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

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. The $N=1$ SKdV 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 them is the square of $L_2$ 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.

and ${\phi }_i,{\phi }_{ij},{\phi }_{ijk},\dots$ are real-valued functions. We define $\overline{\xi }={\sum }_{i=1}^{\mathrm{\infty }}{\phi }_i\overline{e_i}+{\sum }_{ij}{\phi }_{ij}\overline{e_j}\overline{e_i}+{\sum }_{ijk}{\phi }_{ijk}\overline{e_k}\overline{e_j}\overline{e_i}+\cdots$ , where $\overline{e_i}=-e_i$. 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 $\xi \overline{\xi }$, denoted by $\mathcal{P}(\xi \overline{\xi })$, is equal to ${\sum }_i{\phi }_i^2+{\sum }_{ij}{\phi }_{ij}^2+{\sum }_{ijk}{\phi }_{ijk}^2+\cdots$ . In what follows, without loss of generality, we rewrite $\mathcal{P}(\xi \overline{\xi })={\sum }_i{\phi }_i^2+{\sum }_{ij}{\phi }_{ij}^2+{\sum }_{ijk}{\phi }_{ijk}^2+\cdots$ simply as $\mathcal{P}(\xi \overline{\xi })={\sum }_i{\phi }_i^2$.

The system has multi-solitonic solutions, for example, $u(x,t)\equiv \varphi (x,t)=3\mathcal{C}\frac{1}{{cosh}^2(z)}$, $z\equiv \frac{1}{2}{\mathcal{C}}^{\frac{1}{2}}(x-(1+\mathcal{C})t+a)$, $\xi (x,t)=0$, where $\varphi (x,t)$ is the one-soliton solution of KdV equation, a is an arbitrary real number, and $\mathcal{C}>0$.

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 [12, 13].

hence V is $L_2$ norm of $(u,\xi )$. This property is absent for the system with a positive sign on the third term of the first equation in (5).

We may now analyze the stability of the ground state as well as the stability of the one-soliton solution of system (5).

where $d=\frac{1}{2\sqrt{2}}V$ and $e=V+M$. We notice that $d^2+4e\ge 0$.

Consequently, given V and M, from the initial data and a solution satisfying those initial conditions, the ${\parallel (u,\xi )\parallel }_{H_1}$ is bounded by (12). The a priori bound is a strong evidence of the existence of the solution for $0\le t$. 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.

for all $t\ge 0$.

$d_I$ and $d_{II}$ denote two distances to be defined.

We consider now the stability problem of the ground state solution $\hat{u}=0$, $\hat{\xi }=0$.

We take $d_I$ and $d_{II}$ to be the Sobolev norm ${\parallel (u-\hat{u},\xi -\hat{\xi })\parallel }_{H_1}$.

for any given ϵ, provided δ is conveniently chosen.

This argument proves the stability of the ground state solution.

where $\tau u_1$ 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 $u\in H_1(\mathbb{R})$ [5]. $d_{II}$ is related to a stability in the sense that a solution u remains close to $\hat{u}=\varphi$ only in shape but not necessarily in position.

and then we will relax this conditions to get the most general formulation of the stability problem.

In [6], it was proven that the infimum is taken on finite values of y.

where we have used $sup|h|\le \frac{\sqrt{2}}{3}{\parallel h\parallel }_{H_1}$.

This bound completes the proof, in fact, we can make, at $t=0$, $|\mathrm{\Delta }M|$ as small as we wish, and hence, we can always satisfy (14) for any given ϵ.

We will now obtain a lower bound for ${\delta }_{\xi }^{2}M$.

We notice that for $f\in D(H)$, the third term on the right-hand member belongs to $L_2(\mathbb{R})$. ${\psi }_{\lambda }$ are normalized in order to have ${\parallel f\parallel }_{\mathcal{H}}^2=F_1^2+F_2^2+{\int }_0^{\mathrm{\infty }}{F}^2(\lambda )d\rho (\lambda )$. ${\psi }_1$, ${\psi }_2$, ${\psi }_{\lambda }$ are pairwise orthogonal with the internal product in $L_2(\mathbb{R})$.

for any $F(\lambda )$ on the space $L_2({\mathbb{R}}^{+},d\rho )$.

for any $\hat{\lambda }$ in the support of $F(\lambda )$.

Since the $F(\lambda )$ considered is dense in $L_2({\mathbb{R}}^{+},d\rho )$, we conclude that (31) is valid for any $F(\lambda )\in L_2({\mathbb{R}}^{+},d\rho )$.

$〈,〉$ denotes the internal product in the $L_2(\mathbb{R})$ space.

for δ small enough.

The stability proof has then been completed.

which satisfies ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\tilde{\xi }dx=0$.

where $d_{II}[(u,\xi ),(u,\tilde{\xi })]={\parallel F_1{\psi }_1\parallel }_{H_1}$ is conserved by ${\hat{H}}_{\frac{1}{2}}$ and bounded by ${\parallel (u,\xi )\parallel }_{H_1}$ at $t=0$. The solitonic solution $(\varphi ,0)$ is then stable in the sense (13), (14).

Following [1], we considered the breaking of the supersymmetry in the $N=1$ Super KdV system and analyzed a solitonic model in terms of a Clifford algebra-valued field. We showed the stability in the Liapunov 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 [14, 15].

Independently of the original motivation, the breaking of supersymmetry, the system we analyzed 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.