Relating Some Nonlinear Systems to a Cold Plasma Magnetoacoustic System

Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schrödinger equation.


Introduction
Over the past years, points of connection of plasma physics to various nonlinear equations of significant importance have been explored. An initial connection can be traced back to H. Washimi and T. Taniuti [1], for example, who showed that the one-dimensional asymptotic behavior (as t → ∞ ) of small amplitude ion-acoustic waves was described by the Korteweg-deVries equation-following on a parallel track work of C. S. Gardner and G. K. Morikawa [2]. The paper of A. Jeffrey [3] provides for some systematic details on this particular connection, and it includes remarks, for example, on the work of Y. A. Berezin  x t x t ψ ρ − = (2) was obtained, where 0 ρ > is the plasma density and S is a real-valued velocity potential. That is, the velocity field u of the plasma is given by 2 x u S = − . In the present paper we consider for an arbitrary 0 γ < solutions of Equation (1) Here 0 β > and , i k   are unit vectors along the x and z axes. The system (5) is derived by way of a shallow water type approximation of the system is inserted into the second equation [5] [6] [7].
We will work with the following explicit traveling wave solution ( ) 0,u ρ > of the MAS (5) given by A. Gurevich and A. Krylov (G-K) in [8]. For a choice where the second expression for C in (9) follows as Then for the standard Jacobi elliptic function ( ) , dn x κ with elliptic modulus κ [9] ( ) ( ) , .
x t a dn a x vt The choice for 1 δ > in (1) .
In the present case of (10) therefore As before, we see in (11) that S is a potential function for the velocity field u.
Note also that by (3) and (11) we obtain the solution of the RNLS equation (1). The formulas (11), (14), (15) relate the nonlinear systems (4), (12) and the resonance nonlinear Schrödinger Equation (1) to the cold plasma system (5) with solution (10). In Section 2 we relate the solution (10) also to nonlinear systems of Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman and their anti-systems given by the time reversal t t → − . Here we find solutions of these systems that generalized in a non-trivial way those found in [11], for example. By choosing 1 , and choosing 1 0 α = so that 0 C = by (9) and 0 u u = is a constant function in (10), in particular, our r in (14) and v + in (34) of section 2 reduce to the dissipaton e + and shock soliton v + in Sections 2 and 4 of [11]. Attention in Section 4 is turned to further remarks on the cold plasma-2d black hole connection set up in [10]. The main result of that section is the computation of two more plasma dilaton fields such that these combined with the one computed in [10] form a linearly independent set. This too generalizes in a non-trivial way (namely the case 1 0 α ≠ ) a result found in [12].
Finally, in Section 5, we switch from the traveling wave solution ( ) ,u ρ to a plane wave solution of the system (5). Remarkably, its dispersion relation coincides with the dispersion relation obtained from the linearization of the RNLS Equation (1) about a suitably normalized ground state solution 0 ψ .
Throughout, an attempt is made to maintain an expository style in the presentation of the material, for the sake of completeness.

Elliptic Function Solutions of Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman Systems
In addition to the solutions (11), (14) of the Madelung fluid and RD systems (4), (12), and the RNLS solution (15) of (1), all of which were constructed by way of the G-K solution (10), we consider now solutions of nonlinear systems of Broer-Kaup (B-K), Boussinesq, and Hamilton-Jacobi-Bellman (H-J-B), and of their time reversal ( t t → − ) systems. Here, again, the G-K solution (10) of the MAS (5) plays an underlying, subtextual role. derive the B-K system. We provide a sketch of this derivation, for the sake of completeness, and we present a general elliptic function solution.
The continuity equation (5) is one of many such laws. The RDS (12) gives rise to the conservation law for example, for which the particular value  In addition to (16), we also need the conservation law which also follows from (12). Namely, the l.h.s. in (17) Also by (ii), (iii), so that (19) and (20) together give the equations which suggests that one defines Then v r r r r rr r r r r r r r Br s r Similarly, by (12) and (18) again, Putting the pieces together, (Equations (23)-(25)), we have derived from the RDS (12), for 0, 0 r s > < , and B arbitrary (not necessarily the specific value for σ and v ± defined in (18) and (22). The second system in (26) (with the minus signs) corresponds to the time reversal t t → − .
The formulas (27) and (34)  The solution ( ) , v σ + of the first system in (26) vastly generalizes the one ( ) , v ρ + found in [11], where the notation ρ there corresponds to σ here, and where a solution of the second system in (26) is not addressed.
The generalization here is not simply that of elliptic functions over hyperbolic function, but it is in large part due to the freedom to allow 1 α to be non-zero: 1 0 α > . Indeed for 1 0 α = , all results of this paper, and those of [10], simplify greatly-mainly because then 0 C = , by (9). For 1 0 α = , the formulas in (27) and (34), for example, reduce to which, moreover, for 1 which apart from v − are the results that appear in [11], with a different choice of constants-the v + in (36) being the shock soliton mentioned in the introduction. We shall see in Section 3 that in case 1 0 α = , there is also a nice choice for the potential function ( ) , S x t in (11), and thus a nice expression for (13), (14), and for ( ) in (15) exists in this case.

Boussinesq Systems
Given the bulk of details and formulas in Section 2.1, we can glide more easily through this section. A good number of equations for various mathematical models are referred to as Boussinesq equations. It is perhaps best then to consider the systems discussed here more properly as Boussinesq type systems. This Define the "pressure" functions ( ) , As before we will choose One can compute ( ) t p ± also, and in the end, with (39), derive the Boussinesq type systems The second system in (40)

A Hamilton-Jacobi-Bellman System
Given two functions ( ) ( ) 1 2 , , , f x t f x t , consider the Lagrangian density The corresponding Euler-Lagrange equations of motion are for 1, 2 j = , which are immediately computed: x xx xx Now going back to the first B-K equation in (26), which we compare with the first equation in (44), we are obviously motivated to think of 2 f as σ and to take ( ) def 1 2 2 log x f v r x That is, we choose Then (44) is the system where the second equation in (48) is a Hamilton-Jacobi-Bellman (H-J-B) equation, which as remarked in [11] is well-known in the theory of optimal stochastic control for continuous Markov processes [15].
where the bracketed expression here for xx r r was obtained in the first sentence that followed Equation (23). By the definition (22), and the definition of A + again, γ β = − . The useful role of the reaction diffusion system (12) has been noted several times in this section. Typically in physics, chemistry, or biology, for example, a more general system of the form

Further Remarks on the Case 1 0 = α
It was pointed out in Section 2.1 that various formulas presented in this paper (and in the paper [10]) simplify drastically in case 1 0 α = ; in general 1 0 α ≥ .
Here we find in this case, in particular, a concrete expression for the potential function ( ) , S x t (and thus also for the function ( ) ) appearing in some of these formulas. (9) and (10), is a constant function: by straight-forward differentiation of R. By Maple, for example, the differentiation of ρ on the r.h.s. of (53) can be carried out. For ρ given by (10) for a function of integration ( ) f x . Using again that 2  can be plugged into formulas (14), (15), and (51), for example-taking 1 0 α = there, to further explicate the solutions , , , r s A ψ + ; keep in mind the assumption 0 γ < : given in (56). We also have the formulas for ( ) β γ − = , by (13).
Some concluding remarks about the case 1 0 α = pertain to the conservation laws (16) and (17) The result is that where the latter integral formula in (62) is used to compute the integration of

Plasma Metric and Plasma Dilaton Fields
An exact connection of the cold plasma system (5) to a two-dimensional Jackiw-Teitelboim (J-T) black hole was investigated in [10], with the resonance NLS Equation (1)  The results of [10] represent an extension to the non-trivial case 1 0 α ≠ of results in [12]-a case discussed in Sections 2 and 3 here. As indicated in the introduction, two more plasma dilaton fields will be computed in this section, to obtain a set of three linearly independent ones altogether.
The J-T gravitational field equations are a system of equations of which a solution consists of a triple ( ) , , g Φ Λ where g is a pseudo Riemannian metric with local components ij g , ( ) R g is its Ricci scalar curvature, Φ is a real-valued function of the local coordinates ( ) 1 2 , x x in which g is expressed (called a dilaton field), Λ is a cosmological constant (and therefore the scalar curvature is a constant), and where, locally, the Hessian i j ∇ ∇ is given by for the Christoffel symbols k ij Γ of g, of the second kind. , and we point out that the sign convention for scalar curvature that we adopt here (and in [10] [12]) is spelled out on page 182 of [19], for example.
Thus, for example, our ( ) R g is the negative of that employed in [17] [18]. We will also write bh g for the J-T metric given in (66).
As a second example, we consider the plasma metric plasma g g = constructed in [10] by way of the G-K solution in (10) of the MAS (5). Here the local coor- , with the notation ρ here not to be confused with the same notation for the solution ρ in (10) where (again) C is given (9). Obviously this metric is more complicated in structure. For 1 0 α = (so that again 0 C = ) and for the choice 1 2 γ = − (as in [5] [6]) it reduces to the metric (6) in [12], where the notation b there corresponds to 2β here. Some remarks herewith are offered to provide some clarification regarding the "raisons d'être" of the plasma metric formula (67). , u u λ λ for the reaction diffusion system (12), and establish a gauge equivalence of these two nonlinear systems [16].
More precisely, in our specific setup, for a (complex) spectral parameter λ . The equations of motion assertion (69) is equivalent to the assertion that By formulas (34) then  (75)) is more involved. By (18) and (22) In the very specialized case of 1 0, 1 4 v a κ > and with the single condition A clarification regarding how the plasma metric plasma g in (67)  We move on now to the main result in [10], which will allow, in particular, for a direct computation of plasma dilaton fields and thus for solutions of the field as in [10]. There are, however, two other J-T dilaton fields ( ) ( ) not computed in [10], which we compute here.
In fact, one can take