# Integrable motion of anisotropic space curves and surfaces induced by the
Landau-Lifshitz equation

Zhaidary Myrzakulova1,2111Email<EMAIL_ADDRESS>Gulgassyl
Nugmanova1,2222Email<EMAIL_ADDRESS>Kuralay Yesmakhanova1,2333Email:
<EMAIL_ADDRESS>
and Ratbay Myrzakulov1,2444Email<EMAIL_ADDRESS>
1Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics,
Nur-Sultan, 010009, Kazakhstan
2Eurasian National University, Nur-Sultan, 010008, Kazakhstan

###### Abstract

In this paper, we have studied the geometrical formulation of the Landau-
Lifshitz equation (LLE) and established its geometrical equivalent counterpart
as some generalized nonlinear Schrödinger equation. When the anisotropy
vanishes, from this result we obtain the well-known results corresponding for
the isotropic case, i.e. to the Heisenberg ferromagnet equation and the
focusing nonlinear Schrödinger equation. The relations between the LLE and the
differential geometry of space curves in the local and in nonlocal cases are
studied. Using the well-known Sym-Tafel formula, the soliton surfaces induced
by the LLE are briefly considered.

Key words: Integrable equations, Nonlinear Schrödinger equation, Heisenberg
ferromagnet equation, Landau-Lifshitz equation, space curves, soliton
solution, soliton surfaces, nonlocal integrable equations.

###### Contents

  1. 1 Introduction
  2. 2 Landau-Lifshitz equation
    1. 2.1 Equation
    2. 2.2 Lax representation
      1. 2.2.1 Elliptic Lax pair
      2. 2.2.2 Polynomial $4\times 4$ Lax pair
      3. 2.2.3 Polynomial $6\times 6$ Lax pair
    3. 2.3 Hamiltonian structure
    4. 2.4 Different parametrizations
      1. 2.4.1 $(\theta-\varphi)$ \- form
      2. 2.4.2 $w$ \- form
      3. 2.4.3 $(p-q)$ form
      4. 2.4.4 LLE as the Madelung equation
      5. 2.4.5 $\tau_{n}$ \- parametrization
    5. 2.5 Limiting cases
      1. 2.5.1 HFE
      2. 2.5.2 $J_{1}=J_{2}\neq J_{3}$ case
      3. 2.5.3 Sine-Gordon equation
      4. 2.5.4 NLSE
  3. 3 Integrable deformations of anisotropic space curves
  4. 4 Nonlocal Landau-Lifshitz equation
  5. 5 Nonlocal space curves and Nonlocal Serret-Frenet equation
  6. 6 Gauge equivalence between the LLE and NLS-like equation
  7. 7 Anisotropic soliton surfaces induced by the LLE
  8. 8 Some generalizations of the LLE
    1. 8.1 Landau-Lifshitz-Schrödinger equation
    2. 8.2 Myrzakulov-XCIX equation
    3. 8.3 LLE with the self-consistent variable anisotropy
    4. 8.4 Integrable LLE with the external magnetic field
    5. 8.5 Ishimori equation
    6. 8.6 Myrzakulov-IX equation
    7. 8.7 Myrzakulov-VIII equation
    8. 8.8 Myrzakulov-XXXIV equation
    9. 8.9 Myrzakulov-I equation
    10. 8.10 Landau-Lifshitz-Gilbert equation
    11. 8.11 Landau-Lifshitz-Bloch equation
  9. 9 LLE as the Madelung equation. Hydrodynamical formulations
  10. 10 Quantum Heisenberg model
  11. 11 Dispersionless LLE
  12. 12 Self-similar solution of the LLE
  13. 13 Lattice LLE
  14. 14 Conclusion
  15. 15 Acknowledgements

##  1 Introduction

Nonlinear dynamics of one-dimensional anisotropic ferromagnet can be described
by the Landau-Lifshitz equation (LLE) [1]

${\bf S}_{t}={\bf S}\wedge{\bf S}_{xx}+{\bf S}\wedge J{\bf S},$ (1.1)

where $J=diag(J_{1},J_{2},J_{3})$ and $J_{1}\leq J_{2}\leq J_{3}$, ${\bf
S}=(S_{1},S_{2},S_{3})$ is the unit spin vector and ${\bf
S}^{2}=S_{3}^{2}+\epsilon S_{1}^{2}+\epsilon S_{2}^{2}=1$. Here $\epsilon=\pm
1$, which gives rise to two different models corresponding to the $su(2)$
($\epsilon=+1)$ and $su(1,1)$ ($\epsilon=-1)$ cases. The LLE is integrable
[2]-[3]. It admits several integrable and non-integrable extensions and
reductions in 1+1, 2+1 and 3+1 dimensions. The supersymmetric isotropic LLE is
one of such generalizations in 1+1 dimensions [5]. In 2+1 dimensions, the LLE
admits the following integrable spin systems: Ishimori equation [6],
Myrzakulov-I (M-I) equation [7], Myrzakulov-IX (M-IX) equation [8],
Myrzakulov-VIII (M-VIII) equation [9] and so on. The LLE has some interesting
particular and limiting cases [10]-[14]. For example, in the case
$J_{1}=J_{2}=J_{3}$ that is in the isotropic case, from (1.1) we obtain the
following Heisenberg ferromagnet equation (HFE)

${\bf S}_{t}={\bf S}\wedge{\bf S}_{xx}.$ (1.2)

At the same time, in the case $J_{1}=J_{2}$ it is called the LLE with an easy
plan. Also it is well-known that the nonlinear Schrödinger equation (NLSE) and
sine-Gordon equation (SGE) are some limiting cases of LLE under the choice of
the spectral parameter. In this sense, the LLE is a universal integrable model
in 1+1 dimensions. The connection of the LLE (as well as the Bloch equation)
with some types of two-level quantum systems was established in [15]-[16].

The gauge equivalence between some integrable equations is an important tool
in soliton theory [17]. Gauge equivalence of some integrable systems are well
understood both in classical and discrete versions. In particular, it was
shown that the LLE (1.1) is gauge equivalent to some nonlinear Schrödinger-
like equation [18]-[19]. Gauge equivalence of the LLE with an easy plan and
the nonlinear Schrödinger equation was considered in [20]-[21].

There are another type of equivalences between integrable equations which is
called the geometrical equivalence or Lakshmanan equivalence [22] (see, also
[23]-[33] and references therein). The geometrical equivalence is related with
the differential geometry of curves and surfaces. The relation between the
differential geometry of curves and surfaces and integrable systems were
investigated by many authors [23]-[59]. These results give the geometrical
formulation of integrable systems.

Recently, the integrable nonlocal soliton equations were introduced and
studied [60]-[68]. In particular, the nonlocal HFE were studied [69]-[70].
Other interesting problems related with the subjects of this paper are
integrable discrete HFE [71]-[72].

In our previous works, we have studied some integrable and nonintegrable
continuous classical spin chains. In this paper, we continue our
investigations such types of spin chains, namely, we study the LLE. If
exactly, in this paper we study the LLE (1.1) from the geometrical point of
view. We construct the integrable motion of the anisotropic space curves and
anisotropic surfaces induced by the LLE. Also we present the nonlocal versions
of the LLE and its equivalent nonlinear Schrödinger-like equation. The
integrable deformations of nonlocal space curves and the corresponding Serret-
Frenet equation are discussed.

The paper is organized as follows. In Sec. 2, we briefly review some well-
known results on the LLE. The geometrical formulation of the LLE in terms of
anisotropic space curves is studied in Sec. 3. The nonlocal version of the LLE
is presented in Sec. 4. Next, in Sec. 5 we discuss the nonlocal space curves
formalism including the corresponding nonlocal Serret-Frenet equation. In Sec.
6, the gauge equivalence between the LLE and the nonlinear Shcrödinger - like
equation is discussed. In Sec. 7, we briefly consider the anisotropic soliton
surfaces by using the famous Sym-Tafel formula. Some generalizations of the
LLE are presented in Sec. 8. The hydrodynamical formulation of the LLE is
given in Sec. 9. The quantum Heisenberg model is considered in Sec. 10. The
dispersionless LLE is presented in Se. 11 for some particular case. In Sec.
12, the self-similar solutions of the LLE is briefly discussed. The lattice
(discrete) version of the LLE is considered in Sec. 13. We give conclusions in
the last section.

##  2 Landau-Lifshitz equation

In this section we give some basic informations on the LLE like the equation
of motion, three types of Lax pairs, Hamilton formulation, different
parametrizations and limiting cases.

###  2.1 Equation

The LLE has been introduced in [1] and describes the dynamics of the
orientation of the magnetization (or spin) in ferromagnetic materials. It
constitutes nowadays a fundamental tool in the magnetic recording industry,
due to its applications to ferromagnets. The LLE has the form [10]

${\bf S}_{t}={\bf S}\wedge{\bf S}_{xx}+{\bf S}\wedge J{\bf S},$ (2.1)

where $J=diag(J_{1},J_{2},J_{3})$ and $J_{1}\leq J_{2}\leq J_{3}$. In
components it takes the form

$\displaystyle S_{1t}$ $\displaystyle=$ $\displaystyle
S_{2}S_{3xx}-S_{2xx}S_{3}+S_{2}J_{3}S_{3}-J_{2}S_{2}S_{3},$ (2.2)
$\displaystyle S_{2t}$ $\displaystyle=$ $\displaystyle
S_{3}S_{1xx}-S_{3xx}S_{1}+S_{3}J_{1}S_{1}-J_{3}S_{3}S_{1},$ (2.3)
$\displaystyle S_{3t}$ $\displaystyle=$ $\displaystyle
S_{1}S_{2xx}-S_{1xx}S_{2}+S_{1}J_{2}S_{2}-J_{1}S_{1}S_{2},$ (2.4)

or

$\displaystyle S_{1t}$ $\displaystyle=$ $\displaystyle
S_{2}S_{3xx}-S_{2xx}S_{3}-J_{23}S_{2}S_{3},$ (2.5) $\displaystyle S_{2t}$
$\displaystyle=$ $\displaystyle S_{3}S_{1xx}-S_{3xx}S_{1}-J_{31}S_{3}S_{1},$
(2.6) $\displaystyle S_{3t}$ $\displaystyle=$ $\displaystyle
S_{1}S_{2xx}-S_{1xx}S_{2}-J_{12}S_{1}S_{2},$ (2.7)

where $J_{ij}=J_{i}-J_{j},\quad I_{ij}=J_{i}+J_{j}$. We also use the following
form of the LLE

$\displaystyle iS_{t}^{+}$ $\displaystyle=$ $\displaystyle
S^{+}S_{3xx}-S^{+}_{xx}S_{3}+J_{3}S_{3}S^{+}-\frac{1}{2}S_{3}[J_{1}(S^{+}+S^{-})+J_{2}(S^{+}-S^{-})],$
(2.8) $\displaystyle iS^{-}_{t}$ $\displaystyle=$ $\displaystyle
S^{-}_{xx}S_{3}-S^{-}S_{3xx}-J_{3}S_{3}S^{-}+\frac{1}{2}S_{3}[J_{1}(S^{+}+S^{-})-J_{2}(S^{+}-S^{-})],$
(2.9) $\displaystyle iS_{3t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(S^{-}S^{+}_{xx}-S^{-}_{xx}S^{+})-i(J_{2}-J_{1})S_{1}S_{2},$
(2.10)

or

$\displaystyle iS_{t}^{+}$ $\displaystyle=$ $\displaystyle
S^{+}S_{3xx}-S^{+}_{xx}S_{3}+J_{3}S_{3}S^{+}-\frac{1}{2}S_{3}(I_{21}S^{+}-J_{21}S^{-}),$
(2.11) $\displaystyle iS^{-}_{t}$ $\displaystyle=$ $\displaystyle
S^{-}_{xx}S_{3}-S^{-}S_{3xx}-J_{3}S_{3}S^{-}-\frac{1}{2}S_{3}(J_{21}S^{+}-I_{21}S^{-}),$
(2.12) $\displaystyle iS_{3t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(S^{-}S^{+}_{xx}-S^{-}_{xx}S^{+})-\frac{1}{4}J_{12}(S^{+2}-S^{-2}).$
(2.13)

The matrix LLE reads as

$iS_{t}=\frac{1}{2}[S,S_{xx}+S_{J}],$ (2.14)

where

$S=\sum_{k=1}^{3}S_{k}\sigma_{k}=\begin{pmatrix}S_{3}&S^{-}\\\
S^{+}&-S_{3}\end{pmatrix},\quad S^{2}=I,\quad S^{\pm}=S_{1}\pm iS_{2}$ (2.15)
$S_{J}=\sum_{k=1}^{3}J_{k}S_{k}\sigma_{k}=\begin{pmatrix}J_{3}S_{3}&J_{1}S_{1}-iJ_{2}S_{2}\\\
J_{1}S_{1}+iJ_{2}S_{2}&-J_{3}S_{3}\end{pmatrix}.$ (2.16)

Here $\sigma_{j}$ are Pauli matrices:

$\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$
(2.17)

We note that the LLE is a nonlinear dispersive partial differential equation.
The corresponding dispersion relation reads as [11]

$\omega(k)=\pm\sqrt{|k|^{4}+(J_{21}+J_{23})|k|^{2}+J_{21}J_{23}},$ (2.18)

what corresponds to the solutions of the form $e^{i(kx-\omega t)}$ that is to
linear sinusoidal waves of frequency $\omega$ and wavenumber $k$.

###  2.2 Lax representation

There are several forms of the Lax representations of the LLE: elliptic,
$4\times 4$ polynomial and $6\times 6$ polynomial Lax pairs.

####  2.2.1 Elliptic Lax pair

The elliptic Lax representation for the LLE is given by [2]-[3]

$\displaystyle\Phi_{x}$ $\displaystyle=$ $\displaystyle U_{1}\Phi,$ (2.19)
$\displaystyle\Phi_{t}$ $\displaystyle=$ $\displaystyle V_{1}\Phi,$ (2.20)

where

$\displaystyle U_{1}$ $\displaystyle=$
$\displaystyle-i\sum_{k=1}^{3}u_{k}S_{k}\sigma_{k},$ (2.21) $\displaystyle
V_{1}$ $\displaystyle=$ $\displaystyle
2i\sum_{k=1}^{3}\frac{u_{1}u_{2}u_{3}}{u_{k}}S_{k}\sigma_{k}-i\sum_{a,b,c=1}^{3}u_{a}\epsilon_{abc}S_{b}S_{cx}\sigma_{a}.$
(2.22)

Here

$\displaystyle u_{1}=\frac{\rho}{sn(\lambda,k)},\quad u_{2}=\frac{\rho
dn(\lambda,k)}{sn(\lambda,k)},\quad u_{3}=\frac{\rho
cn(\lambda,k)}{sn(\lambda,k)}$ (2.23)

and

$\displaystyle
k=\sqrt{\frac{J_{2}-J_{1}}{J_{3}-J_{1}}},\quad\rho=\frac{1}{2}\sqrt{J_{3}-J_{1}},\quad
0\leq k\leq 1,\quad\rho\geq 0.$ (2.24)

Note that $sn(\lambda,k),\quad cn(\lambda,k),\quad dn(\lambda,k)$ are Jacobi
elliptic functions having the following limit cases

$\displaystyle sn(\lambda,0)$ $\displaystyle=$ $\displaystyle\sin\lambda,\quad
cn(\lambda,0)=\cos\lambda,\quad dn(\lambda,0)=1,$ (2.25) $\displaystyle
sn(\lambda,1)$ $\displaystyle=$ $\displaystyle\tanh\lambda,\quad
cn(\lambda,1)=dn(\lambda,1)=\frac{1}{\cosh\lambda}$ (2.26)

and

$\displaystyle u_{a}^{2}-u_{b}^{2}=\frac{1}{4}(J_{b}-J_{a}).$ (2.27)

####  2.2.2 Polynomial $4\times 4$ Lax pair

The $4\times 4$ polynomial Lax pair for the LLE has the form [73]-[75]

$\displaystyle U_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}A_{1}(\lambda+\widetilde{J}),$ (2.28) $\displaystyle
V_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\frac{1}{2}A_{1}-[A,A_{1x}]+\frac{1}{2}A_{2J})(\lambda+\widetilde{J}),$
(2.29)

where

$\displaystyle A_{1}=\frac{1}{2}\begin{pmatrix}0&S_{1}&S_{2}&S_{3}\\\
-S_{1}&0&S_{3}&-S_{2}\\\ -S_{2}&-S_{3}&0&S_{1}\\\
-S_{3}&S_{2}&-S_{1}&0\end{pmatrix},\quad
A_{2J}=\frac{1}{2}\begin{pmatrix}0&J_{1}S_{1}&J_{2}S_{2}&-J_{3}S_{3}\\\
-J_{1}S_{1}&0&J_{3}S_{3}&J_{2}S_{2}\\\
-J_{2}S_{2}&-J_{3}S_{3}&0&-J_{1}S_{1}\\\
J_{3}S_{3}&-J_{2}S_{2}&J_{1}S_{1}&0\end{pmatrix},$
$\displaystyle\widetilde{J}=\begin{pmatrix}-J_{1}-J_{2}+J_{3}&0&0&0\\\
0&-J_{1}+J_{2}-J_{3}&0&0\\\ 0&0&J_{1}-J_{2}-J_{3}&0\\\
0&0&0&J_{1}+J_{2}+J_{3}\end{pmatrix}.$ (2.30)

####  2.2.3 Polynomial $6\times 6$ Lax pair

The $6\times 6$ polynomial Lax pair for the LLE has the form [73]-[75]

$\displaystyle U_{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2}\begin{pmatrix}M({\bf S})&0\\\
0&0\end{pmatrix}+\frac{1}{2}\begin{pmatrix}0&0\\\ &-M(J{\bf
S})\end{pmatrix}+\frac{1}{2}ad_{\widetilde{J}}\begin{pmatrix}M({\bf S})&0\\\
0&0\end{pmatrix},$ (2.31) $\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda^{2}}{4}\begin{pmatrix}M({\bf S})&0\\\
0&0\end{pmatrix}+\frac{\lambda}{2}\\{\begin{pmatrix}M({\bf S}\times{\bf
S}_{x})&0\\\
&0\end{pmatrix}+\frac{1}{2}ad_{\widetilde{J}}\begin{pmatrix}M({\bf S})&0\\\
0&0\end{pmatrix}\\}+K,$ (2.32)

where $ad_{f}g=[f,g]$ and

$\displaystyle M({\bf S})=\begin{pmatrix}0&S_{3}&-S_{2}\\\ -S_{3}&0&S_{1}\\\
S_{2}&-S_{1}&0\end{pmatrix},\quad\widetilde{J}=\begin{pmatrix}0&J\\\
J&0\end{pmatrix},\quad[M({\bf A}),M({\bf B})]=-M({\bf A}\times{\bf B}),$
(2.33) $\displaystyle K=-\frac{1}{4}\begin{pmatrix}M(J^{2}{\bf S})&0\\\
0&2M(J({\bf S}\times{\bf
S}_{x}))\end{pmatrix}+\frac{1}{4}ad_{J}\begin{pmatrix}M({\bf S}\times{\bf
S}_{x})&0\\\ 0&M(J{\bf S})\end{pmatrix}.$ (2.34)

From the compatibility condition $U_{1t}-V_{1x}+[U_{1},V_{1}]=0$ we obtain

$\displaystyle M({\bf S})_{t}+[M({\bf S}),M({\bf S})_{xx}]+[M({\bf S},M(J{\bf
S})]=0,$ (2.35)

which is the LLE.

###  2.3 Hamiltonian structure

The LLE can be written in the Hamiltonian form as [10]

$\displaystyle{\bf S}_{t}=\\{H,{\bf S}\\},$ (2.36)

where the Hamilton function is given by

$\displaystyle H=\frac{1}{2}\int\left({\bf
S}_{x}^{2}-J_{1}S_{1}^{2}-J_{2}S_{2}^{2}-J_{3}S_{3}^{2}\right)dx.$ (2.37)

The Poisson bracket for the two functionals $A,B$ has the form

$\displaystyle\\{A,B\\}=-\sum_{a,b,c=1}^{3}\epsilon_{abc}\int\frac{\delta
A}{\delta S_{a}}\frac{\delta B}{\delta S_{b}}S_{c}dxdy.$ (2.38)

In particular, for the components of the spin vector the corresponding Poisson
brackets read as

$\displaystyle\\{S_{a}(x),S_{b}(y)\\}=-\epsilon_{abc}S_{c}(x)\delta(x-y).$
(2.39)

###  2.4 Different parametrizations

The LLE can be written in the different forms that follow from the different
parametrizations of the spin vector ${\bf S}$. In this subsection we going to
present some of these forms of the LLE.

####  2.4.1 $(\theta-\varphi)$ \- form

Consider the following angle parametrization of the LLE

$\displaystyle S^{+}=e^{i\varphi}\sin\theta,\quad
S^{-}=e^{-i\varphi}\sin\theta,\quad S_{3}=\cos\theta,$ (2.40)

where $\theta(x,t),\varphi(x,t)$ are some real functions. Then the LLE takes
the form

$\displaystyle\theta_{t}+\varphi_{xx}\sin\theta+2\varphi_{x}\theta_{x}\cos\theta+J_{2}\sin\theta\sin\varphi$
$\displaystyle=$ $\displaystyle 0,$ (2.41)
$\displaystyle\varphi_{t}\sin\theta-\theta_{xx}+(\varphi_{x}^{2}+J_{3}-J_{1}\cos\varphi)\sin\theta\cos\theta$
$\displaystyle=$ $\displaystyle 0.$ (2.42)

####  2.4.2 $w$ \- form

Let us introduce the new complex function $w$ as

$\displaystyle w=\frac{S^{+}}{1+S_{3}}=\tan\frac{\theta}{2}e^{i\varphi}.$
(2.43)

Then in terms of this function, the LLE takes the form

$\displaystyle
iw_{t}+w_{xx}-\frac{2\bar{w}w_{x}^{2}}{1+|w|^{2}}-\frac{1}{2(1+|w|^{2})}\\{[(2J_{3}-I_{21})(1-|w|^{2}+$
$\displaystyle J_{12}(w^{2}-\bar{w}^{2})]w+J_{21}(1-|w|^{2})\bar{w}\\}=0.$
(2.44)

####  2.4.3 $(p-q)$ form

We now consider the following parametrization of the spin vector [76]

$\displaystyle S_{1}=p(q^{2}-1)+q,\quad S_{2}=ip(q^{2}+1)+iq,\quad
S_{3}=2pq+1.$ (2.45)

In terms of these new variables $(p-q)$, the LLE reads as

$\displaystyle p_{t_{1}}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial q}=p_{xx}-2(p^{2}q_{x})_{x},$ (2.46) $\displaystyle q_{t_{1}}$
$\displaystyle=$ $\displaystyle\frac{\partial H}{\partial
p}=-q_{xx}-2pq_{x}^{2},$ (2.47)

where

$\displaystyle
H=p_{x}q_{x}-p^{2}(q_{x}^{2}+r(q))-\frac{1}{12}r^{{}^{\prime\prime}}(q)-\frac{1}{2}pr^{{}^{\prime}}(q),\quad
r^{V}=0,\quad r=-\frac{1}{4}{\bf S}\cdot J{\bf S},\quad t_{1}=it.$ (2.48)

####  2.4.4 LLE as the Madelung equation

To study some properties of some nonlinear differential equations is very
useful their hydrodynamical forms. The same conclusion is correct for the LLE.
Let us find its hydrodynamical form. To derive the hydrodynamical form of the
LLE, we consider the following Madelung like transformation [14]

$\displaystyle S^{+}=\sqrt{1-S_{3}^{2}}e^{-i(\phi-0.5\pi)}.$ (2.49)

Then the LLE (1.1) takes the form

$\displaystyle\phi_{t}+\left(\frac{S_{3x}}{1-S_{3}^{2}}\right)_{x}-\frac{S_{3}S_{3x}^{2}}{(1-S_{3}^{2})^{2}}+\left(\phi_{x}^{2}-J_{23}+J_{21}\sin^{2}\phi\right)S_{3}$
$\displaystyle=$ $\displaystyle 0,$ (2.50) $\displaystyle
S_{3t}-\left((1-S_{3}^{2})\phi_{x}\right)_{x}+0.5J_{21}(1-S_{3}^{2})\sin^{2}2\phi$
$\displaystyle=$ $\displaystyle 0.$ (2.51)

This is the Madelung equation form of the LLE (1.1). Note that this Madelung
form will be essential in the study of solutions of the LLE.

####  2.4.5 $\tau_{n}$ \- parametrization

Let us present one of very interesting parametrizations of the spin vector.
Let the spin matrix $S$ has the form

$\displaystyle S=g^{-1}\sigma_{3}g,$ (2.52)

where $g$ satisfies the following equations

$\displaystyle g_{x}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&\tau_{n-1}\tau_{n}^{-1}\\\
\tau_{n+1}\tau_{n}^{-1}&0\end{pmatrix}g,$ (2.53) $\displaystyle g_{t}$
$\displaystyle=$ $\displaystyle
i\begin{pmatrix}\tau_{n-1}\tau_{n+1}\tau_{n}^{-2}&-(\tau_{n-1}\tau_{n}^{-1})_{x}\\\
(\tau_{n+1}\tau_{n}^{-1})_{x}&-\tau_{n-1}\tau_{n+1}\tau_{n}^{-2}\end{pmatrix}g.$
(2.54)

The compatibility condition of this set of linear equations gives

$\displaystyle(D_{t}-D_{x}^{2})\tau_{n+1}\cdot\tau_{n}$ $\displaystyle=$
$\displaystyle 0,$ (2.55) $\displaystyle
D_{x}^{2}\tau_{n}\cdot\tau_{n}-2\tau_{n+1}\cdot\tau_{n-1}$ $\displaystyle=$
$\displaystyle 0.$ (2.56)

It is the second member of the TL hierarchy (see e.g. [77] and references
therein). On the other hand, the spin matrix $S$ (2.52) obeys the HFE (as
$t\rightarrow it$)

$2iS_{t}=[S,S_{xx}].$ (2.57)

At the same time the following functions

$\displaystyle r=\tau_{n+1}\tau_{n}^{-1},\quad q=\tau_{n-1}\tau_{n}$ (2.58)

solves the NLSE (2.61)-(2.62).

###  2.5 Limiting cases

In this subsection we want to present some well-known particular (limiting)
cases of the LLE (see, e.g. Refs. [10],[14]).

####  2.5.1 HFE

In the isotropic case $J_{1}=J_{2}=J_{3}$, the LLE turn to the HFE

${\bf S}_{t}={\bf S}\wedge{\bf S}_{xx}.$ (2.59)

The corresponding Lax pair of the HFE has the form

$U_{2}=-i\lambda S,\quad V_{2}=-2i\lambda^{2}S+\lambda SS_{x}.$ (2.60)

It is well-known that the HFE (1.2) or (2.59) is gauge [17] and geometrically
[22] equivalent to the following NLSE

$\displaystyle iq_{t}+q_{xx}-2rq^{2}$ $\displaystyle=$ $\displaystyle 0,$
(2.61) $\displaystyle ir_{t}-r_{xx}+2qr^{2}$ $\displaystyle=$ $\displaystyle
0$ (2.62)

with the Lax pair

$U_{1}=-i\lambda\sigma_{3}+Q,\quad V_{1}=-2i\lambda^{2}\sigma_{3}+\lambda
V_{1}+V_{0}.$ (2.63)

Here

$r=-\epsilon\bar{q},\quad\sigma_{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad Q=\begin{pmatrix}0&q\\\ r&0\end{pmatrix},\quad
V_{1}=2Q,\quad V_{0}=i\begin{pmatrix}-rq&q_{x}\\\ -r_{x}&rq\end{pmatrix},$
(2.64)

where $bar$ stands for the complex conjugation and $\epsilon=\pm 1$ signals
the focusing $(+)$ and defocusing $(-)$ nonlinearity. Let us introduce the new
variables as

$u=rq,\quad v=-\ln q.$ (2.65)

Then the NLSE takes the form

$\displaystyle iu_{t}-u_{xx}-2(uv_{x})_{x}$ $\displaystyle=$ $\displaystyle
0,$ (2.66) $\displaystyle iv_{t}+v_{xx}-v_{x}^{2}+2u$ $\displaystyle=$
$\displaystyle 0$ (2.67)

which is the Toda lattice (see e.g. Refs. [76]-[77] and references therein).

####  2.5.2 $J_{1}=J_{2}\neq J_{3}$ case

When $J_{1}=J_{2}\neq J_{3}$, the equation (1.1) is called the LLE for a spin
chain with an easy plan.

####  2.5.3 Sine-Gordon equation

To get the sine-Gordon equation (SGE) from the LLE, we assume that [10]

$\displaystyle{\bf S}^{2}$ $\displaystyle=$ $\displaystyle R^{2},\quad
S_{1}=-\frac{\beta\pi}{2},\quad
S_{2}=\sqrt{R^{2}-\frac{\beta^{2}\pi^{2}}{4}}\sin\frac{\varphi\pi}{2},\quad
S_{3}=\sqrt{R^{2}-\frac{\beta^{2}\pi^{2}}{4}}\cos\frac{\varphi\pi}{2},$ (2.68)
$\displaystyle J_{2}$ $\displaystyle=$ $\displaystyle J_{1}+1,\quad
J_{3}=J_{2}+m^{2}R^{-2}=1+J_{1}+m^{2}R^{-2}.$ (2.69)

Then the LLE takes the form

${\bf S}_{t}=R^{-2}{\bf S}\wedge{\bf S}_{xx}+{\bf S}\wedge J{\bf S},$ (2.70)

so that in the limit $R\rightarrow\infty$, it turns to the SGE

$\displaystyle\pi$ $\displaystyle=$ $\displaystyle\varphi_{t},$ (2.71)
$\displaystyle\pi_{t}$ $\displaystyle=$
$\displaystyle\varphi_{xx}-\frac{m^{2}}{\beta}\sin\beta\varphi$ (2.72)

or

$\displaystyle\varphi_{tt}-\varphi_{xx}+\frac{m^{2}}{\beta}\sin\beta\varphi=0.$
(2.73)

####  2.5.4 NLSE

Let us now we derive the NLSE from the LLE. To do that, we assume [10]

$\displaystyle{\bf S}^{2}=1,\quad
S^{+}=\sqrt{2\eta}qe^{2i\kappa\eta^{-1}t},\quad
S_{3}=\sqrt{1-2\eta|q|^{2}},\quad J_{2}=J_{1},\quad
J_{3}=J_{1}-2\kappa\eta^{-1},\quad\kappa>0.$ (2.74)

In this case, from the LLE (1.1) as $\eta\rightarrow 0$ we obtain the NLSE

$\displaystyle iq_{t}+q_{xx}+2\epsilon|q|^{2}q=0.$ (2.75)

##  3 Integrable deformations of anisotropic space curves

In this section, our goal is to present the geometrical formulation of the LLE
in terms of space curves. To the end, we now consider the space curve in the
Euclidean space $R^{3}$. Such space curves governed by the following Serret-
Frenet equation (SFE) and its temporal counterpart equation (see e.g. [22],
[55] and references therein)

$\begin{split}\begin{pmatrix}e_{1}\\\ e_{2}\\\
e_{3}\end{pmatrix}_{x}=C\begin{pmatrix}e_{1}\\\ e_{2}\\\
e_{3}\end{pmatrix}\end{split},$ (3.1) $\begin{split}\begin{pmatrix}e_{1}\\\
e_{2}\\\ e_{3}\end{pmatrix}_{t}=D\begin{pmatrix}e_{1}\\\ e_{2}\\\
e_{3}\end{pmatrix}\end{split},$ (3.2)

where

$C=\begin{pmatrix}0&\kappa&\sigma\\\ -\kappa&0&\tau\\\
-\sigma&-\tau&0\end{pmatrix},\quad D=\begin{pmatrix}0&\omega_{3}&\omega_{2}\\\
-\omega_{3}&0&\omega_{1}\\\ -\omega_{2}&-\omega_{1}&0\end{pmatrix}.$ (3.3)

Here the functions $\kappa$ is called normal curvature, $\sigma$ is called
geodesic curvature, $\tau$ is called geodesic torsion and $\omega_{j}$ are
some real functions. The later functions must be expressed in terms of
$\kappa,\sigma,\tau$ and their derivatives. The compatibility conditions ${\bf
e}_{jxt}={\bf e}_{jtx}$ of the equations (3.1)-(3.2) reads as

$C_{t}-D_{x}+[C,D]=0.$

For the elements this equation gives

$\kappa_{t}=\omega_{3x}-\tau\omega_{2}+\sigma\omega_{1},$ (3.4)
$\sigma_{t}=\omega_{2x}-\kappa\omega_{1}+\tau\omega_{3},$ (3.5)
$\tau_{t}=\omega_{1x}-\sigma\sigma\omega_{3}+\kappa\omega_{2}.$ (3.6)

Now let us we assume that

$\kappa=0,\quad\sigma=r-q,\quad\tau=-i(r+q),\quad r=-\bar{q},$ (3.7)
$\omega_{1}=q_{x}-r_{x}-i(q_{21}+q_{12}),\quad\omega_{2}=-i(r_{x}+q_{x})+q_{21}-q_{12},\quad\omega_{3}=-2rq-2iq_{11}.$
(3.8)

Substituting these expressions of $\kappa,\sigma,\tau,\omega_{j}$ into
(3.4)-(3.6), we obtain the following nonlinear Schrodinger-like equation

$\displaystyle iq_{t}+q_{xx}-2rq^{2}-iR_{12}$ $\displaystyle=$ $\displaystyle
0,$ (3.9) $\displaystyle ir_{t}-r_{xx}+2qr^{2}-iR_{21}$ $\displaystyle=$
$\displaystyle 0,$ (3.10) $\displaystyle q_{11x}-(qq_{21}-rq_{12})$
$\displaystyle=$ $\displaystyle 0,$ (3.11)

where

$\displaystyle R_{12}=q_{12x}+2qq_{11},\quad R_{21}=q_{21x}-2rq_{11}.$ (3.12)

It is the case to mention that the system of equations (3.9)-(3.11) is
integrable in the sense that it admits the following Lax representation
[18]-[19]

$\displaystyle\Psi_{x}$ $\displaystyle=$ $\displaystyle U_{2}\Psi,$ (3.13)
$\displaystyle\Psi_{t}$ $\displaystyle=$ $\displaystyle V_{2}\Psi,$ (3.14)

where

$\displaystyle U_{2}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}0&q\\\
r&0\end{pmatrix},\quad Q=\begin{pmatrix}q_{11}&q_{12}\\\
q_{21}&-q_{11}\end{pmatrix},\quad R=\begin{pmatrix}0&R_{12}\\\
R_{21}&0\end{pmatrix}=Q_{x}-[U_{2},Q],\quad R_{21}=-\bar{R}_{12},$ (3.15)
$\displaystyle V_{2}$ $\displaystyle=$
$\displaystyle-i(U_{2}^{2}+U_{2x})\sigma_{3}+Q=\begin{pmatrix}-irq+q_{11}&iq_{x}+q_{12}\\\
-ir_{x}+q_{21}&irq-q_{11}\end{pmatrix}.$ (3.16)

Let us present here some useful formulas. Let $\Psi$ has the form

$\displaystyle\Psi=\begin{pmatrix}g_{1}&-\bar{g}_{2}\\\
g_{2}&\bar{g}_{1}\end{pmatrix},$ (3.17)

where $g_{j}$ are some complex functions. From (3.13) follows that

$\displaystyle[det(\Psi)]_{x}=(|g_{1}|^{2}+|g_{2}|^{2})_{x}=0,$ (3.18)

that is $|g_{1}|^{2}+|g_{2}|^{2}=c=const$. Usually we assume $c=1$, so that
$|g_{1}|^{2}+|g_{2}|^{2}=1$. Then the function $R$ has the form [18]

$\displaystyle R=\Psi[(\phi S)_{x}-\frac{i}{2}S_{Jx}]\Psi^{-1}$ (3.19)

or

$\displaystyle R=\Psi[(\phi
S)_{x}-\frac{i}{2}S_{Jx}]\Psi^{-1}=iq[J_{1}(2Im(\bar{g}_{2}\bar{g}_{1}))^{2}+J_{2}(2Re(\bar{g}_{2}\bar{g}_{1}))^{2}+J_{3}(|g_{2}|^{2}-|g_{1}|^{2})]-$
$\displaystyle
i[J_{1}(g_{1}^{2}+\bar{g}_{2}^{2})Im(\bar{g}_{1}^{2}q-\bar{g}_{2}^{2}\bar{q})+J_{2}(g_{1}^{2}-\bar{g}_{2}^{2})Re(\bar{g}_{1}^{2}q-\bar{g}_{2}^{2}\bar{q})+2J_{3}Re(\bar{g}_{2}g_{1})(\bar{g}_{2}g_{1}\bar{q}+g_{2}\bar{g}_{1}q)],$
(3.20)

where $\phi=J_{1}S_{1}^{2}+J_{2}S_{2}^{2}+J_{3}S_{3}^{2}$ and we used the
formula $S_{x}=\Psi^{-1}[\sigma_{3},U_{2}]\Psi$. Now from (3.1)-(3.2), taking
into account the last relation, we obtain the following equation for the unit
vector ${\bf e}_{3}$:

${\bf e}_{3t}={\bf e}_{3}\times\left({\bf e}_{3xx}+J{\bf e}_{3}\right),$
(3.21)

which is nothing but the LLE (1.1) after the identification ${\bf
e}_{3}\equiv{\bf S}$.

Theorem 1. The Landau-Lifshitz equation (1.1) is geometrically equivalent to
the nonlinear Schrödinger-like equation (3.9)-(3.11). In the isotropic case,
this geometrical equivalence is reduced to the well-known geometrical
equivalence between the HFE and the NLSE [22].

##  4 Nonlocal Landau-Lifshitz equation

In this section we want to present the nonlocal version of the LLE. It is
well-known that in the nonlocal case, the unit spin vector ${\bf
S}=(S_{1},S_{2},S_{3})$ is the complex-valued vector [69]. The complex-valued
spin vector ${\bf S}$ induced that the unit vectors ${\bf e}_{j}$ become also
complex-valued. It means that the curvature $\kappa(t,x)$, the torsion
$\tau(t,x)$ and $\omega_{j}$ are complex-valued functions [68]. Since in the
nonlocal case, the spin vector ${\bf S}$ is the complex-valued vector
function, so that we may decompose it as ${\bf S}={\bf M}+i{\bf L}$, where
${\bf M}$ and ${\bf L}$ are already real valued vector functions [69]. These
new real vectors satisfy the following relations

$\displaystyle{\bf M}^{2}-{\bf L}^{2}=1,\quad{\bf M}\cdot{\bf L}=0.$ (4.1)

From the LLE (1.1) follows that these real valued vector functions ${\bf M}$
and ${\bf L}$ satisfy the following set of coupled equations:

$\displaystyle{\bf M}_{t}$ $\displaystyle=$ $\displaystyle{\bf M}\wedge{\bf
M}_{xx}-{\bf L}\wedge{\bf L}_{xx}+{\bf M}\wedge J{\bf M}-{\bf L}\wedge J{\bf
L},$ (4.2) $\displaystyle{\bf L}_{t}$ $\displaystyle=$ $\displaystyle{\bf
M}\wedge{\bf L}_{xx}+{\bf L}\wedge{\bf M}_{xx}+{\bf L}\wedge J{\bf M}+{\bf
M}\wedge J{\bf L}.$ (4.3)

It is the desired nonlocal LLE. Note that in the isotropic case that is when
$J_{1}=J_{2}=J_{3}$ this nonlocal LLE reduces to the following nonlocal HFE
[69]

$\displaystyle{\bf M}_{t}$ $\displaystyle=$ $\displaystyle{\bf M}\wedge{\bf
M}_{xx}-{\bf L}\wedge{\bf L}_{xx},$ (4.4) $\displaystyle{\bf L}_{t}$
$\displaystyle=$ $\displaystyle{\bf M}\wedge{\bf L}_{xx}+{\bf L}\wedge{\bf
M}_{xx}.$ (4.5)

##  5 Nonlocal space curves and Nonlocal Serret-Frenet equation

As we mentioned in the previous section, in the nonlocal case, the unit
vectors ${\bf e}_{j}$ and the functions $\kappa,\tau,\sigma,\omega_{j}$ are
the complex-valied variables. As a consequence of these statements, the
corresponding space curves describe by the nonlocal Serret-Frenet equation
(SFE). To derive the nonlocal SFE, first we decompose the unit vectors ${\bf
e}_{j}$ and functions $\kappa,\sigma,\tau,\omega_{j}$ into the real and image
parts as

$\displaystyle{\bf e}_{1}$ $\displaystyle=$ $\displaystyle{\bf m}_{1}+i{\bf
l}_{1},\quad{\bf e}_{2}={\bf m}_{2}+i{\bf l}_{2},\quad{\bf e}_{3}={\bf
m}_{3}+i{\bf l}_{3},$ (5.1) $\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\kappa_{1}+i\kappa_{2},\quad\sigma=\sigma_{1}+i\sigma_{2},\quad\tau=\tau_{1}+i\tau_{2},$
(5.2) $\displaystyle\omega_{1}$ $\displaystyle=$
$\displaystyle\omega_{11}+i\omega_{12},\quad\omega_{2}=\omega_{21}+i\omega_{22},\quad\omega_{3}=\omega_{31}+i\omega_{32},$
(5.3)

where ${\bf m}_{j},{\bf l}_{j}$ are the real valued vector functions and
$\kappa_{j},\sigma_{j},\tau_{j},\omega_{ij}$ are real functions. From

$\displaystyle{\bf e}_{j}^{2}=1,\quad{\bf e}_{i}\cdot{\bf e}_{j}=0(i\neq
j),\quad{\bf e}_{i}\wedge{\bf e}_{j}=\epsilon_{ijk}{\bf e}_{k}$ (5.4)

follows that the vectors ${\bf m}_{j},{\bf l}_{j}$ satisfy the following
relations

$\displaystyle{\bf m}_{j}^{2}-{\bf l}^{2}_{j}=1,\quad{\bf m}_{j}\cdot{\bf
l}_{j}=0,\quad{\bf m}_{i}\wedge{\bf m}_{j}-{\bf l}_{i}\wedge{\bf
l}_{j}=\epsilon_{ijk}{\bf m}_{k},\quad{\bf m}_{i}\wedge{\bf l}_{j}+{\bf
l}_{i}\wedge{\bf m}_{j}=\epsilon_{ijk}{\bf l}_{k}$ (5.5)

and for the case $i\neq j$ we have

$\displaystyle{\bf m}_{i}\cdot{\bf m}_{j}-{\bf l}_{i}\cdot{\bf
l}_{j}=0,\quad{\bf m}_{i}\cdot{\bf l}_{j}+{\bf m}_{j}\cdot{\bf l}_{i}=0.$
(5.6)

We now ready to write the nonlocal SFE which follow from (3.1)-(3.2) and have
the forms

$\begin{pmatrix}{\bf m}_{1}\\\ {\bf m}_{2}\\\ {\bf
m}_{3}\end{pmatrix}_{x}=C_{1}\begin{pmatrix}{\bf m}_{1}\\\ {\bf m}_{2}\\\ {\bf
m}_{3}\end{pmatrix}-C_{2}\begin{pmatrix}{\bf l}_{1}\\\ {\bf l}_{2}\\\ {\bf
l}_{3}\end{pmatrix},\quad\begin{pmatrix}{\bf l}_{1}\\\ {\bf l}_{2}\\\ {\bf
l}_{3}\end{pmatrix}_{x}=C_{1}\begin{pmatrix}{\bf l}_{1}\\\ {\bf l}_{2}\\\ {\bf
l}_{3}\end{pmatrix}+C_{2}\begin{pmatrix}{\bf m}_{1}\\\ {\bf m}_{2}\\\ {\bf
m}_{3}\end{pmatrix},$ (5.7) $\begin{pmatrix}{\bf m}_{1}\\\ {\bf m}_{2}\\\ {\bf
m}_{3}\end{pmatrix}_{t}=D_{1}\begin{pmatrix}{\bf m}_{1}\\\ {\bf m}_{2}\\\ {\bf
m}_{3}\end{pmatrix}-D_{2}\begin{pmatrix}{\bf l}_{1}\\\ {\bf l}_{2}\\\ {\bf
l}_{3}\end{pmatrix},\quad\begin{pmatrix}{\bf l}_{1}\\\ {\bf l}_{2}\\\ {\bf
l}_{3}\end{pmatrix}_{t}=D_{1}\begin{pmatrix}{\bf l}_{1}\\\ {\bf l}_{2}\\\ {\bf
l}_{3}\end{pmatrix}+D_{2}\begin{pmatrix}{\bf m}_{1}\\\ {\bf m}_{2}\\\ {\bf
m}_{3}\end{pmatrix},$ (5.8)

where $C=C_{1}+iC_{2},\quad D=D_{1}+iD_{2}$ that is

$C=\begin{pmatrix}0&\kappa_{1}&\sigma_{1}\\\ -\kappa_{1}&0&\tau_{1}\\\
-\sigma_{1}&-\tau_{1}&0\end{pmatrix}+i\begin{pmatrix}0&\kappa_{2}&\sigma_{2}\\\
-\kappa_{2}&0&\tau_{2}\\\
-\sigma_{2}&-\tau_{2}&0\end{pmatrix},D=\begin{pmatrix}0&\omega_{31}&\omega_{21}\\\
-\omega_{31}&0&\omega_{11}\\\
-\omega_{21}&-\omega_{11}&0\end{pmatrix}+i\begin{pmatrix}0&\omega_{32}&\omega_{22}\\\
-\omega_{32}&0&\omega_{12}\\\ -\omega_{22}&-\omega_{12}&0\end{pmatrix}.$ (5.9)

The compatibility conditions ${\bf m}_{jxt}={\bf m}_{jtx},\quad{\bf
l}_{jxt}={\bf l}_{jtx}$ of these equations read as

$C_{1t}-D_{1x}+[C_{1},D_{1}]-[C_{2},D_{2}]=0,\quad
C_{2t}-D_{2x}+[C_{1},D_{2}]+[C_{2},D_{1}]=0.$ (5.10)

In components these equations take the following forms

$\displaystyle\kappa_{1t}$ $\displaystyle=$
$\displaystyle\omega_{31x}-(\tau_{1}\omega_{21}-\tau_{2}\omega_{22})+(\sigma_{1}\omega_{11}-\sigma_{2}\omega_{12}),$
(5.11) $\displaystyle\kappa_{2t}$ $\displaystyle=$
$\displaystyle\omega_{32x}-(\tau_{1}\omega_{22}-\tau_{2}\omega_{21})+(\sigma_{2}\omega_{11}+\sigma_{2}\omega_{11}),$
(5.12) $\displaystyle\sigma_{1t}$ $\displaystyle=$
$\displaystyle\omega_{21x}-(\kappa_{1}\omega_{11}-\kappa_{2}\omega_{12})+(\tau_{1}\omega_{31}-\tau_{2}\omega_{32}),$
(5.13) $\displaystyle\sigma_{2t}$ $\displaystyle=$
$\displaystyle\omega_{22x}-(\kappa_{1}\omega_{12}-\kappa_{2}\omega_{12})+(\tau_{1}\omega_{32}+\tau_{2}\omega_{31}),$
(5.14) $\displaystyle\tau_{1t}$ $\displaystyle=$
$\displaystyle\omega_{11x}-(\sigma_{1}\omega_{31}-\sigma_{2}\omega_{32})+(\kappa_{1}\omega_{21}-\kappa_{2}\omega_{22}).$
(5.15) $\displaystyle\tau_{2t}$ $\displaystyle=$
$\displaystyle\omega_{12x}-(\sigma_{1}\omega_{32}+\sigma_{2}\omega_{31})+(\kappa_{1}\omega_{22}+\kappa_{2}\omega_{22}).$
(5.16)

Now we set

$\kappa=0,\quad\sigma=r-q,\quad\tau=-i(r+q),\quad r=-\bar{q},$ (5.17)
$\omega_{1}=q_{x}-r_{x}-i(q_{21}+q_{12}),\quad\omega_{2}=-i(r_{x}+q_{x})+q_{21}-q_{12},\quad\omega_{3}=-2rq-2iq_{11},$
(5.18)

where $r=\nu\bar{q}(\epsilon_{1}x,\epsilon_{2}t)$. As in the local case, we
can show that the nonlocal LLE (4.2)-(4.3) is the gauge and geometrical
equivalent to the following nonlocal nonlinear Schrödinger-like equation

$\displaystyle iq_{t}+q_{xx}-2rq^{2}-iR_{12}(r,q)$ $\displaystyle=$
$\displaystyle 0,$ (5.19) $\displaystyle ir_{t}-r_{xx}+2qr^{2}-iR_{21}(r,q)$
$\displaystyle=$ $\displaystyle 0,$ (5.20) $\displaystyle
q_{11x}-(qq_{21}-rq_{12})$ $\displaystyle=$ $\displaystyle 0,$ (5.21)

where

$\displaystyle R_{12}(r,q)=q_{12x}+2qq_{11},\quad
R_{21}(r,q)=q_{21x}-2rq_{11},\quad
r=\nu\bar{q}(\epsilon_{1}x,\epsilon_{2}t),\quad\nu=\pm
1,\quad\epsilon_{j}^{2}=1.$ (5.22)

##  6 Gauge equivalence between the LLE and NLS-like equation

The gauge equivalent counterpart of the LLE was constructed in [18]. For the
noncompact case, the LLE (1.1) is called the modified LLE, the corresponding
gauge equivalent was studied in [19]. Here let us we consider the relation
between the solutions of the LLE (1.1) and the nonlinear Schrödinger-like
equation (3.9)-(3.11). Let $S$ is the solution of the LLE (1.1) and $g=\Psi$
is the solution linear equations (3.13)-(3.14). Then we have

$S=g^{-1}\sigma_{3}g,\quad
g=A(S+\sigma_{3})=\begin{pmatrix}f_{1}(S_{3}+1)&f_{1}S^{-}\\\
f_{2}S^{+}&-f_{2}(S_{3}+1)\end{pmatrix},\quad A=\begin{pmatrix}f_{1}&0\\\
0&f_{2}\end{pmatrix}=diag(e^{i\gamma},e^{-i\gamma}),$ (6.1)

where $\gamma$ is a real function, $det(g)=1$ and $\Delta=-2(S_{3}+1)$. Since

$(S+\sigma_{3})^{-1}=-\frac{1}{\Delta}\begin{pmatrix}S_{3}+1&S^{-}\\\
S^{+}&-(S_{3}+1)\end{pmatrix},\quad\Delta=\det(S+\sigma_{3})=-2(S_{3}+1)$
(6.2)

we obtain

$(S+\sigma_{3})^{-1}(S+\sigma_{3})=-\frac{1}{\Delta}\begin{pmatrix}S_{3}+1&S^{-}\\\
S^{+}&-(S_{3}+1)\end{pmatrix}\begin{pmatrix}S_{3}+1&S^{-}\\\
S^{+}&-(S_{3}+1)\end{pmatrix}=I.$ (6.3)

Similarly, we calculate

$g^{-1}=(S+\sigma_{3})^{-1}A^{-1}=-\frac{1}{\Delta}\begin{pmatrix}S_{3}+1&S^{-}\\\
S^{+}&-(S_{3}+1)\end{pmatrix}\begin{pmatrix}e^{-i\gamma}&0\\\
0&e^{i\gamma}\end{pmatrix}=-\frac{1}{\Delta}(S+\sigma_{3})A^{-1}.$ (6.4)

To find the desired relation we recall that

$S_{x}=g^{-1}[\sigma_{3},U]g=2g^{-1}\begin{pmatrix}0&q\\\ -r&\end{pmatrix}g.$
(6.5)

Hence we get

$\begin{pmatrix}0&q\\\
-r&0\end{pmatrix}=\frac{1}{2}A(S+\sigma_{3})S_{x}(S+\sigma_{3})^{-1}A^{-1}=-\frac{1}{2\Delta}A(S+\sigma_{3})S_{x}(S+\sigma_{3})A^{-1}=-\frac{1}{2\Delta}AEA^{-1}$
(6.6)

and

$(S+\sigma_{3})S_{x}(S+\sigma_{3})=E=\begin{pmatrix}0&e_{12}\\\
e_{21}&0\end{pmatrix}=2\begin{pmatrix}0&S^{-}S_{3x}-(S_{3}+1)S^{-}_{x}\\\
S^{+}S_{3x}-(S_{3}+1)S^{+}_{x}&0\end{pmatrix}.$ (6.7)

Hence we obtain

$\displaystyle\begin{pmatrix}0&q\\\
-r&0\end{pmatrix}=\frac{1}{4(S_{3}+1)}\begin{pmatrix}0&e_{12}e^{2i\gamma}\\\
e_{21}e^{-2i\gamma}&0\end{pmatrix}=$
$\displaystyle\frac{1}{2(S_{3}+1)}\begin{pmatrix}0&[S^{-}S_{3x}-(S_{3}+1)S^{-}_{x}]e^{2i\gamma}\\\
[S^{+}S_{3x}-(S_{3}+1)S^{+}_{x}]e^{-2i\gamma}&0\end{pmatrix},$ (6.8)

or

$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2(S_{3}+1)}e_{12}e^{2i\gamma}=\frac{1}{2(S_{3}+1)}[S^{-}S_{3x}-(S_{3}+1)S^{-}_{x}]e^{2i\gamma},$
(6.9) $\displaystyle r$ $\displaystyle=$
$\displaystyle-\frac{1}{2(S_{3}+1)}e_{21}e^{-2i\gamma}=-\frac{1}{2(S_{3}+1)}[S^{+}S_{3x}-(S_{3}+1)S^{+}_{x}]e^{-2i\gamma},$
(6.10)

where $r=-\bar{q}$. In terms of $\theta-\varphi$ variables these expressions
take the form

$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{(i\varphi_{x}\sin\theta-\theta_{x})}{2}e^{-i\int\varphi_{x}\cos\theta
dx},$ (6.11) $\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{(i\varphi_{x}\sin\theta+\theta_{x})}{2}e^{i\int\varphi_{x}\cos\theta
dx},$ (6.12)

where $\epsilon=\pm 1$. Hence we obtain

$\displaystyle
rq=\frac{\varphi_{x}^{2}\sin^{2}\theta+\theta^{2}_{x}}{4}=\frac{1}{4}{\bf
S}_{x}^{2}.$ (6.13)

##  7 Anisotropic soliton surfaces induced by the LLE

In this section, our aim is to present the anisotropic soliton surfaces
induced by the LLE (1.1). To do that, let us recall that the position vector
${\bf r}=(r_{1},r_{2},r_{3})$ of the soliton surface satisfies the certain two
equations. Note that in this approach, the matrix form of the position vector
defines by the Sym-Tafel formula

$\displaystyle
r=r_{1}\sigma_{1}+r_{2}\sigma_{2}+r_{3}\sigma_{3}=\begin{pmatrix}r_{3}&r^{-}\\\
r^{+}&-r_{3}\end{pmatrix}=\Phi^{-1}\Phi_{\lambda}.$

Hence we get the following two equations:

$\displaystyle r_{x}$ $\displaystyle=$
$\displaystyle\Phi^{-1}U_{\lambda}\Phi,$ (7.1) $\displaystyle r_{t}$
$\displaystyle=$ $\displaystyle\Phi^{-1}V_{\lambda}\Phi,$ (7.2)

where

$\displaystyle U_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{i\rho}{sn^{2}(\lambda,k)}\left[cn(\lambda,k)dn(\lambda,k)S_{1}\sigma_{1}+cn(\lambda,k)S_{2}\sigma_{2}+dn(\lambda,k)S_{3}\sigma_{3}\right],$
(7.3) $\displaystyle V_{\lambda}$ $\displaystyle=$ $\displaystyle
i\left[(u_{2}u_{3})_{\lambda}S_{1}\sigma_{1}+(u_{1}u_{3})_{\lambda}S_{2}\sigma_{2}+(u_{1}u_{2})_{\lambda}S_{3}\sigma_{3}\right]-i\sum_{a,b,c=1}^{3}u_{a\lambda}\epsilon_{abc}S_{b}S_{cx}\sigma_{a}.$
(7.4)

Here we used the following formulas

$\displaystyle[sn(\lambda)]_{\lambda}=cn(\lambda,k)dn(\lambda,k),\,[cn(\lambda)]_{\lambda}=-sn(\lambda,k)dn(\lambda,k),\,[dn(\lambda)]_{\lambda}=-k^{2}sn(\lambda,k)cn(\lambda,k)$
(7.5)

and

$\displaystyle sn^{2}(\lambda)+cn^{2}(\lambda)=1,\quad
dn^{2}(\lambda)+k^{2}sn^{2}(\lambda)=1,\quad
dn^{2}(\lambda)-k^{2}cn^{2}(\lambda)=k^{\prime 2}.$ (7.6)

We now can write the first fundamental form of the desired anisotropic soliton
surface

$\displaystyle I={\bf r}_{x}^{2}dx^{2}+2{\bf r}_{x}\cdot{\bf r}_{t}dxdt+{\bf
r}_{t}^{2}dt^{2},$ (7.7)

where

$\displaystyle{\bf r}_{x}^{2}=\frac{1}{2}tr(r_{x}^{2}),\quad{\bf
r}_{x}\cdot{\bf r}_{t}=\frac{1}{2}tr(r_{x}r_{t}),\quad{\bf
r}_{t}^{2}=\frac{1}{2}tr(r_{t}^{2}).$ (7.8)

Similarly, we can construct the second fundamental form of the soliton surface
corresponding to the LLE. In our case, the well-known Sym-Tafel formula has
the form

$r=\Phi^{-1}\Phi_{\lambda}=\begin{pmatrix}r_{3}&r^{-}\\\
r^{+}&-r_{3}\end{pmatrix}.$ (7.9)

Using the following expressions

$\Phi_{\lambda}=\begin{pmatrix}\phi_{1\lambda}&-\bar{\phi}_{2\lambda}\\\
\phi_{2\lambda}&\bar{\phi}_{1\lambda}\end{pmatrix},\quad\Phi^{-1}=\frac{1}{\det\Phi}\begin{pmatrix}\bar{\phi}_{1}&\bar{\phi}_{2}\\\
-\phi_{2}&\phi_{1}\end{pmatrix},\quad\det\Phi=|\phi_{1}|^{2}+|\phi_{2}|^{2},$
(7.10)

we finally have

$r=\frac{1}{\det\Phi}\begin{pmatrix}\bar{\phi}_{1}\phi_{1\lambda}+\bar{\phi}_{2}\phi_{2\lambda}&-\bar{\phi}_{1}\bar{\phi}_{2\lambda}+\bar{\phi}_{2}\bar{\phi}_{1\lambda}\\\
-\phi_{2}\phi_{1\lambda}+\phi_{1}\phi_{2\lambda}&\phi_{2}\bar{\phi}_{2\lambda}+\phi_{1}\bar{\phi}_{1\lambda}\end{pmatrix}.$
(7.11)

Hence for the components of the position vector ${\bf r}=(r_{1},r_{2},r_{3})$
we obtain

$r^{+}=r_{1}+ir_{2}=\frac{-\phi_{2}\phi_{1\lambda}+\phi_{1}\phi_{2\lambda}}{\det\Phi},\quad
r^{-}=\frac{-\bar{\phi}_{1}\bar{\phi}_{2\lambda}+\bar{\phi}_{2}\bar{\phi}_{1\lambda}}{\det\Phi},\quad
r_{3}=\frac{\bar{\phi}_{1}\phi_{1\lambda}+\bar{\phi}_{2}\phi_{2\lambda}}{\det\Phi}$
(7.12)

or

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\frac{-\phi_{2}\phi_{1\lambda}+\phi_{1}\phi_{2\lambda}-\bar{\phi}_{1}\bar{\phi}_{2\lambda}+\bar{\phi}_{2}\bar{\phi}_{1\lambda}}{2\det\Phi},$
(7.13) $\displaystyle r_{2}$ $\displaystyle=$
$\displaystyle\frac{-\phi_{2}\phi_{1\lambda}+\phi_{1}\phi_{2\lambda}+\bar{\phi}_{1}\bar{\phi}_{2\lambda}-\bar{\phi}_{2}\bar{\phi}_{1\lambda}}{2i\det\Phi},$
(7.14) $\displaystyle r_{3}$ $\displaystyle=$
$\displaystyle\frac{\bar{\phi}_{1}\phi_{1\lambda}+\bar{\phi}_{2}\phi_{2\lambda}}{\det\Phi}.$
(7.15)

Let us now we construct the soliton surface corresponding to the 1-soliton
solution of the LLE (1.1) which we presented in the previous section. In this
case, the components of the position vector are given by (7.13)-(7.15), where

$\displaystyle\phi_{1}$ $\displaystyle=$ $\displaystyle
c_{1}e^{-i(u_{3}x-2u_{1}u_{2}t)},\quad\phi_{2}=c_{2}e^{i(u_{3}x-2u_{1}u_{2}t)},\quad\bar{\phi}_{1}=\bar{c}_{1}e^{-\bar{\chi}},\quad\bar{\phi}_{2}=\bar{c}_{2}e^{\bar{\chi}}$
(7.16) $\displaystyle\phi_{1\lambda}$ $\displaystyle=$
$\displaystyle-i[u_{3\lambda}x-2(u_{1}u_{2})_{\lambda}t]\phi_{01},\quad\phi_{2\lambda}=i[u_{3\lambda}x-2(u_{1}u_{2})_{\lambda}t]\phi_{02}.$
(7.17)

Thus in this section we have presented the soliton surface given by the
position vector ${\bf r}$ corresponding to the 1-soliton solution of the LLE.

##  8 Some generalizations of the LLE

There are several integrable and nonintegrable generalizations of the LLE. In
this section we will present some of them.

###  8.1 Landau-Lifshitz-Schrödinger equation

One of generalizations is the Landau-Lifshitz-Schrödinger equation (LLSE)
which has the form [78]

$\displaystyle
iS_{t}+\frac{1}{2}[S,S_{xx}+S_{J}]+2Q_{x}+i[S,F_{0}]-[Q,SS_{x}]$
$\displaystyle=$ $\displaystyle 0,$ (8.1) $\displaystyle
Q_{t}-F_{0x}+[Q,F_{0}]$ $\displaystyle=$ $\displaystyle 0,$ (8.2)

where

$F_{0}=V_{0}+\begin{pmatrix}a&b\\\ c&-a\end{pmatrix},\quad
Q=\begin{pmatrix}0&q\\\ r&0\end{pmatrix},\quad
V_{0}=i\begin{pmatrix}-rq&q_{x}\\\ -r_{x}&rq\end{pmatrix},\quad
r=\epsilon\bar{q}.$ (8.3)

In the isotropic case $J_{1}=J_{2}=J_{3}$, this LLSE takes the form

$\displaystyle iS_{t}+\frac{1}{2}[S,S_{xx}]+2Q_{x}+i[S,F_{0}]-[Q,SS_{x}]$
$\displaystyle=$ $\displaystyle 0,$ (8.4) $\displaystyle
Q_{t}-F_{0x}+[Q,F_{0}]$ $\displaystyle=$ $\displaystyle 0.$ (8.5)

This isotropic LLSE admits the following LR [78]

$\displaystyle Z_{x}$ $\displaystyle=$ $\displaystyle U_{3}Z,$ (8.6)
$\displaystyle Z_{t}$ $\displaystyle=$ $\displaystyle V_{3}Z,$ (8.7)

with the Lax pair

$\displaystyle U_{3}$ $\displaystyle=$ $\displaystyle-i\lambda S+Q,$ (8.8)
$\displaystyle V_{3}$ $\displaystyle=$
$\displaystyle-2i\lambda^{2}S+\lambda(SS_{x}+2Q)+F_{0}.$ (8.9)

###  8.2 Myrzakulov-XCIX equation

Another integrable generalization of the LLE is the Myrzakulov-XCIX (M-XCIX)
equation [24]-[25]. The anisotropic M-XCIX equation reads as

$\displaystyle
iS_{t}+\frac{1}{2}[S,S_{xx}]-4i\lambda_{0}S_{x}+\frac{1}{\lambda_{0}+\omega}[S,W]+\frac{1}{2}[S,S_{J}]$
$\displaystyle=$ $\displaystyle 0,$ (8.10) $\displaystyle
iW_{x}+({\lambda_{0}+\omega})[S,W]$ $\displaystyle=$ $\displaystyle 0,$ (8.11)

where $\omega=const$, $S={\Sigma}^{3}_{j=1}S_{j}(x,y,t)\sigma_{j}$ is a matrix
analogue of the spin vector, $W$ is the potential with the matrix form
$W={\Sigma}^{3}_{j=1}W_{j}(x,y,t)\sigma_{j}$. In the isotropic case
$J_{1}=J_{2}=J_{3}$ that is the isotropic M-XCIX equation

$\displaystyle iS_{t}+\frac{1}{2}[S,S_{xx}]+\frac{1}{\omega}[S,W]$
$\displaystyle=$ $\displaystyle 0,$ (8.12) $\displaystyle iW_{x}+\omega[S,W]$
$\displaystyle=$ $\displaystyle 0,$ (8.13)

is integrable by the IST. Its LR can be written in the form [24]-[25]

$\displaystyle\Phi_{x}$ $\displaystyle=$ $\displaystyle U_{4}\Phi,$ (8.14)
$\displaystyle\Phi_{t}$ $\displaystyle=$ $\displaystyle V_{4}\Phi,$ (8.15)

where the Lax pair $U_{4}$ and $V_{4}$ have the form

$\displaystyle U_{4}$ $\displaystyle=$ $\displaystyle-i\lambda S,$ (8.16)
$\displaystyle V_{4}$ $\displaystyle=$ $\displaystyle\lambda^{2}V_{2}+\lambda
V_{1}+\left(\frac{i}{\lambda+\omega}-\frac{i}{a}\right)W.$ (8.17)

Here

$\displaystyle V_{2}=-2iS,\quad V_{1}=SS_{x}.$ (8.18)

The M-XCIX equation (8.12)-(8.13) is gauge equivalent to the following SMBE
[24]-[25]

$\displaystyle iq_{t}+q_{xx}+2\delta|q|^{2}q-2ip$ $\displaystyle=$
$\displaystyle 0,$ (8.19) $\displaystyle p_{x}-2i\omega p-2\eta q$
$\displaystyle=$ $\displaystyle 0,$ (8.20)
$\displaystyle\eta_{x}+\delta(q^{*}p+p^{*}q)$ $\displaystyle=$ $\displaystyle
0,$ (8.21)

where $q(x,t),p(x,t)$ are complex functions, $\eta(x,t)$ is a real function,
$\omega$ is the real contant. This system is integrable by the IST. The
corresponding LR is given by

$\displaystyle\Psi_{x}$ $\displaystyle=$ $\displaystyle U_{5}\Psi,$ (8.22)
$\displaystyle\Psi_{t}$ $\displaystyle=$ $\displaystyle 2\lambda
U_{5}\Psi+B\Psi,$ (8.23)

where $U_{5}$ and $B$ have the forms

$\displaystyle U_{5}$ $\displaystyle=$
$\displaystyle-i\lambda\sigma_{3}+U_{0},$ (8.24) $\displaystyle B$
$\displaystyle=$ $\displaystyle B_{0}+\frac{i}{\lambda+\omega}B_{-1}.$ (8.25)

Here

$\displaystyle U_{0}=\begin{pmatrix}0&q\\\ -q^{*}&0\end{pmatrix},\quad
B_{0}=-i\delta|q|^{2}\sigma_{3}+i\begin{pmatrix}0&q_{x}\\\ \delta
q^{*}_{x}&0\end{pmatrix},\quad B_{-1}=\begin{pmatrix}\eta&-p\\\ -\delta
p^{*}&-\eta\end{pmatrix},$ (8.26)

where $*$ means a complex conjugate and $\delta=\pm 1$, so that $\delta=+1$
corresponds to the attractive interaction and $\delta=-1$ to the repulsive
interaction respectively.

###  8.3 LLE with the self-consistent variable anisotropy

Let us consider the variable anisotropy case that is
$J_{j}=\frac{1}{\lambda_{0}+\omega}K_{j}(x,t)$. Then the LLE (1.1) takes the
form

$\displaystyle{\bf S}_{t}+{\bf S}\wedge{\bf S}_{xx}+{\bf S}\wedge K{\bf S}=0,$
(8.27)

where $K(x,t){\bf S}=(K_{1}(x,t)S_{1}+K_{2}(x,t)S_{2}+K_{3}(x,t)S_{3}$. This
equation is integrable if we additionally demand that the quantity $K{\bf S}$
to satisfy some additional equation, say, the following equation

$\displaystyle(K{\bf S})_{x}+\mu{\bf S}\wedge K{\bf S}=0.$ (8.28)

Then the full set of equations has the form

$\displaystyle{\bf S}_{t}+{\bf S}\wedge{\bf S}_{xx}+{\bf S}\wedge K{\bf S}$
$\displaystyle=$ $\displaystyle 0,$ (8.29) $\displaystyle(K{\bf
S})_{x}+\mu{\bf S}\wedge K{\bf S}$ $\displaystyle=$ $\displaystyle 0.$ (8.30)

###  8.4 Integrable LLE with the external magnetic field

The previous examples of the generalized LLE were in the absence of an
external magnetic field. We now consider the LLE with the external magnetic
field ${\bf H}$ which can be written as

$\displaystyle{\bf S}_{t}+{\bf S}\wedge{\bf S}_{xx}+{\bf S}\wedge J{\bf
S}+{\bf S}\wedge{\bf H}$ $\displaystyle=$ $\displaystyle 0,$ (8.31)

where ${\bf H}=(H_{1},H_{2},H_{3})$ is the external magnetic field. We now
demand that in this LLE with the external magnetic field, the vector field
${\bf H}$ is variable and satisfies the following additional equation

$\displaystyle{\bf H}_{x}+\mu{\bf S}\wedge{\bf H}=0,$ (8.32)

where $\mu=const$. Thus we obtain the following set of the closed equations

$\displaystyle{\bf S}_{t}+{\bf S}\wedge{\bf S}_{xx}+{\bf S}\wedge J{\bf
S}+{\bf S}\wedge{\bf H}$ $\displaystyle=$ $\displaystyle 0,$ (8.33)
$\displaystyle{\bf H}_{x}+\mu{\bf S}\wedge{\bf H}$ $\displaystyle=$
$\displaystyle 0,$ (8.34)

which is, in fact, the LLE with the self-consistent external magnetic field
${\bf H}$. This LLE with the self-consistent external magnetic field in the
isotropic case reads as

$\displaystyle{\bf S}_{t}+{\bf S}\wedge{\bf S}_{xx}+{\bf S}\wedge{\bf H}$
$\displaystyle=$ $\displaystyle 0,$ (8.35) $\displaystyle{\bf H}_{x}+\mu{\bf
S}\wedge{\bf H}$ $\displaystyle=$ $\displaystyle 0.$ (8.36)

This equation is integrable.

###  8.5 Ishimori equation

The famous anisotropic Ishimori equation (IE) has the form [6]

$\displaystyle{\bf S}_{t}-{\bf S}\wedge({\bf S}_{xx}+\alpha^{2}{\bf
S}_{yy})-u_{y}{\bf S}_{x}-u_{x}{\bf S}_{y}-{\bf S}\wedge J{\bf S}$
$\displaystyle=$ $\displaystyle 0,$ (8.37) $\displaystyle
u_{xx}-\alpha^{2}u_{yy}+2\alpha^{2}{\bf S}\cdot({\bf S}_{x}\wedge{\bf S}_{y})$
$\displaystyle=$ $\displaystyle 0.$ (8.38)

The isotropic IE is integrable and reads as

$\displaystyle{\bf S}_{t}-{\bf S}\wedge({\bf S}_{xx}+\alpha^{2}{\bf
S}_{yy})-u_{y}{\bf S}_{x}-u_{x}{\bf S}_{y}$ $\displaystyle=$ $\displaystyle
0,$ (8.39) $\displaystyle u_{xx}-\alpha^{2}u_{yy}+2\alpha^{2}{\bf S}\cdot({\bf
S}_{x}\wedge{\bf S}_{y})$ $\displaystyle=$ $\displaystyle 0,$ (8.40)

or

$\displaystyle iw_{t}+w_{xx}-\alpha
w_{yy}-\frac{2\bar{w}(w_{x}^{2}-w_{y}^{2})}{1+|w|^{2}}+i\beta(u_{x}w_{y}-u_{y}w_{x})$
$\displaystyle=$ $\displaystyle 0,$ (8.41) $\displaystyle u_{xx}+\nu
u_{yy}-\frac{8Im(w_{x}w_{y})}{1+|w|^{2}}$ $\displaystyle=$ $\displaystyle 0.$
(8.42)

###  8.6 Myrzakulov-IX equation

The anisotropic Myrzakulov-IX (M-IX) equation is given by

$\displaystyle{\bf S}_{t}-{\bf S}\wedge M_{1}{\bf S}+iA_{2}{\bf
S}_{x}+iA_{1}{\bf S}_{y}-{\bf S}\wedge J{\bf S}$ $\displaystyle=$
$\displaystyle 0,$ (8.43) $\displaystyle M_{2}u-2\alpha^{2}{\bf S}\cdot({\bf
S}_{x}\wedge{\bf S}_{y})$ $\displaystyle=$ $\displaystyle 0,$ (8.44)

where $M_{j}$ and $A_{j}$ are some operators [8]-[9]. The isotropic M-IX
equation is integrable. Note that the isotropic M-IX equation is equivalent to
the following equation [79]-[81]

$\displaystyle iq_{t}+M_{1}q+vq$ $\displaystyle=$ $\displaystyle 0,$ (8.45)
$\displaystyle ip_{t}-M_{1}p-vp$ $\displaystyle=$ $\displaystyle 0,$ (8.46)
$\displaystyle M_{2}v+2M_{1}(pq)$ $\displaystyle=$ $\displaystyle 0.$ (8.47)

###  8.7 Myrzakulov-VIII equation

The M-IX equation admits some integrable reductions. For example, as $a=b=-1$
it reduces to the anisotropic Myrzakulov-VIII (M-VIII) equation. The M-VIII
reads as [8]-[9]

$\displaystyle iS_{t}+\frac{1}{2}[S,S_{xx}]+iuS_{x}+[S,S_{J}]$
$\displaystyle=$ $\displaystyle 0,$ (8.48) $\displaystyle
u_{x}+u_{y}+\frac{1}{4i}tr(S[S_{x},S_{y}])$ $\displaystyle=$ $\displaystyle
0.$ (8.49)

Note that the isotropic M-VIII equation is integrable. Its gauge equivalent
counterpart has the form [79]-[81]

$\displaystyle iq_{t}+q_{xx}+vq$ $\displaystyle=$ $\displaystyle 0,$ (8.50)
$\displaystyle ip_{t}-p_{xx}-vp$ $\displaystyle=$ $\displaystyle 0,$ (8.51)
$\displaystyle v_{x}+v_{y}+2(pq)_{y}$ $\displaystyle=$ $\displaystyle 0.$
(8.52)

###  8.8 Myrzakulov-XXXIV equation

The (1+1)-dimensional anisotropic Myrzakulov-XXXIV (M-XXXIV) equation is given
by [8]-[9]

$\displaystyle{\bf S}_{t}-{\bf S}\wedge{\bf S}_{xx}+u{\bf S}_{x}-{\bf S}\wedge
J{\bf S}$ $\displaystyle=$ $\displaystyle 0,$ (8.53) $\displaystyle
u_{t}+u_{x}-\beta({\bf S}_{x}^{2})_{x}$ $\displaystyle=$ $\displaystyle 0.$
(8.54)

In the isotropic case, the M-XXXIV equation is integrable and is equivalent to
the Yajima-Oikawa equation [9]

$\displaystyle iq_{t}+q_{xx}-vq$ $\displaystyle=$ $\displaystyle 0,$ (8.55)
$\displaystyle v_{t}+v_{x}+\delta(|q|^{2})_{x}$ $\displaystyle=$
$\displaystyle 0.$ (8.56)

###  8.9 Myrzakulov-I equation

The anisotropic Myrzakulov-I (M-I) equation is given by [7]

$\displaystyle{\bf S}_{t}-{\bf S}\wedge{\bf S}_{xy}-u{\bf S}_{x}-{\bf S}\wedge
J{\bf S}$ $\displaystyle=$ $\displaystyle 0,$ (8.57) $\displaystyle u_{x}+{\bf
S}\cdot({\bf S}_{x}\wedge{\bf S}_{y})$ $\displaystyle=$ $\displaystyle 0.$
(8.58)

In the isotropic case, it takes the form

$\displaystyle{\bf S}_{t}-{\bf S}\wedge{\bf S}_{xy}-u{\bf S}_{x}$
$\displaystyle=$ $\displaystyle 0,$ (8.59) $\displaystyle u_{x}+{\bf
S}\cdot({\bf S}_{x}\wedge{\bf S}_{y})$ $\displaystyle=$ $\displaystyle 0.$
(8.60)

or

$\displaystyle iw_{t}+w_{xy}-iuw_{x}-\frac{2\bar{w}w_{x}w_{y}}{1+|w|^{2}}$
$\displaystyle=$ $\displaystyle 0,$ (8.61) $\displaystyle
u_{x}+\frac{2i(w_{x}\bar{w}_{y}-\bar{w}_{x}w_{y})}{(1+|w|^{2})^{2}}$
$\displaystyle=$ $\displaystyle 0.$ (8.62)

The isotropic M-I equation is integrable by the IST method and equivalent to
the following (2+1)-dimensional NLSE

$\displaystyle iq_{t}+q_{xy}-vq$ $\displaystyle=$ $\displaystyle 0,$ (8.63)
$\displaystyle v_{x}+2(|q|^{2})_{y}$ $\displaystyle=$ $\displaystyle 0.$
(8.64)

Note that in 1+1 dimension that is when $y=x$, this NLSE reduces to the usual
NLSE

$\displaystyle iq_{t}+q_{xx}+2|q|^{2}q=0.$ (8.65)

###  8.10 Landau-Lifshitz-Gilbert equation

From the physical point of view, one of very important generalizations of the
LLE is the so-called Landau-Lifshit-Gilbert equation (LLGE). It was proposed
in 1955 by T. Gilbert as a modification of the LLE (1.1) to incorporate a
damping term. This LLGE is given by [13]

$\displaystyle{\bf S}_{t}+\beta{\bf S}\wedge{\bf H}_{eff}+\alpha{\bf
S}\wedge{\bf S}\wedge{\bf H}_{eff}=0,$ (8.66)

where the vector function ${\bf H}_{eff}$ is the effective magnetic field as
some kind derivative of the magnetic energy of the material. For more details,
we refer to the survey [14] and references therein.

###  8.11 Landau-Lifshitz-Bloch equation

The Landau-Lifshitz-Bloch equation (LLBE) is one of important generalizations
of the LLE. It describes the magnetization dynamics of magnetic particles at
high temperatures without the restriction of a fixed magnetization length and
thus allows for its longitudinal relaxation. The LLBE is given by (see i.g.
[82])

$\displaystyle{\bf u}_{t}=c_{1}{\bf u}\wedge{\bf H}_{eff}+c_{2}{\bf
u}\wedge({\bf u}\wedge{\bf H}_{eff})+c_{3}\frac{1}{|{\bf u}|^{2}}({\bf
u}\cdot{\bf H}_{eff}){\bf u},$ (8.67)

where ${\bf u}$ is the spin polarization, $c_{j}$ are constants, ${\bf
H}_{eff}$ is effective field.

##  9 LLE as the Madelung equation. Hydrodynamical formulations

To study some properties of the LLE, it is very useful its hydrodynamical
form. To derive the hydrodynamical form of the LLE we consider the following
Madelung like transformation [14]

$\displaystyle S^{+}=\sqrt{1-S_{3}^{2}}e^{-i(\phi-0.5\pi)}.$ (9.1)

Then the LLE (1.1) takes the form

$\displaystyle\phi_{t}+\left(\frac{S_{3x}}{1-S_{3}^{2}}\right)_{x}-\frac{S_{3}S_{3x}^{2}}{(1-S_{3}^{2})^{2}}+\left(\phi_{x}^{2}-J_{23}+J_{21}\sin^{2}\phi\right)S_{3}$
$\displaystyle=$ $\displaystyle 0,$ (9.2) $\displaystyle
S_{3t}-\left((1-S_{3}^{2})\phi_{x}\right)_{x}+0.5J_{21}(1-S_{3}^{2})\sin^{2}2\phi$
$\displaystyle=$ $\displaystyle 0.$ (9.3)

This is the Madelung equation form of the LLE (1.1). Note that this Madelung
form will be essential in the study of solutions of the LLE. It is interesting
to present the Madelung equation for the gauge and geometrical equivalent of
the LLE, namely, for the NLS like equation (3.9)-(3.11). Let
$q=\sqrt{\rho}e^{i\varphi}$. For simplicity, the equation (3.9)-(3.11) we
write in the form

$\displaystyle iq_{t}+q_{xx}-vq=0,$ (9.4)

where

$\displaystyle v=2|q|^{2}+iF_{12},\quad F_{12}=-R_{12}/q,\quad
F_{21}=-R_{21}.$ (9.5)

Then for the equation (3.9)-(3.11), the corresponding Madelung equations takes
the form

$\displaystyle\varphi_{t}+\varphi\varphi_{x}+\left(v-\frac{(\sqrt{\rho})_{xx}}{2\sqrt{\rho}}\right)$
$\displaystyle=$ $\displaystyle 0,$ (9.6)
$\displaystyle\rho_{t}+(\rho\varphi)_{x}$ $\displaystyle=$ $\displaystyle 0.$
(9.7)

##  10 Quantum Heisenberg model

It is well-known that the LLE (1.1) is the continuous classical limit of the
quantum Heisenberg XYZ model:

$\displaystyle\hat{H}=-\frac{1}{2}\sum_{j=1}^{N}\left(J_{1}\hat{S}_{1j}\hat{S}_{1(j+1)}+J_{2}\hat{S}_{2j}\hat{S}_{2(j+1)}+J_{3}\hat{S}_{3j}\hat{S}_{3(j+1)}+h\hat{S}_{3j}\right)$
(10.1)

where $h$ is the external magnetic field, $J_{j}$ are real-valued coupling
constants. Three particular cases of this model: i) if $J_{1}=J_{2}=J_{3}$,
the model is called quantum Heisenberg XXX model; ii) if $J_{1}=J_{2}\neq
J_{3}$, it is the quantum Heisenberg XXZ model; iii) if $J_{1}\neq J_{2}\neq
J_{3}$, this model is called the uantum Heisenberg XYZ model. Note that
quantum Heisenberg XYZ model is one of examples of the so-called integrable
quantum models. Such integrable quantum models can be solved by the quantum
inverse scattering methods and/or by the Bethe ansatz. Finally we present the
well-known result, namely, the spin operators $\hat{S}_{j}$ can be written in
terms of auxiliary oscillators $a_{j}$ and their conjugates $a_{j}^{+}$ as

$\displaystyle\hat{S}^{+}_{j}=a_{j},\quad\hat{S}=a_{j}^{+}(1-a_{j}^{+}a_{j}),\quad\hat{S}_{3j}=\frac{1}{2}-a_{j}^{+}a_{j}.$
(10.2)

##  11 Dispersionless LLE

Let us present the dispersionless LLE. The vector LLE (1.1) is equivalent to
the only one equation for the complex function $S^{+}(x,t)$. This equation has
the form (2.11) that is

$\displaystyle
iS_{t}^{+}=S^{+}S_{3xx}-S^{+}_{xx}S_{3}+J_{3}S_{3}S^{+}-\frac{1}{2}S_{3}(I_{21}S^{+}-J_{21}S^{-}),$
(11.1)

where $S^{-}=\bar{S}^{+},\quad S_{3}=\sqrt{1-|S^{+}|^{2}}$. Let us consider
the transformation

$\displaystyle S^{+}=\sqrt{u}e^{i\epsilon^{-1}s},$ (11.2)

where $u=1-S_{3}^{2}$ and $s$ are some real functions, $\epsilon$ is some real
parameter. In our case, the equation (11.1) takes the form

$\displaystyle i\epsilon
S_{t}^{+}=\epsilon^{2}\left(S^{+}S_{3xx}-S^{+}_{xx}S_{3}\right)+J_{3}S_{3}S^{+}-\frac{1}{2}S_{3}(I_{21}S^{+}-J_{21}S^{-}).$
(11.3)

For simplicity, let us consider the case $J_{1}=J_{2}$ that is the LLE with an
easy plan. In this case, the equation (11.3) takes the form

$\displaystyle i\epsilon
S_{t}^{+}=\epsilon^{2}\left(S^{+}S_{3xx}-S^{+}_{xx}S_{3}\right)+J_{3}S_{3}S^{+}-J_{1}S_{3}S^{+}.$
(11.4)

Hence and using the transformation (11.2) we obtain the following set of
equations

$\displaystyle s_{t}+\sqrt{1-u}(s_{x}^{2}+J_{31})$ $\displaystyle=$
$\displaystyle 0,$ (11.5) $\displaystyle
u_{t}+2\sqrt{1-u}(s_{xx}u+s_{x}u_{x})$ $\displaystyle=$ $\displaystyle 0,$
(11.6)

or

$\displaystyle v_{t}+[\sqrt{1-u}(v^{2}+J_{31})]_{x}$ $\displaystyle=$
$\displaystyle 0,$ (11.7) $\displaystyle u_{t}+2(uv)_{x}\sqrt{1-u}$
$\displaystyle=$ $\displaystyle 0,$ (11.8)

where $v=s_{x}$. It is the desired dispersionless LLE (dLLE) for the easy plan
case. Note that this dLLE is integrable. Finally, we note that the
dispersionless isotropic LLE reads as

$\displaystyle v_{t}+[v^{2}\sqrt{1-u}]_{x}$ $\displaystyle=$ $\displaystyle
0,$ (11.9) $\displaystyle u_{t}+2(uv)_{x}\sqrt{1-u}$ $\displaystyle=$
$\displaystyle 0.$ (11.10)

##  12 Self-similar solution of the LLE

In the theory of the nonlinear differential equations, their self-similar
solutions play an important role [83]-[89]. For that reason, in this section
we want just briefly mention some basic facts about the self-similar
solutions, as example, for the LLGE (for details see e.g. [89]). Note that the
LLGE contents the LLE as the particular case. In 1+1 dimension, the LLGE we
write as [89]

$\displaystyle{\bf S}_{t}-\beta{\bf S}\wedge{\bf S}_{xxx}+\alpha{\bf
S}\wedge({\bf S}\wedge{\bf S}_{xx})=0.$ (12.1)

Recall that the solution of the LLGE is called self-similar, if it satisfies
the condition:

$\displaystyle{\bf S}(x,t)={\bf S}(\lambda x,\lambda^{2}t),$ (12.2)

i.e the LLGE is invariant under this scaling. In [89] were studied the
following two types self-similar solutions:
i) the expander self-similar solution

$\displaystyle{\bf S}(x,t)={\bf f}\left(\frac{x}{\sqrt{t-T}}\right),$ (12.3)

ii) the shrinker self-similar solution

$\displaystyle{\bf S}(x,t)={\bf f}\left(\frac{x}{\sqrt{T-t}}\right).$ (12.4)

##  13 Lattice LLE

Almost all main integrable systems have their mirror side in the form of the
lattice or discrete integrable equations [90]-[92]. Here we want briefly
present some basic facts of the lattice version of the LLE (LLLE). It has the
form (see e.g. [10] and references therein)

$\displaystyle\frac{d\Gamma^{(n)}_{\alpha}}{dt}=\\{H_{LLLE},\Gamma^{(n)}_{\alpha}\\},$
(13.1)

where $\Gamma^{(n)}_{\alpha}$ are some dynamical variables and (for details
see e.g. [10])

$\displaystyle H_{LLLE}=\sum_{n=1}^{N}\log
h(\Gamma^{(n)}_{\alpha},\Gamma^{(n+1)}_{\alpha}).$ (13.2)

The isotropic LLLE that is the discrete HFE is given by ([90]-[91])

$\displaystyle{\bf S}_{jt}=2\left[\frac{{\bf S}_{j+1}\wedge{\bf S}_{j}}{1+{\bf
S}_{j+1}\cdot{\bf S}_{j}}-\frac{{\bf S}_{j}\wedge{\bf S}_{j-1}}{1+{\bf
S}_{j}\cdot{\bf S}_{j-1}}\right].$ (13.3)

##  14 Conclusion

In this paper, we have established that the LLE (1.1) is geometrical
equivalent to the generalized nonlinear Schrödinger like equation
(3.9)-(3.11). When the anisotropy vanishes, from this result follows the well-
known results corresponding for the continuous isotropic spin chain, i.e. the
HFE (1.2). We have studied the relation between the LLE and the differential
geometry of space curves in the local and nonlocal cases. Also the soliton
surfaces induced by the LLE are briefly considered using the well-known Sym-
Tafel formula.

##  15 Acknowledgements

This work was supported by the Ministry of Education and Science of
Kazakhstan, Grant AP08856912.

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