We may now analyze the stability of the ground state as well as the stability of the one-soliton solution of system (5).
We start the analysis by considering an a priori bound for the solutions of system (5). We denote by the Sobolev norm
(8)
We obtain
(9)
we now use
(10)
it yields
(11)
From (11), it follows that
(12)
where and . We notice that .
Consequently, given V and M, from the initial data and a solution satisfying those initial conditions, the is bounded by (12). The a priori bound is a strong evidence of the existence of the solution for . 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]: , a solution of (5), is stable if given ϵ, there exists δ such that for any solution of (5), satisfying at
(13)
then
(14)
for all .
and denote two distances to be defined.
We consider now the stability problem of the ground state solution , .
We take and to be the Sobolev norm .
We get
and
At , we then have, using (13),
Consequently, from the a priori bound (12), we obtain for any
for any given ϵ, provided δ is conveniently chosen.
This argument proves the stability of the ground state solution.
We now consider the stability of the one-soliton solution , . 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
(15)
(16)
where denotes the group of translations along the x-axis. is a distance on a metric space obtained by identifying the translations of each [5]. is related to a stability in the sense that a solution u remains close to only in shape but not necessarily in position.
We first assume that
(17)
and
(18)
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 , by
In [6], it was proven that the infimum is taken on finite values of y.
We obtain
(19)
where we have used and that ϕ is the soliton solution of KdV equation, and hence it satisfies
We denote . We then have
(21)
where we have used .
Coming back to ΔM, we get
(22)
At , we will assume that
(23)
hence
(24)
ΔM is a conserved quantity on the space of solutions of system (5). By taking δ small enough in (23), we can make ΔM as small as we wish. The second step in the proof of stability is to argue that at any , ΔM satisfies the bound
(25)
This bound completes the proof, in fact, we can make, at , as small as we wish, and hence, we can always satisfy (14) for any given ϵ.
We decompose ΔM into
(26)
where
(27)
Using the same argument as in [5, 6], we obtain
(28)
and
We will now obtain a lower bound for .
We consider the operator with domain in the Hilbert space . It is a symmetric essentially self-adjoint operator. We denote with the same letter H its self-adjoint extension. It has two eigenvalues and and a continuous spectrum . The eigenfunctions are proportional to and , respectively, where . The spectral theorem for self-adjoint operators ensures the existence of a unitary transformation from the domain in the Hilbert space ℋ to . In the case in which , this unitary transformation may be realized in terms of the eigenfunctions and the hypergeometric functions which satisfy point to point
for , but do not belong to . Under the unitary map can be expressed
where belongs to . The action of H in ℋ corresponds to the multiplication by λ in :
We notice that for , the third term on the right-hand member belongs to . are normalized in order to have . , , are pairwise orthogonal with the internal product in .
If we denote , , it follows that
its left-hand member is zero, hence
(29)
for any on the space .
We consider first to have support in the complement of a neighborhood of . We may then consider , where is the characteristic function with value one in the interval and 0 otherwise. We have
(30)
for any in the support of .
It then follows, by decomposing , that
(31)
Since the considered is dense in , we conclude that (31) is valid for any .
We may always decompose 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 space.
The eigenvalues satisfy (using the min-max theorem)
We thus get
From (27), and using , we obtain
and introducing a parameter
Consequently, using , we have
for any β satisfying . In particular, for ,
(33)
From (28) and (33), we get
(34)
and from (21),
Finally,
(35)
where
We now use an argument in [5, 6] to show, for δ small enough, that
Since
we then obtain
for δ small enough.
The stability proof has then been completed.
We may now relax the assumptions (17) and (18). The assumption (17) may be removed by an application of the triangle inequality as is [5]. While (18) may be relaxed by considering
which satisfies .
Using again the triangle inequality for the distance , we get
where is conserved by and bounded by at . The solitonic solution is then stable in the sense (13), (14).