This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schrödinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schrödinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schrödinger–Hirota–Maxwell–Bloch equations, along with their Lax pairs.

Integrable and non-integrable spin systems [

Though a straightforward generalization of the (1 + 1)-dimensional Heisenberg spin system to (2 + 1) dimensions is not integrable [

The plan of the paper is as follows. In

There exists a few integrable spin systems in (2 + 1) dimensions in the literature [

The well-known Ishimori equation has the form [

The Ishimori equation is one of the integrable (2 + 1)-dimensional extensions of the following celebrated integrable (1 + 1)-dimensional continuum Heisenberg ferromagnetic spin equation (HFE) [

Note that the gauge/geometric equivalent counterpart of the Ishimori equation is the Davey–Stewartson equation [

It is one of the (2 + 1)-dimensional integrable extensions of the nonlinear Schrödinger equation (NSE)

Different properties of the Ishimori and Davey–Stewartson equations are well studied in the literature [

Both Ishimori and DSequations admit Lax pairs and, so, are linearizable. They admit a zero-curvature representation and, so, are integrable in the Lax sense. The Riemann–Hilbert problems associated with the linear eigenvalue problems have been analyzed [

As the second example of the integrable spin systems in (2 + 1) dimensions, we here present some details of the Myrzakulov-I equation (M-I) [

Often, we write this equation in the following form

The M-I equation has the following Lax representation

Note that for this equation, the eigenvalue satisfies the equation

Some properties of the M-I equation were studied, for example, in [

Additionally, here, we can note that Lax representations involving linear problems where the eigenvalue evolves as a function of time (and even as a function of x) are already well known [

Another example of the integrable spin systems in (2 + 1) dimensions is the so-called Myrzakulov–Lakshmanan I (ML-I) equation [

It has the following Lax representation

The Myrzakulov–Lakshmanan I equation is another integrable (2 + 1)-dimensional extension of the (1 + 1)-dimensional Heisenberg ferromagnet equation (10). The ML-I equation admits the well-known two integrable reductions: the HFE (10) as

Note that in this case, the spectral parameter λ obeys the equations:

The physically-important (2 + 1)-dimensional Heisenberg ferromagnet equation can be written as

It is a very important system from a physical application point of view. However, unfortunately, this equation is not integrable [

In this and the next two sections, we will present a new class of integrable spin systems in (2 + 1) dimensions by introducing a vector potential interacting with the spin field self-consistently in addition to the scalar potential considered in

Some comments on the reduction of the ML-II equation are in order. First, we note that if we put

Therefore, the ML-II equation is one of the potential (2 + 1)-dimensional integrable extensions of the M-XCIX equation.

The ML-II Equations (37)–(39) are integrable in the sense that they can be associated with a linear eigenvalue problem and that they admit a Lax representation. The corresponding Lax representation can be written in the form

We also note that here, also, the spectral parameter obeys the equation

Let us find the gauge equivalent counterpart of the ML-II Equations (37)–(39). It is not difficult (see

Next, we consider the reduction

Note that the spin-equivalent counterpart of System (68)–(70) is given by

It is nothing but the (1 + 1)-dimensional M-XCIX equations (43) and (44), which is well known to be integrable [

Note that the (2 + 1)-dimensional SMBE (63)–(65) admits the following integral of motion

In fact, from System (63)–(65), it follows that

This result with the gauge equivalence gives us the following integral of motion of the ML-II equation

Now, we want to present another new integrable spin system in 2 + 1dimensions, namely the so-called the Myrzakulov–Lakshmanan III (ML-III) equation, which contains a vector and two scalar potentials. Its form is given as

As an integrable equation, the ML-III equation admits a Lax representation. It is given by

Finally, we note that for the ML-III equation, the spectral parameter satisfies the following nonlinear evolution equation

Let us now consider some reductions of the ML-III Equation (84)–(87).

In this case, the ML-III equation reduces to the following principal chiral equation (see, e.g., [

It is integrable in the sense that it admits the Lax representation. The corresponding Lax representation follows from Equations (89) and (90) as

In this case, we get the following integrable equation

This case corresponds to the M-LXIV equation, which reads as (see, e.g., [

It is not difficult to verify that the gauge-equivalent counterpart of the ML-III equation has the form

This equation can be considered as the general (2 + 1)-dimensional complex modified Korteweg–de Vries–Maxwell–Bloch equation (cmKdVMBE), as it is one of the (2 + 1)-dimensional generalizations of the (1 + 1)-dimensional cmKdVMB equation (see, e.g., [

Now, we assume that

The above set of Equations (122)–(126) is the reduced form of the (2 + 1)-dimensional cmKdVMB equation. It admits the following integrable reduction, if

It is the usual (2 + 1)-dimensional cmKdV equation.

In (1 + 1) dimensions, that is if

Its Lax representation reads as

Note that the (1 + 1)-dimensional cmKdVMBE (130)–(132) itself admits the following integrable reductions.

(i) The (1 + 1)-dimensional complex mKdV equation is obtained when

(ii) The following (1 + 1)-dimensional equation arises when

(iii) On the other hand, the following (1 + 1)-dimensional equation results for

Our third new integrable spin system is the Myrzakulov–Lakshmanan IV (ML-IV) equation, which is a higher order spin evolution equation,

First let us present the corresponding Lax representation of the ML-IV Equations (150)–(153). It has the form

Now, we present some reductions of the LM-IV Equations (150)–(153).

Let us put

It is nothing but the principal chiral equation noted previously, which is integrable.

Next, we consider the case

It is the ML-II Equations (37)–(39).

Our next example is the case

It is nothing but the ML-III Equations (84)–(87).

Now, we put

The last example is the case

It is the M-XCIV equation (see, e.g., [

The gauge equivalent counterpart of the ML-IV Equations (150)–(153) has the form

We designate this set of equations as the (2 + 1)-dimensional Hirota–Maxwell–Bloch equation (HMBE) for the reason that when

Now, we assume that

It is the (2 + 1)-dimensional HMBE (compare with the HMBE from [

(i) For the case

(ii) The (2 + 1)-dimensional complex mKdV equation is obtained for the choice

(iii) The (2 + 1)-dimensional Schrödinger–Maxwell–Bloch equation results when

(iv) The (2 + 1)-dimensional complex mKdV-Maxwell–Bloch equation is obtained for

(v) The following (2 + 1)-dimensional equation is obtained for

Its Lax representation reads as

Note that the (1 + 1)-dimensional HMBE (206)–(208) admits the following integrable reductions.

(i) The NSLE for

(ii) The (1 + 1)-dimensional complex mKdV equation for

(iii) The (1 + 1)-dimensional Schrödinger–Maxwell–Bloch equation for

(iv) The (1 + 1)-dimensional complex mKdV-Maxwell–Bloch equation for

(v) The following (1 + 1)-dimensional equation is obtained for

It can be rewritten in the following form

(vi) The following (1 + 1)-dimensional equation is obtained for

Spin systems are fascinating nonlinear dynamical systems. In particular, integrable spin systems have much relevance in applied ferromagnetism and nanomagnetism. More interestingly, integrable spin systems have a close connection to the nonlinear Schrödinger family of equations. In this paper, we have introduced three specific cases of (2 + 1)-dimensional integrable spin systems, which we designated as the Myrzakulov–Lakshmanan II, III and IV equations, where additional scalar potentials or vector potentials interact in specific ways with the spin fields. Through appropriate gauge or geometric equivalence, we have identified the three equivalent (2 + 1)-dimensional nonlinear Schrödinger family of equations along with their Lax pairs. These equations, in turn, encompass a large class of the interesting (2 + 1)-dimensional family of NLSequations. Regarding both the (2 + 1)-dimensional spin and NLS family of equations, an extremely interesting question is to investigate what the the physical applications of these new equations are, for example in nonlinear optics (for their (1 + 1)-dimensional analogues, see, e.g., [

The work of Muthusamy Lakshmanan is supported by DST–IRHPA research project. The work of Muthusamy Lakshmanan is also supported by a DAE Raja Ramanna Fellowship.

All authors have equally contributed to every aspect of the paper.

The authors declare no conflict of interest.

In the previous sections, we presented some new integrable spin systems in 2 + 1 dimensions. Furthermore, we presented their equivalent counterparts in terms of the NLS family of equations. Here, in this Appendix, we want to demonstrate that between the former and latter systems, a gauge equivalence takes place. As an example, we consider the ML-IV equation, as the other two systems are its particular cases. Let

Finally, we would like to note that the nonlinear equations considered in this paper correspond to the nonisospectral problem. In fact, for example, for the more general ML-IV equation, the spectral parameter obeys the equation

Let us in more detail consider the case when

Here, some comments are in order.

(i) This last equation has, for example, the following particular solution:

(ii) In the more general case, Equation (B2) admits the solution

(iii) Equation (B2) is in fact the dispersionless KdV equation, where the KdV equation itself has the form

(iv) Equation (B2) is sometimes called the Riemann equation [

(v) Equation (B2) is integrable. The corresponding Lax representation is given by (see, e.g., [