alexa
Reach Us +44-7480-724769
A non-abeliann nonlinear Schr ̈odinger equation and countable superposition of solitons 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on Medical, Pharma, Engineering, Science, Technology and Business

A non-abeliann nonlinear Schr ̈odinger equation and countable superposition of solitons 1

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

Abstract

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

Introduction and main results

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 equation by U1 in the scalar equation yields a non-abelian NLS system

equation (1.1)

We interpret U1, U2 as functions depending on the real variables x, t with values in the spaces equation of bounded linear operators mapping between Banach spacesequation

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

Theorem 1. Let E1, E2 be Banach spaces andequation Assume thatequationequation are differentiable operator-functions solving the base equationsequation Then

equation (1.2)

solve the non-abelian NLS system (1.1) whereverequation 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 equation 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 (kj)j be a bounded sequence with equation sequences satisfying the growth condition

equation (1.3)

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

equation

where equation Moreover, u is a solution of the NLS equationequationequation

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

An operator equation governing the NLS

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

equation (2.1)

and show that it has a traveling wave solution.

Theorem 3. Let equation withequation Assume thatequation is an operatorvalued function anti-commuting with J and solving the base equations equation

Then

equation (2.2)

solves the operator equation (2.1) wherever equation is invertible.

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

equation (2.3)

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

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 equation is differentiable andequation Using the base equations we thus infer

equation

Analogously, one checks equation

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

equation

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

equation (2.4)

Proof. We need the following auxiliary identity

equation

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

equation

Proof of Theorem 3. Using Lemma 1 we get

equation

Thus, applying successively Lemma 3 and Lemma 2,

equation

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

equation

Solution formulas for the NLS

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

equation (3.1)

To formulate our result we recall that a one-dimensional operator equation can be written as equation with appropriateequation where the mapequation is defined byequation (andequation denotes the evaluation of the functional a onequation

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

equation (3.2)

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

equation(3.3)

with the operator-functions equation the vector-functionsequation whereequation provided the denominator does not vanish.

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

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

equation

solve the non-abelian NLS system (1.1).

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

equation

where equation Similarly,equation

We now show that

equation (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

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

equation

Using the multiplicity property of a determinant and the fact that on the finite-dimensional operators the generalized determinant coincides with the standard determinant2

equation

and similarly for the other identity in (3.3).

Countable superposition of solitons

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 equation to be one of the classical sequence spacesequation and, for a given bounded sequenceequation withequation for all j, we defineequation to be the diagonal operators generated by equation respectively.

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

equation

and equation whereequation denotes the quasi-Banach ideal of operators factorizing through first an L1-space, then a Hilbert space, and finally an L-space.

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

Since equation 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 equationThis 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].

Acknowledgement

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

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

2in particular equation for one-dimensional endomorphismsequation

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Relevant Topics

Article Usage

  • Total views: 11886
  • [From(publication date):
    September-2008 - Dec 09, 2019]
  • Breakdown by view type
  • HTML page views : 8063
  • PDF downloads : 3823
Top