system, including potential, harmonic and separable types of solutions, are proposed. A technique for reduction of the Weierstrass-Enneper system to decoupled linear equations, by subjecting it to certain differential constraints, is presented as well. New elementary and doubly periodic solutions are found, among them kinks, bumps and multi-soliton solutions

CP2S\mathbb {C}P^{2S} Sigma Models Described Through Hypergeometric Orthogonal Polynomials

International audienceThe main objective of this paper is to establish a new connection between the Hermitian rank-1 projector solutions of the Euclidean $\mathbb {C}P^{2S}$ sigma model in two dimensions and the particular hypergeometric orthogonal polynomials called Krawtchouk polynomials. We show that any Veronese subsequent analytical solutions of the $\mathbb {C}P^{2S}$ model, defined on the Riemann sphere and having a finite action, can be explicitly parametrized in terms of these polynomials. We apply these results to the analysis of surfaces associated with $\mathbb {C}P^{2S}$ models defined using the generalized Weierstrass formula for immersion. We show that these surfaces are homeomorphic to spheres in the $\mathfrak {su}(2s+1)$ algebra and express several other geometrical characteristics in terms of the Krawtchouk polynomials. Finally, a connection between the $\mathfrak {su}(2)$ spin-s representation and the $\mathbb {C}P^{2S}$ model is explored in detail

On the Solutions of the CP1 Model in (2+1) Dimensions,

We use the methods of group theory to reduce the equations of motion of the $CP^{1}$ model in (2+1) dimensions to sets of two coupled ordinary differential equations. We decouple and solve many of these equations in terms of elementary functions, elliptic functions and Painlev{\'e} transcendents. Some of the reduced equations do not have the Painlev{\'e} property thus indicating that the model is not integrable, while it still posesses many properties of integrable systems (such as stable ``numerical'' solitons).Comment: 28 page