Dispersionless Limits of Integrable Generalized Heisenberg Ferromagnet   Equations

Zhaidary Myrzakulova; Gulgassyl Nugmanova; Kuralay Yesmakhanova and; Ratbay Myrzakulov

arXiv:1903.09195·nlin.SI·March 25, 2019

Dispersionless Limits of Integrable Generalized Heisenberg Ferromagnet Equations

Zhaidary Myrzakulova, Gulgassyl Nugmanova, Kuralay Yesmakhanova and, Ratbay Myrzakulov

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TL;DR

This paper investigates the dispersionless limits of certain integrable generalized Heisenberg ferromagnet equations, extending previous work on similar magnetic systems to new classes of equations.

Contribution

It introduces dispersionless limits for a broader class of integrable spin systems, specifically generalized Heisenberg ferromagnet equations, expanding the understanding of their behavior.

Findings

01

Derived dispersionless equations for generalized Heisenberg ferromagnet models

02

Extended previous dispersionless analysis to new integrable spin systems

03

Provided mathematical framework for future studies of magnetic equations

Abstract

This paper is a continuation of our previous work in which we studied a dispersionless limits of some integrable spin systems (magnetic equations). Now, we shall present dispersionless limits of some integrable generalized Heisenberg ferromagnet equations.

Equations99

A_{t} = A \land A_{xx},

A_{t} = A \land A_{xx},

A = (A_{1}, A_{2}, A_{3}), A^{2} = 1.

A = (A_{1}, A_{2}, A_{3}), A^{2} = 1.

Ψ_{x}

Ψ_{x}

Ψ_{t}

U_{1}

U_{1}

V_{2}

\displaystyle A=\left(\begin{array}[]{cc}A_{3}&A^{-}\\ A^{+}&-A_{3}\end{array}\right),\quad A^{\pm}=A_{1}\pm iA_{2}.

\displaystyle A=\left(\begin{array}[]{cc}A_{3}&A^{-}\\ A^{+}&-A_{3}\end{array}\right),\quad A^{\pm}=A_{1}\pm iA_{2}.

A_{t} + N \land A_{xx} + J (2 u A_{x} + u_{x} A) + (B_{2} A)_{x}

A_{t} + N \land A_{xx} + J (2 u A_{x} + u_{x} A) + (B_{2} A)_{x}

u_{t} + 2 u u_{x} + (B_{2} u)_{x} - (ln A_{3})_{xxx} - {[(ln A_{3})_{x}]^{2}}_{x}

B_{2} = δ (A_{1 x} A_{2} - A_{1} A_{2 x}) - A_{3}^{2} χ_{x} = 0.5 i δ (A_{x}^{+} A^{-} - A^{+} A_{x}^{-}) - A_{3}^{2} u .

B_{2} = δ (A_{1 x} A_{2} - A_{1} A_{2 x}) - A_{3}^{2} χ_{x} = 0.5 i δ (A_{x}^{+} A^{-} - A^{+} A_{x}^{-}) - A_{3}^{2} u .

A_{1 t} + A_{2 xx} + (B_{2} A_{1})_{x}

A_{1 t} + A_{2 xx} + (B_{2} A_{1})_{x}

A_{2 t} - A_{1 xx} + (B_{2} A_{2})_{x}

A_{3 t} + 2 A_{3 x} χ_{x} + A_{3} χ_{xx} + (B_{2} A_{3})_{x}

χ_{t} - (ln A_{3})_{xx} - [(ln A_{3})_{x}]^{2} + χ_{x}^{2} + B_{2} χ_{x}

A_{1 t} + A_{2 xx} + (B_{2} A_{1})_{x}

A_{1 t} + A_{2 xx} + (B_{2} A_{1})_{x}

A_{2 t} - A_{1 xx} + (B_{2} A_{2})_{x}

A_{3 t} + 2 A_{3 x} u + A_{3} u_{x} + (B_{2} A_{3})_{x}

u_{t} - (ln A_{3})_{xxx} - {[(ln A_{3})_{x}]^{2}}_{x} + 2 u u_{x} + (B_{2} u)_{x}

i A_{t}^{+} + A_{xx}^{+} + i (B_{2} A^{+})_{x}

i A_{t}^{+} + A_{xx}^{+} + i (B_{2} A^{+})_{x}

A_{3 t} + 2 A_{3 x} χ_{x} + A_{3} χ_{xx} + (B_{2} A_{3})_{x}

χ_{t} - (ln A_{3})_{xx} - {[(ln A_{3})_{x}]^{2}} + χ_{x}^{2} + B_{2} χ_{x}

i A_{t}^{+} + A_{xx}^{+} + i (B_{2} A^{+})_{x}

i A_{t}^{+} + A_{xx}^{+} + i (B_{2} A^{+})_{x}

A_{3 t} + 2 A_{3 x} u + A_{3} u_{x} + (B_{2} A_{3})_{x}

u_{t} - (ln A_{3})_{xxx} - {[(ln A_{3})_{x}]^{2}}_{x} + 2 u u_{x} + (B_{2} u)_{x}

B_{2} = 0.5 i δ (A_{x}^{+} A^{-} - A^{+} A_{x}^{-}) - A_{3}^{2} u = 0.5 i δ (A_{x}^{+} A^{-} - A^{+} A_{x}^{-}) - A_{3}^{2} χ_{x} .

B_{2} = 0.5 i δ (A_{x}^{+} A^{-} - A^{+} A_{x}^{-}) - A_{3}^{2} u = 0.5 i δ (A_{x}^{+} A^{-} - A^{+} A_{x}^{-}) - A_{3}^{2} χ_{x} .

Ψ_{x}

Ψ_{x}

Ψ_{t}

U_{1}

U_{1}

V_{2}

K

K

K_{1}

A = δ A^{+} A_{x}^{-} + A_{3} (A_{3 x} - i u_{x} A_{3}) .

A = δ A^{+} A_{x}^{-} + A_{3} (A_{3 x} - i u_{x} A_{3}) .

A^{+} = w e^{\frac{i}{ϵ} S},

A^{+} = w e^{\frac{i}{ϵ} S},

A_{3} = 1 - δ w .

A_{3} = 1 - δ w .

w_{t} + 2 (S_{x} w)_{x} - [(A_{3}^{2} χ_{x} + δ w S_{x}) w]_{x}

w_{t} + 2 (S_{x} w)_{x} - [(A_{3}^{2} χ_{x} + δ w S_{x}) w]_{x}

S_{t} + S_{x}^{2} - S_{x} [A_{3}^{2} χ_{x} + δ w S_{x}]_{x}

χ_{t} + χ_{x}^{2} - χ_{x} [A_{3}^{2} χ_{x} + δ w S_{x}]

w_{t} + 2 (S_{x} w)_{x} - [(A_{3}^{2} u + δ w S_{x}) w]_{x}

w_{t} + 2 (S_{x} w)_{x} - [(A_{3}^{2} u + δ w S_{x}) w]_{x}

S_{t} + S_{x}^{2} - S_{x} [A_{3}^{2} u + δ w S_{x}]_{x}

u_{t} + 2 u u_{x} - [u (A_{3}^{2} u + δ w S_{x})]_{x}

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Taxonomy

TopicsNonlinear Waves and Solitons · Advanced Mathematical Physics Problems · Algebraic structures and combinatorial models

Full text

Dispersionless Limits of Integrable Generalized Heisenberg Ferromagnet Equations

Zhaidary Myrzakulova***Email: [email protected], Gulgassyl Nugmanova†††Email: [email protected], Kuralay Yesmakhanova‡‡‡Email: [email protected],

and Ratbay Myrzakulov§§§Email: [email protected]

Eurasian International Center for Theoretical Physics and

Department of General & Theoretical Physics,

Eurasian National University, Astana, 010008, Kazakhstan

Abstract

This paper is a continuation of our previous work in which we studied a dispersionless limits of some integrable spin systems. Now, we shall present dispersionless limits of some integrable generalized Heisenberg ferromagnet equations

1 Introduction

One of classical equations integrable through inverse scattering transform is the famous Heisenberg ferromagnet equation (HFE) [1]-[2]

[TABLE]

where

[TABLE]

The Lax representation (LR) of the HFE reads as

[TABLE]

where

[TABLE]

Here

[TABLE]

After discovery the integrability of the HFE were constructed several class integrable and nonintegrable generalized HFE in 1+1 and 2+1 dimensions (see e.g. [15]-[50] and references therein). Integrable dispersionless equations play important role in modern physics and mathematics. In this context, dispersionless limits of some integrable spin systems were found [51]-[52]. In the present paper, we intend to consider dispersionless limits of some integrable generalized Heisenberg ferromagnet equations, e.g., the Gerdjikov-Mikhailov-Valchev equation (GMVE).

The paper is organized as follows. Next section introduces the so-called M-? equation and its Lax representation (LR). In section 3, we present the dispersionless limit of this equation. The dispersionless limit of the GMVE is presented in section 4. Section 5 contains some further discussion and remarks.

2 M-CI equation

Consider the following M-CI equation (which is a spin system with the self-consistent potential)

[TABLE]

where ${\bf A}=(A_{1},A_{2},A_{3})$ is a unit spin vector that is ${\bf A}^{2}=\delta(A_{1}^{2}+A_{2}^{2})+A_{3}^{2}=1$ ( $\delta^{2}=1$ ), $u=\chi_{x}$ is a real function (scalar potential), ${\bf N}=(0,0,-1)$ , $J=diag(0,0,1)$ , $A^{\pm}=A_{1}\pm iA_{2}$ and

[TABLE]

In components, the M-CI equation takes the form

[TABLE]

or

[TABLE]

We can write the M-CI equation also in the following form

[TABLE]

or

[TABLE]

where $A^{\pm}=A_{1}\pm iA_{2}$ and

[TABLE]

The M-CI equation is integrable. It LR has the form

[TABLE]

where

[TABLE]

Here

[TABLE]

where

[TABLE]

We note that to get this LR of the M-CI equation, we have used the corresponding LR of the GMVE [15].

3 M-CII equation

Now we want to derive the dispersionless limit of the M-CI equation, using, for example, the equations (19)-(21). Consider the transformation

[TABLE]

where

[TABLE]

Substituting the expression (33) into (19)-(21), in the dispersionless limit, we get the following M-CII equation

[TABLE]

or

[TABLE]

We can write this M-CII equation also in the following form

[TABLE]

These three forms of the M-CII equation are the equivalent forms of the dispersionless limit of the M-CI equation. Note that in the above equations, $A_{3}$ is given by the expression (34).

4 Dispersionless limit of GMVE

Consider the GMVE [15]

[TABLE]

where

[TABLE]

and

[TABLE]

Let us now introduce the Madelung tranformations

[TABLE]

Then from (46) we have

[TABLE]

Substituting the expressions (48)-(49) into the GMVE (44)-(45), in the dispersionless limit, we obtain the following M-CIII equation

[TABLE]

or

[TABLE]

where $v=S_{x}$ and

[TABLE]

Thus the M-CIII equation is the dispersionless limit of the GMVE. The LR of the M-CIII equation reads as

[TABLE]

or

[TABLE]

where

[TABLE]

5 Conclusion

In this paper, we have considered some generalized Heisenberg ferromagnet equations which are integrable by IST method. The dispersionless limits of these equations were found. Also we have shown that the M-CI equation is equivalent to the GMVE.

6 Acknowledgements

This work was supported in part by the Ministry of Edication and Science of Kazakhstan under grant 0118RK00935 as well as by grant 0118RK00693.

Bibliography52

The reference list from the paper itself. Each links out to its DOI / PubMed record.

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