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**Cornelia SCHIEBOLD ^{*}**

Department of Mathematics, Mid Sweden University, S-851 70 Sundsvall, Sweden

- *Corresponding Author:
- C. Schiebold

Department of Mathematics Mid Sweden

University, S-851 70 Sundsvall, Sweden

**E-mail:**[email protected]

**Received date:** December 17, 2007; **Revised date:** March 27, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We study solutions of an operator-valued NLS and apply our results to construct countable superpositions of solitons for the scalar NLS.

In the present note we explain how to construct solutions of the NLS via the study of a corresponding non-abelian system. Non-abelian integrable systems are a very active field of recent research (cf [5, 8,10,11] to mention only a few closely related to our work). From our point of view, they provide the appropriate framework to apply functional analytic techniques to the original scalar equations. The operator theoretic part of this note is based on [9], where the whole AKNS is treated uniformly. We point out several sharpenings which become possible for the individual equation at hand.

Concerning applications, we have to restrict to a concise discussion of the countable superpositions of solitons. This topic was initiated by Gesztesy and collaborators with results for several other equations [6,7]. Their approach requires involved computations along the lines of ISM. Our method leads to much shorter arguments as hard analysis is replaced by advanced functional analysis. For more applications, like the asymptotic description of multipole solutions, the reader is referred to [9].

Replacing u by by U_{1} in the scalar yields a non-abelian
NLS system

(1.1)

We interpret U_{1}, U_{2} as functions depending on the real variables x, t with values in the spaces of bounded linear operators mapping between Banach spaces

In Section 2, we find soliton-like solutions of (1.1).

**Theorem 1. **Let E_{1}, E_{2} be Banach spaces and Assume that are differentiable operator-functions solving the base equations Then

(1.2)

solve the non-abelian NLS system (1.1) wherever are both invertible.

Since the proof of this result is mainly algebraic and does not use specific properties of bounded operators, more general formulations for functions with values in appropriate algebras are possible. We stated the above version for sake of concreteness. The special form of the solution (1.2) was obtained as a consequence of more general considerations about the noncommutative AKNS system in [9]. A reformulation closer to the familiar scalar solution is given in Theorem 3.

In Section 3 we derive solution formulas for the NLS which still depend on arbitrary operator parameters Such solution formulas are actually very general (see [2,4] for the KdV case). As a concrete application, we construct countable superpositions of solitons in Sec. 4.

**Theorem 2.** *Let (k _{j})_{j} be a bounded sequence with sequences
satisfying the growth condition*

(1.3)

where E is one of the classical Banach spaces and is its topological dual. Then the following spectral determinants of the infinite matrices are well-defined

where Moreover, u is a solution of the NLS equation

If we truncate sequences by requiring the formula of Theorem 2 describes an Nsoliton uN. Then u is their limit for Notice that neither the existence nor the solution property of u are clear a priori.

Let E be a Banach space and with For an operator-valued function we consider the non-commutative partial differential equation

(2.1)

and show that it has a traveling wave solution.

**Theorem 3.** Let with Assume that is an operatorvalued
function anti-commuting with J and solving the base equations

Then

(2.2)

solves the operator equation (2.1) wherever is invertible.

For the proof we introduce in addition the operator-valued function

(2.3)

**Lemma 1. ***The derivative of the operator-valued functions given in
(2.2), (2.3) with respect to x is *

**Proof. **First we recall the non-abelian differentiation rule for inverse operators. If T = T(x)
is differentiable with respect to x and invertible for all is differentiable and Using the base equations we thus infer

Analogously, one checks

In the same way one can calculate the derivative with respect to the time variable. Note that here the fact that [A, J] = 0 and {L, J} = 0 is crucial. We omit the proof.

**Lemma 2. ***The derivative of the operator-valued function U = U(x, t) given in (2.2) with respect
to t is*

**Lemma 3.** For the operator-functions U, V in (2.2), (2.3), the following identity holds

(2.4)

**Proof. **We need the following auxiliary identity

which is applied in the third step of the succeeding calculation to replace the terms in the first and in the last large brackets.

**Proof of Theorem 3.** Using Lemma 1 we get

Thus, applying successively Lemma 3 and Lemma 2,

**Proof of Theorem 1. **We obtain Theorem 1 by applying Theorem 3 to

Next we explain how Theorem 1 can be used to extract explicit solution formulas for the scalar NLS system

(3.1)

To formulate our result we recall that a one-dimensional operator can be written as with appropriate where the map is defined by (and denotes the evaluation of the functional a on

**Proposition 1.** Let E1, E2 be Banach spaces and Assume that there
are operators belonging to a quasi-Banach ideal A admitting
a continuous determinant ±, which satisfy the one-dimensionality conditions

(3.2)

with functionals and vectors and Then a solution of the NLS system (3.1) is given by

(3.3)

with the operator-functions the vector-functions where provided the denominator does not vanish.

**Remark 1.** We want to stress that the one-dimensionality condition (3.2) can always be met
provided (Minkowski sum), see also [3]. The normalization is only chosen for convenience. It suffices to assume

**Proof. **The main argument of the proof is contained in the Step 2 where a solution of the scalar
system is constructed from a solution of the non-abelian system by cross-evaluation.

**Step 1: **Applying Theorem 1, it can be immediately checked that the operator-functions

solve the non-abelian NLS system (1.1).

**Step 2: **As a consequence of the one-dimensionality condition (3.2),

where Similarly,

We now show that

(3.4)

solve (3.1). Indeed, evaluating the first equation of the operator system (1.1) on the vector c2, we obtain the vector-equation

Applying the functional a1, we get the first equation of the system (3.1). Similarly for the second equation of (3.1).

**Step 3: **It remains to verify the solution formula in terms of the determinant available on
the underlying quasi-Banach ideal. To this end we first note

Using the multiplicity property of a determinant and the fact that on the finite-dimensional
operators the generalized determinant coincides with the standard determinant^{2}

and similarly for the other identity in (3.3).

As an application we study solutions of the NLS arising from our solution formula by plugging in diagonal operators on sequence spaces. The resulting solution class describes the superposition of countably many solitons.

Choose to be one of the classical sequence spaces and, for a given bounded sequence with for all j, we define to be the diagonal operators generated by respectively.

Let be sequences satisfying the growth condition (1.3) and set Then the one-dimensionality condition (3.2) can be solved explicitly by

and where denotes the quasi-Banach ideal of operators factorizing
through first an L_{1}-space, then a Hilbert space, and finally an L_{∞}-space.

Indeed, are the diagonal operators generated by the sequences and defined on the standard basis by

Since admits a continuous (even spectral) determinant det¸ [1], the solution formula of Proposition 1 can be applied. Moreover one can check that the particular choices above guarantee This yields Theorem 2.

**Remark 2.** The results of this section can be easily generalized to construct also countable superpositions
of multipole solutions of the NLS. Moreover it can be shown that all these solutions
are globally regular. For details see [9].

I would like to thank the referee for careful reading and valuable remarks leading to a more suggestive presentation of the main results.

^{1}Presented at the 3^{rd} Baltic-Nordic Workshop “Algebra, Geometry, and Mathematical Physics“, G¨oteborg, Sweden, October 11–13, 2007.

^{2}in particular for one-dimensional endomorphisms

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