Stability of solitonic solutions of Super KdV equations under Susy breaking conditions

A supersymmetric breaking procedure for N=1 Super KdV, preserving the positivity of the hamiltonian as well as the existence of solitonic solutions, is implemented. The resulting integrable system is shown to have nice stability properties.


Introduction
The breaking of supersymmetry in physical systems is always an interesting aspect to analyze. We consider a solitonic system [1] arising from the breaking of supersymmetry in the N = 1 Super KdV system [2,3]. In the latter there is only one hamiltonian structure in distinction to the bi-hamiltonian one in the Korteweg-de Vries (KdV) system. Its hamiltonian however is not manifestly positive. Nevertheless the quantum formulation of the theory yields a manifestly positive definite self-adjoint operator. The stability of the ground state of the system is then assured from it. We considered in [1] the supersymmetry breaking of the SKdV system by changing the Grassmann algebra structure of the SKdV formulation to a Clifford algebra one. This susy breaking mechanism has already appeared in several works, see for example [4]. One may then obtain a solitonic system with the same evolution equation for the new Clifford algebra valued field as one had for the odd Grassmann algebra valued one in the SKdV system and what is more important with a bounded from below hamiltonian. The system presents solitonic solutions although the infinite set of local conserved quantities of SKdV breaks down. We will show in this work that this solitonic system has nice stability properties.
The stability in the sense of Liapunov of the one-soliton solution of KdV equation was first proven by Benjamin [5] and Bona [6]. The proof makes use of the first few conserved quantities of the KdV equation. In particular the fact that one of then is the square of the L 2 norm is relevant in their argument. The use of conserved quantity was also considered in a stability argument by Boussinesq [7]. In this paper we make use of this main idea to prove stability of the one-soliton solution of the coupled equation, with fields valued on a Clifford algebra, derived from the supersymmetric breaking of the N = 1 SKdV equation.

SKdV and the breaking of supersymmetry
The fields u(x, t) and ξ(x, t) describing N = 1 SKdV equations [2] take values on the even and odd part of a Grassmann algebra respectively. The N = 1 SKdV equations are This system of partial differential equations has infinite local conserved charges as well as infinite non-local conserved charges [2,8,9,10,11]. The first few local ones are To break supersymmetry we consider the fields u and ξ taking values on a Clifford algebra instead of being Grassmann algebra valued. We thus take u to be a real valued field while ξ to be an expansion in terms of the generators e i , i = 1, . . . of the Clifford algebra: where e i e j + e j e i = −2δ ij (4) and ϕ i , ϕ ij , ϕ ijk , . . . are real valued functions. We defineξ = ∞ i=1 ϕ iēi + ij ϕ ijējēi + ijk ϕ ijkēkējēi + · · · whereē i = −e i . We denote as in superfield notation the body of the expansion those terms associated with the identity generator and the soul the remaining ones. Consequently the body of ξξ, denoted by P(ξξ), is equal to In what follows, without loss of generality, we rewrite P(ξξ) = Σ i ϕ 2 i + Σ ij ϕ 2 ij + Σ ijk ϕ 2 ijk + · · · simply as P(ξξ) = Σ i ϕ 2 i . The system of partial differential equations arising from the breaking of supersymmetry which has the required properties as discussed in [1] is The system (5) has the following conserved chargeŝ The system has multi-solitonic solutions, for example, is the one-soliton solution of KdV equation, a is an arbitrary real number and C > 0.
This system with a change of sign in the third term of the right hand member of the first equation of system (5) was considered in [13,14].
It will be important in the following stability argument that hence V is the L 2 norm of (u, ξ). This property is absent for the system with a positive sign on the third term of the first equation in (5).

Stability properties of the system
We may now analize the stability of the ground state as well as the stability of the onesoliton solution of the system (5). We start the analysis by considering an a priori bound for the solutions of system (5). We denote by H 1 the Sobolev norm: We obtain we now use it yields From (11) it follows We notice that d 2 + 4e ≥ 0. Consequently, given V and M from the initial data and a solution satisfying those initial conditions, the (u, ξ) H 1 is bounded by (12). The a priori bound is a strong evidence of the existence of the solution for 0 ≤ t. We will consider this existence problem elsewhere. In this work we assume the existence of the solution and its continuous dependence on the initial data under smooth enough assumptions on the initial perturbation.
We consider the stability in the sense of Liapunov. In particular we take the same definition as in [5]: (û,ξ), a solution of (5), is stable if given ǫ there exists δ such that for any solution (u, ξ) of (5), satisfying at t = 0 then d II (u, ξ) , û,ξ < ǫ for all t ≥ 0. d I and d II denote two distances to be defined. We consider now the stability problem of the ground state solutionû = 0,ξ = 0. We take d I and d II to be the Sobolev norm (u −û, ξ −ξ) H 1 .

We get
At t = 0 we then have, using (13), Consequently, from the a priori bound (12), we obtain for any t ≥ 0 (u, ξ) H 1 < ǫ for any given ǫ, provided δ is conveniently chosen. This argument proves the stability of the ground state solution.
We now consider the stability of the one-soliton solutionû = φ,ξ = 0. The proof of stability is based on estimates for the u field which are analogous to the one presented in [5,6] while a new argument will be given for the ξ field. The distances we will use are where τ u 1 denotes the group of translations along the x-axis. d II is a distance on a metric space obtained by identifying the translations of each u ∈ H 1 (R) [5]. d II is related to a stability in the sense that a solution u remains close toû = φ only in shape but not necessarily in position. We first assume that and and then we will relax this conditions to get the most general formulation of the stability problem.
Following [5] we define where a is defined, for each t ≥ 0,by In [6] it was proven that the infimum is taken on finite values of y. We obtain, where we have used C (V (u, ξ) − V (φ, 0)) = 0 and that φ is the soliton solution of KdV equation and hence it satisfies We then have where we have used sup |h| ≤ At t = 0 we will assume that ∆M is a conserved quantity on the space of solutions of the system (5). By taking δ small enough in (23) we can make ∆M as small we wish. The second step in the proof of stability is to argue that at any t ≥ 0 ∆M satisfies the bound This bound completes the proof, in fact we can make, at t = 0, |∆M| as small as we wish and hence we can always satisfy (14) for any given ǫ. We decompose ∆M into where Using the same argument as in [5,6] we obtain and We will now obtain a lower bound for δ 2 ξ M. We consider the operator H = − d 2 dx 2 − 1 2 φ with domain in the Hilbert space L 2 (R). It is a symmetric essentially self-adjoint operator. We denote with the same letter H its self-adjoint extension. It has two eigenvalues λ 1 = −C and λ 2 = − C 4 and a continuous spectrum [0, ∞). The eigenfunctions are proportional to 1 cosh 2 z and sinh z cosh 2 z respectively, where z = 1 2 C 1 2 . The spectral theorem for self-adjoint operators ensures the existence of an unitary transformation from the domain D(H) in the Hilbert space H to L 2 (R, dρ). In the case in which H = L 2 (R) this unitary transformation may be realized in terms of the eigenfunctions ψ 1 , ψ 2 ∈ L 2 (R) and the hypergeometric functions ψ λ (x) which satisfy point to point Hψ λ = λψ λ for λ > 0, but do not belong to L 2 (R). Under the unitary map f (x) ∈ L 2 (R) can be expressed where F (λ) belongs to L 2 (R + , dρ). The action of H in H corresponds to the multiplication by λ in L 2 (R + , dρ): We notice that for f ∈ D(H) the third term on the right hand member belongs to L 2 (R). ψ λ are normalized in order to have f 2 . ψ 1 , ψ 2 , ψ λ are pairwise orthogonal with the internal product in L 2 (R).
If we denote g( for any F (λ) on the space L 2 (R + , dρ). We consider first F (λ) to have support in the complement of a neighborhood of λ = 0. We may then consider λF (λ) = χ(λ) where χ(λ) is the characteristic function with value one in the interval (λ − ǫ,λ + ǫ) and 0 otherwise. We have for anyλ in the support of F (λ). It then follows, by decomposing F (λ), that Since the F (λ) considered is dense in L 2 (R + , dρ) we conclude that (31) is valid for any F (λ) ∈ L 2 (R + , dρ). We may always decompose ξ ∈ L 2 (R) as and use (18) together with (31) to obtain We now consider the Rayleigh quotient of any ξ in the domain of H: , denotes the internal product in the L 2 (R) space. The eigenvalues satisfy (using the min-max theorem) We thus get From (27), and using λ 2 = − C 4 , we obtain and introducing a parameter 0 ≤ β ≤ 1

Conclusions
Following [1] we considered the breaking of the supersymmetry in the N = 1 Super KdV system and analize a solitonic model in terms of a Clifford algebra valued field. We showed the stability in the Lyapunov extended sense of the ground state and the one-soliton solution of the integrable model. The approach introduced in [5] to prove the stability of the solitonic solutions of KdV equation and extended here for Clifford valued fields may also be used in the stability analysis of supersymmetric solitons in the bosonization scheme in [15,16]. Independently of the original motivation, the breaking of supersymmetry, the system we analized in this paper is interesting because it contains the same soliton solutions as the KdV equation but is more realistic in the sense that the symmetry associated to the infinite number of conserved charges of KdV is broken.