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

Abstract

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.Running Title: Supersymmetric system in Riemann invariants PACS: primary 02.20. Sv, 12.60.Jv, 02.30.Jr, Keywords: Riemann invariants, Lie superalgebra, invariant solutions. * email address: grundlan@crm.umontreal.ca † email address: hariton@crm.umontreal.ca 1 I Introduction A Historical BackgroundThe concept of differential invariants was first introduced by B. Riemann in 1858 in his classical works 1, 2 concerning the Euler equations for an ideal fluid flow in two independent variables u t + uu x + p ′ (ρ) ρ ρ x = 0, ρ t + uρ x + ρu x = 0, ρ > 0.Here, u is the local velocity of the fluid, the pressure p is assumed to be a differentiable function of the density ρ and p ′ = dp/dρ. Riemann investigated the formulation and mathematical correctness of problems involving the propagation and superposition of waves described by equation (1). Next, he constructed rank-2 solutions corresponding to the "superposition" of two waves propagating with local velocity u = ± (p ′ Riemann studied the asymptotic behavior of initial localized disturbances (i.e. initial data with compact support) corresponding to the above-mentioned waves. Consequently, he demonstrated that even for sufficiently small initial data, after some finite time T , these waves could be separated again in such a way that waves of the same type as those assumed in the initial data could be observed. Riemann noticed that the solution for systems of hydrodynamic type (3), even with arbitrarily smooth initial data, usually(could not be continued indefinitely in time. After a certain finite time T the solutions blew 2 up (more precisely, the first derivatives of the considered solution become unbounded after some finite time T > 0). So, for times t > T , smooth solutions to the Cauchy problem do not exist. This phenomenon is known as the gradient catastrophe. 3, 4 Furthermore, Riemann was interested in extending the solution in some more generalized sense beyond the time T of the blow up. On the basis of the conservation laws for the mass, energy and momentum, Riemann 2 and later Hugoniot 5 introduced the concept of weak solutions (non-continuous) in the form of shock waves. Based on this concept he proved some laws connecting the wave front velocity and parameters of the fluid state before and behi...

N = 2 supersymmetric extension of a hydrodynamic system in Riemann invariants

“…A systematic classification in terms of conjugacy classes was performed for the one-dimensional subalgebras, resulting in a list of 401 nonequivalent classes of subalgebras. It is interesting and significant to note that the classification is much more extensive for the N = 2 supersymmetric extension than for its N = 1 counterpart [28]. Consequently, a complete symmetry reduction analysis of our supersymmetric hydrodynamic system would lead to very large classes of invariant solutions.…”

“…The Lie symmetry algebra of the classical hydrodynamic system (12) shares in common with its classical Schrödinger counterpart time and space translations together with a Galilean-type boost and a dilation in time and space [28,29]. The classical hydrodynamic algebra also contains a second dilation involving the fields and an inverse boost, while the classical Schrödinger algebra contains a phase shift and a special conformal-type transformation.…”

“…The method of symmetry reduction for Grassmann-valued equations allows us to determine exact analytic solutions of our generalized supersymmetric model. In a previous article [28], we constructed an N = 1 extension of the hydrodynamic type system (12) through the use of a fermionic superfield. Since comparatively little is known concerning such extensions, we propose to construct an N = 2 supersymmetric extension of the system (12) in which the superfield is bosonic.…”

Extended symmetry analysis of an isothermal no-slip drift flux model

“…, u n ) but not of the derivatives of the latter. Such systems and their natural generalizations to the case of more than two independent variables are known as (translation-noninvariant nonhomogeneous) hydrodynamictype systems and are a subject of intense research, see for example [4,6,18,20,21,31,36,43,44] and references therein. An important class of such systems is given by evolutionary translation-invariant systems of hydrodynamic type in two independent variables, for which A is the n × n unit matrix, the vector C vanishes, and the matrix B depends only on dependent variables.…”

Invariant solutions of the supersymmetric version of a two-phase fluid flow system

“…Recently, supersymmetric extensions of hydrodynamic-type systems has become a subject of intensive research (see e.g. [8,24,25,29,30,31,32]). In view of the above, the objective of this paper is to construct a supersymmetric extension of the twophase fluid flow equation (1.7) and to study its supersymmetry properties.…”