Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

In this paper we employ a "direct method" in order to obtain rank-k solutions of any hyperbolic system of first order quasilinear differential equations in many dimensions. We discuss in detail the necessary and sufficient conditions for existence of these type of solutions written in terms of Riemann invariants. The most important characteristic of this approach is the introduction of specific first order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method of solving multi-dimensional systems of PDEs. We have demonstrated the usefulness of our approach through several examples of hydrodynamic type systems; new classes of solutions have been obtained in a closed form.Comment: 30 page

Multidimensional simple waves in fully relativistic fluids

A special version of multi--dimensional simple waves given in [G. Boillat, {\it J. Math. Phys.} {\bf 11}, 1482-3 (1970)] and [G.M. Webb, R. Ratkiewicz, M. Brio and G.P. Zank, {\it J. Plasma Phys.} {\bf 59}, 417-460 (1998)] is employed for fully relativistic fluid and plasma flows. Three essential modes: vortex, entropy and sound modes are derived where each of them is different from its nonrelativistic analogue. Vortex and entropy modes are formally solved in both the laboratory frame and the wave frame (co-moving with the wave front) while the sound mode is formally solved only in the wave frame at ultra-relativistic temperatures. In addition, the surface which is the boundary between the permitted and forbidden regions of the solution is introduced and determined. Finally a symmetry analysis is performed for the vortex mode equation up to both point and contact transformations. Fundamental invariants and a form of general solutions of point transformations along with some specific examples are also derived.Comment: 21 page

Supersymmetric version of a hydrodynamic system in Riemann invariants and its solutions

In this paper, a supersymmetric extension of a system of hydrodynamic type equations involving Riemann invariants is formulated in terms of a superspace and superfield formalism. The symmetry properties of both the classical and supersymmetric versions of this hydrodynamical model are analyzed through the use of group-theoretical methods applied to partial differential equations involving both bosonic and fermionic variables. More specifically, we compute the Lie superalgebras of both models and perform classifications of their respective subalgebras. A systematic use of the subalgebra structures allow us to construct several classes of invariant solutions, including travelling waves, centered waves and solutions involving monomials, exponentials and radicals.Comment: 30 page

Conditional symmetries and Riemann invariants for inhomogeneous hydrodynamic-type systems

A new approach to the solution of quasilinear nonelliptic first-order systems of inhomogeneous PDEs in many dimensions is presented. It is based on a version of the conditional symmetry and Riemann invariant methods. We discuss in detail the necessary and sufficient conditions for the existence of rank-2 and rank-3 solutions expressible in terms of Riemann invariants. We perform the analysis using the Cayley-Hamilton theorem for a certain algebraic system associated with the initial system. The problem of finding such solutions has been reduced to expanding a set of trace conditions on wave vectors and their profiles which are expressible in terms of Riemann invariants. A couple of theorems useful for the construction of such solutions are given. These theoretical considerations are illustrated by the example of inhomogeneous equations of fluid dynamics which describe motion of an ideal fluid subjected to gravitational and Coriolis forces. Several new rank-2 solutions are obtained. Some physical interpretation of these results is given.Comment: 19 page

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

Plasma produced by a laser in a medium with convection and free surface satisfying a Hamilton-Jacobi equation

A nonlinear diffusion equation is considered which models the temperature distribution in a laser-sustained plasma subject to wind. As certain parameters are small a singular perturbation problem arises and the method of matched asymptotic expansions is applied to approximate the solution. An essential role in this problem is played by the plasma front. This is a free surface separating the plasma and non-plasma phases. One of the main results is that we derive a 1st order nonlinear P.D.E. of Hamilton-Jacobi type, which describes certain parts of the free surface in the stationary case. This equation is analysed for various wind directions. It appears that for the initial conditions for these parts of this free surface there are different possibilities depending on the wind direction. We show further that the solution of the Hamilton-Jacobi equation can contain singularities of corner type. Furthermore, the effect of wind on the stability region of the stationary full plasma solution is analysed. The method of analysis presented in this paper is not restricted to the cone-like geometry or the specific form of the non-linearity of the problem considered here, but has potentially a much wider scope. However, the case under study in this paper is certainly representative for the effects that have to be taken into account

Plasma produced by a laser in a medium with convection and free surface satisfying a Hamilton-Jacobi equation

A nonlinear diffusion equation is considered which models the temperature distribution in a laser-sustained plasma subject to wind. As certain parameters are small a singular perturbation problem arises and the method of matched asymptotic expansions is applied to approximate the solution. An essential role in this problem is played by the plasma front. This is a free surface separating the plasma and non-plasma phases. One of the main results is that we derive a 1st order nonlinear P.D.E. of Hamilton-Jacobi type, which describes certain parts of the free surface in the stationary case. This equation is analysed for various wind directions. It appears that for the initial conditions for these parts of this free surface there are different possibilities depending on the wind direction. We show further that the solution of the Hamilton-Jacobi equation can contain singularities of corner type. Furthermore, the effect of wind on the stability region of the stationary full plasma solution is analysed. The method of analysis presented in this paper is not restricted to the cone-like geometry or the specific form of the non-linearity of the problem considered here, but has potentially a much wider scope. However, the case under study in this paper is certainly representative for the effects that have to be taken into account