Relativistic DNLS and Kaup-Newell Hierarchy

By the recursion operator of the Kaup-Newell hierarchy we construct the relativistic derivative NLS (RDNLS) equation and the corresponding Lax pair. In the nonrelativistic limit $c \rightarrow \infty$ it reduces to DNLS equation and preserves integrability at any order of relativistic corrections. The compact explicit representation of the linear problem for this equation becomes possible due to notions of the $q$-calculus with two bases, one of which is the recursion operator, and another one is the spectral parameter.


Introduction
The Derivative NLS (DNLS) equation was introduced in plasma physics as descriptive of weakly nonlinear and dispersive parallel MHD waves [10,11]. The equation is integrable and belongs to the Kaup-Newell (KN) hierarchy [4,5]. As a model of one-dimensional anyons it was studied in [1,3,9]. Solving problem of chiral soliton in quantum potential, the SL(2, R) version of this equation was derived from the Chern-Simons gauge theory in 2 + 1 dimensions [8]. The SL(2, R) version of DNLS, also known as DRD or resonant DNLS, has solutions in the form of soliton resonances with chiral properties [6,13]. The SL(2, R) KN hierarchy, reformulated in [14] by modified spectral problem, was applied in [2,7] to obtain resonance solitons for modified KP equation.
In the present paper we are going to apply DNLS hierarchy to construct the relativistic DNLS equation as an integrable nonlinearization of the semirelativistic Schrödinger equation. The relativistic version of the NLS equation, based on the Zakharov-Shabat hierarchy was derived before in [12]. We show that the RDNLS equation in the nonrelativistic limit reduces to the DNLS equation, and at any order of relativistic corrections 1/c 2 it produces an integrable model. To represent the linear problem in a compact explicit form we use notions of q-number and q-derivative from the q-calculus with two bases, one of which is the recursion operator, and another one is the spectral parameter.
The paper is organized as follows. In Section 2 we review SL(2, R) KN hierarchy in formulation of [14]. This formulation is linear in the spectral parameter, in contrast to the original KN paper with the quadratic dependence. We found the first one very convenient for our calculations, and problem of equivalency with the second one still requires to be clarified. We represent the corresponding linear problem in terms of q-calculus with two bases: one of which is the recursion operator and another one is the spectral parameter. For this hierarchy in Section 3 we This paper is a contribution to the Special Issue on Symmetries and Integrability of Difference Equations. The full collection is available at http://www.emis.de/journals/SIGMA/SIDE12.html arXiv:1704.04078v2 [nlin.SI] 25 Jul 2017 derive equation and the linear problem with an arbitrary dispersion. Section 4 is devoted to the DNLS hierarchy. In Section 4.1 we study DNLS hiearchy as integrable deformations of the linear Schrödinger equation and the corresponding linear hierarchy of higher-order equations. Arbitrary dispersive DNLS and corresponding linear problem are the subject of Section 4.2. In Section 5 we introduce the relativistic DNLS, its nonrelativistic reductions and integrable corrections at any order. Implications for the soliton solutions and the relativistic character of time dependence are discussed briefly in conclusions.

SL(2, R) KN hierarchy
Here we briefly review the KN hierarchy in the form of [14], with the linear dependence on the spectral parameter for U . The linear problem for the zero curvature equation is determined by the system of linear equations It gives us the system of equations By expanding in spectral parameter from the last equations we get recursions Then, as follows b 1 = q. From (2.5) we have and combining (2.3) and (2.4) d dx In terms of we get This relation can be rewritten as where the recursion operator is Substituting to (2.1) and (2.2) we get the N -th flow of SL(2, R) KN hierarchy [14] q As was shown in [14], this set of equations possesses infinitely many commuting symmetries associated with every flow t N . This allows us to combine these flows in an arbitrary linear combination form to construct new equations defined on this hierarchy.
Unfortunately in [14] no explicit form of the Lax pair in terms of recursion operator is given. Up to our knowledge it is also not known before for the AKNS hierarchy, and for that case it was first derived in [12]. Here we are going to complete this part and give simple and compact form of the Lax pair for the KN hierarchy. For this reason we need to introduce some notions from q-calculus, which allows us to get explicit formula for the Lax pair. For NLS hierarchy it was constructed before in [12].
The nonsymmetric q-number with one base is defined by and it is a particular case of the pq-number with two bases We can extend these definitions to operator q-numbers with a base as an operator. In particular we will use next notations, for the nonsymmetric operator q-number and for the two bases operator q-number For the Lax pair members we have where c N +1 = 0 and b 0 = 0. Explicitly or by combining terms as In terms of (2.7) then we get or by using (2.8) and recursion operator (2.9) Finally, due to (2.11) we can rewrite the result in a short form as an operator q-number or as a q-derivative, with the scaled recursion operator L/λ as a base, where the operator q-derivative is defined as In explicit form we have In a similar way, due to (2.6) we obtain It can be rewritten in terms of the operator q-number and the q-derivative as follows

Arbitrary dispersion and KN hierarchy
As we have seen in previous section, the N -th flow of SL(2, R) KN hierarchy is described by equation which is equivalent to the zero-curvature condition Now, let us introduce a new time variable t, determined by this hierarchy as Then, equations of motion are where function is the symbol of operator F (L). The linear problem and the zero-curvature condition corresponding to (3.2) are determined by the related sum of equations (3.1) we find expression as In terms of the operator q-derivative (2.12), the connection components become

RD system
As an example we consider the linear function case F (x) = x and corresponding equations of motion This gives the derivative reaction-diffusion system (DRD) with the linear problem Then for A we have and thus As a result, we get the following zero curvature potentials This DRD system was studied in [6,13] by Hirota's bilinear method and the soliton solutions with resonant interaction were derived. Combined with the next hierarchy flow, it produces the MKP-II equation, for which the resonant solitons was obtained and studied as well in [2,7].

DNLS hierarchy
The DNLS hierarchy can be derived from SL(2, R) KN hierarchy (2.10) by formal substitutions. As a first step we replace operator L by so that the hierarchy (2.10) becomes Then, we substitute where κ 2 = ±1. So, the hierarchy can be rewritten as By introducing finally we get the DNLS hierarchy with the recursion operator

Integrable deformation of linear Schrödinger equation
In the linear limit, which can be formally taken as κ 2 → 0, the recursion operator becomes just half of the momentum operator and the first flow of hierarchy reduces to the linear Schrödinger equation and its complex conjugate, with = 1, m = 1, Then, for the nonlinear case κ 2 = 1 we get the following first flow equation which is just the DNLS equation The above consideration allows us to consider DNLS as the specific nonlinearization of the linear Schrödinger equation. Starting from the classical dispersion with Hamiltonian function by the first quantization rules E → i ∂/∂t, p → −i ∂/∂x, we get the linear Schrödinger equation Combining this equation and its complex conjugate as in (4.4) ( = 1, m = 1), we can rewrite it in the form Then, the DNLS model, as a nonlinearization of the linear Schrödinger equation, appears by replacement of the linear recursion operator M 0 (4.3) by the nonlinear one, the operator M in (4.2), so that This procedure can be extended to the DNLS hierarchy. In the formal limit κ 2 → 0 of DNLS hierarchy (4.1) we get the linear Schrodinger hierarchy It corresponds to the first quantization of a classical system with dispersion Then, following in the opposite direction, from this dispersion we get the linear Schrödinger hierarchy. By nonlinearization in (4.5) we replace M 0 by M and obtain the DNLS hierarchy: It is convenient to have it in the present form, reflecting the same power for the classical momentum and the recursion operator.

Arbitrary dispersive DNLS
Now we are ready to consider a system with an arbitrary classical dispersion E(p) and the Hamiltonian function In a more general case it is possible to have summation also in negative powers of p as a Laurent series expansion, which will require to use the negative DNLS hierarchy flows. The first quantized linear Schrödinger equation corresponding to this dispersion is Combining it with its complex conjugate we have or in terms of operator M 0 (4.3) Shortly it is By replacing M 0 → M, finally we get the DNLS nonlinearization of the linear Schrödinger equation with arbitrary form of dispersion (4.6), We notice that another type of nonlinearization of the linear Schrödinger equation, based on the NLS hierarchy was derived in [12].

DNLS hierarchy linear problem
For the N -th flow we start with Replacing L by M L = 1 0 0 κ 2 M 1 0 0 κ 2 finally we get

Similar calculations for A give
Here we like to emphasize that these expressions are written in terms of two base operator pq-numbers with basis M and λ

Arbitrary dispersion linear problem
To construct the linear problem for arbitrary dispersive DNLS (4.7) with dispersion (4.6) we first expand Then we define new time variable and the linear problem with Substituting expressions obtained in previous section we find

Relativistic DNLS
As an application of the above procedure, in this section we consider the relativistic DNLS equation determined by semirelativistic dispersion H(p) = m 2 c 4 + p 2 c 2 = mc 2 1 + p 2 m 2 c 2 .
Following to this procedure we find then RDNLS as

Conlusions
In the present paper we have derived Lax representation for the SL(2, R) KN hierarchy and DNLS hierarchy by using operator q-numbers and q-derivatives with two bases, one of which is the recursion operator and the another one is the spectral parameter. These compact expressions allowed us to derive DNLS with an arbitrary dispersion. Choosing the semirelativistic form of the dispersion we have constructed relativistic DNLS, which becomes DNLS in nonrelativistic limit and it is integrable at any order of relativistic corrections to DNLS. Since the U operator for this equation is in the same form as for DNLS, the spectral characteristics for both models without time evolution would be the same. But in time evolution of solitons we will have now the relativistic form of dispersion.