# Group properties and solutions for the 1D Hall MHD system in the cold plasma
approximation

Andronikos Paliathanasis
Institute of Systems Science, Durban University of Technology
PO Box 1334, Durban 4000, Republic of South Africa
Instituto de Ciencias Físicas y Matemáticas,
Universidad Austral de Chile, Valdivia, Chile
Email<EMAIL_ADDRESS>
###### Abstract

We study the Lie point symmetries and the similarity transformations for the
partial differential equations of the nonlinear one-dimensional
magnetohydrodynamic system with the Hall term known as HMHD system. For this
1+1 system of partial differential equations we find that is invariant under
the action of a seventh dimensional Lie algebra. Furthermore, the one-
dimensional optimal system is derived while the Lie invariants are applied for
the derivation of similarity transformations. We present different kinds of
oscillating solutions.

Keywords: Magnetohydrodynamic; plasma physics; Lie symmetries; invariants;
similarity solutions.

##  1 Introduction

Lie point symmetries play a significant role in the study on nonlinear
differential equations. The novelty of Lie point symmetries is that they
provide invariant functions which define similarity transformations such that
to write the differential equation in an equivalent simplest form. The latter
can be done in two-ways, either by reducing the number of independent
variables, in the case of partial differential equations, or by reduce the
order of the differential equation, in the case of ordinary differential
equations. Furthermore, Lie symmetries can be used to classify differential
equations with common group properties and when it is feasible to find an
equivalent transformation between different kinds of differential equations
invariants under the elements of same Lie algebra.

In the theory of fluid dynamics Lie symmetries has been widely studied and
applied in various problems, and they have been used for the determination and
the classification of these hydrodynamic systems. Applications of the Lie
symmetries on the Shallow-water equations with or without the Coriolis force
have been studied before in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The group
properties of the two-phase flows system were studied in [13, 14, 15]. In the
presence of an electromagnetic field, and specifically in Plasma physics Lie
symmetries has been applied before. The group properties for the ideal
equations of magnetohydrodynamic (MHD) were studied in detailed in [16]. The
Lie point symmetries for the MHD convection flow and heat transfer of an
incompressible viscous nanofluid past a semi-infinite vertical stretching
sheet in the presence of thermal stratification were studied in [17]. The
Grad–Shafranov equation which is and equilibrium equation in ideal MHD was
studied on the existence of point symmetries in [18]. The relation of the Lie
point symmetries and Noether symmetries and the Lagrangian map in MHD were
investigated before in [19, 20, 21, 22]. On the other hand, new solutions were
found recently for the Pulsar equation in [23].

In this work we are interested on the study of the group-properties of the MHD
equations where the Hall term effect is included. In ideal MHD the Hall term
is very small and usually is neglected, thus Hall term play a significant role
in the study of the magnetic reconnection due to its ability to accurately
describe plasmas with large magnetic field gradients [24], while other
important applications of the Hall term in MHD can be found for instance in
[25, 26, 27, 28].

The Hall MHD (HMHD) equations without any pressure term of the ions or any
electron pressure are [24]

$\displaystyle\rho_{,t}+\nabla\left(\rho\mathbf{u}\right)$
$\displaystyle\mathbf{=}$ $\displaystyle 0\mathbf{,}$ (1)
$\displaystyle\left(\rho\mathbf{u}\right)_{,t}+\nabla\left(\rho\mathbf{u~{}u}+\frac{\left|B\right|^{2}}{2}-\mathbf{BB}\right)$
$\displaystyle=$ $\displaystyle 0,$ (2)
$\displaystyle\mathbf{B}_{,t}-\nabla\times\left(-u\times
B+\frac{\xi_{0}}{\rho}\left(\mathbf{\nabla\times B}\right)\mathbf{\times
B}\right)$ $\displaystyle=$ $\displaystyle 0.$ (3)

where $\xi_{0}$ is the coefficient parameter for the Hall term, while when
$\xi_{0}=0$ the equations of MHD are recovered. The zero pressure
consideration is also known as cold plasma approximation. The dimensionless
parameter $\xi_{0}$ is the ion inertial scale or skin depth defined as
$\xi_{0}=\frac{c}{\omega_{i}L}$, where $c$ is the speed of light, $\omega_{i}$
is the ion plasma frequency and $L$ is the scale length of the plasma. The
Alfén speed is defined as $V_{A}=\frac{B_{0}}{\sqrt{\rho_{0}\mu}}$ where for
the vacuum permeability we have assumed $\mu=1$. If someone considered
parallel propagation on the magnetic field the HMHD system for large values of
the Hall term $\xi_{0}$ the Alfvén waves propagates in the fast manifold by
the derivative nonlinear Schrödinger equation (DNLS) [30, 31, 32]. As it was
found before in [33] the DNLS is an integrable equation while its algebraic
properties has been studied before in [34]. Specifically in [34] the triple
degenerate nonlinear Schrödinger system (TDNLS) was studied. TDNLS arises from
wave propagation along the magnetic field, when the gas sound speed matches
the Alfvén speed, that is, the slow and Alfvén speeds coincide. Because the
original DNLS equation has a singular, divergent coefficient for the nonlinear
term at this limit, a modified perturbation approach has been considered in
order the difference between the Alfvén speed and sound speed to be assumed
small in the perturbation analysis.

In a highly ionized plasma the Hall effect follows because of the difference
in electron and ion inertia. Specifically, ions are incapable to follow the
magnetic fluctuations at frequencies higher than their cyclotron, while
electrons remain coupled to the magnetic field lines. An important
characteristic of the HMHD system is the it admits a Hamiltonian formulation.
In [35] the authors defined a set of canonical variables to describe an
equivalent canonical Hamiltonian system with the HMHD system, while this
property was used to recover the MHD limit. An alternative approach on the
construction of the noncanonical Poisson brackets for the HMHD system can be
found in [36].

We continue by considering that the system is one dimensional and the magnetic
field is constant on the direction $x$, thus let us assume
$\rho=\rho\left(t,x\right)$,
$u=u^{1}\left(t,x\right)\partial_{x}+u^{2}\left(t,x\right)\partial_{y}+u^{3}\left(t,x\right)\partial_{z}~{}$and
$B=B_{0}^{1}\partial_{x}+B^{2}\left(t,x\right)\partial_{y}+B^{3}\left(t,x\right)\partial_{z}$
such that the HMHD equations are simplified in the following form [29]

$\displaystyle\rho_{,t}+\left(\rho u^{1}\right)_{,x}$ $\displaystyle=$
$\displaystyle 0,$ (4) $\displaystyle\left(\rho
u^{1}\right)_{,t}+\left(\rho\left(u^{1}\right)^{2}+\frac{1}{2}\left(\left(B^{2}\right)^{2}+\left(B^{3}\right)^{2}\right)\right)_{,x}$
$\displaystyle=$ $\displaystyle 0,$ (5) $\displaystyle\left(\rho
u^{2}\right)_{,t}+\left(\rho u^{1}u^{2}-B^{1}B^{2}\right)_{,x}$
$\displaystyle=$ $\displaystyle 0,$ (6) $\displaystyle\left(\rho
u^{3}\right)_{,t}+\left(\rho u^{1}u^{3}-B^{1}B^{3}\right)_{,x}$
$\displaystyle=$ $\displaystyle 0,$ (7) $\displaystyle
B_{,t}^{2}-\left(-\left(u^{1}B^{2}-u^{2}B_{0}^{1}\right)+\xi_{0}\frac{B_{0}^{1}}{\rho}B_{,x}^{3}\right)_{,x}$
$\displaystyle=$ $\displaystyle 0,$ (8) $\displaystyle
B_{,t}^{3}+\left(-\left(u^{3}B_{0}^{1}-u^{1}B^{3}\right)+\xi_{0}\frac{B_{0}^{1}}{\rho}B_{,x}^{2}\right)_{,x}$
$\displaystyle=$ $\displaystyle 0.$ (9)

where for the $x$-component from the Faraday’s equaiton it follows
$B_{0,t}^{1}=0$ and $B_{0,x}^{1}=0$. For the latter system we study the
admitted Lie point symmetries, the invariants of the admitted Lie algebra are
investigated as also the invariants are applied for the derivation of
similarity solutions. The plan of the paper is as follows.

In Section 2 we apply Lie’s theory and we derive the infinitesimal generators,
i.e. the Lie symmetries, for the one-parameter point transformation which
leaves the system (4)-(9) invariant. Specifically, we found that the system
admits seven Lie point symmetries, one symmetry less from the same system in
the ideal MHD without the Hall-term. The commutators and the Adjoint
representation for the admitted Lie symmetries are determined which are used
to determine the one-dimensional optimal system. In Section 3 we demonstrate
the use of the Lie symmetries by presenting the application of some similarity
transformations which lead to integrable reduced systems. We recover previous
results for the existence of solitary waves as also we find new oscillating
solutions. Finally, in Section 4 we summarize our results and we draw our
conclusions.

##  2 Lie symmetries for the 1D HMHD equations

Consider the infinitesimal one-parameter point transformation

$\displaystyle\bar{t}$ $\displaystyle=$ $\displaystyle
t+\varepsilon\xi^{1}\left(t,x,\rho,\mathbf{u},B^{2},B^{3}\right),~{}\bar{x}=x+\varepsilon\xi^{2}\left(t,x,\rho,\mathbf{u},B^{2},B^{3}\right),$
(10) $\displaystyle\bar{\rho}$ $\displaystyle=$
$\displaystyle\rho+\varepsilon\eta^{1}\left(t,x,\rho,\mathbf{u},B^{2},B^{3}\right),~{}\bar{u}^{1}=u^{1}+\varepsilon\eta^{2}\left(t,x,\rho,\mathbf{u},B^{2},B^{3}\right),$
(11) $\displaystyle\bar{u}^{2}$ $\displaystyle=$ $\displaystyle
u^{2}+\varepsilon\eta^{3}\left(t,x,\rho,\mathbf{u},B^{2},B^{3}\right),~{}\bar{u}^{2}=u+\varepsilon\eta^{4}\left(t,x,\rho,\mathbf{u},B^{2},B^{3}\right),$
(12) $\displaystyle B^{2}$ $\displaystyle=$ $\displaystyle
B^{2}+\varepsilon\eta^{5}\left(t,x,\rho,\mathbf{u},B^{2},B^{3}\right),B^{3}=B^{3}+\varepsilon\eta^{6}\left(t,x,\rho,\mathbf{u},B^{2},B^{3}\right).$
(13)

with
$\mathbf{X}=\xi^{1}\partial_{t}+\xi^{2}\partial_{x}+\eta^{1}\partial_{\rho}+\eta^{2}\partial_{u^{1}}+\eta^{3}\partial_{u^{2}}+\eta^{4}\partial_{u^{3}}+\eta^{5}\partial_{B^{2}}+\eta^{6}\partial_{B^{3}}$.
Hence, we shall say that the HMHD system $\mathbf{H\equiv 0}$ defined by the
equations (4)-(9) is invariant under the Action of the latter one-parameter
point transformation if and only if

$\mathbf{X}^{\left[1\right]}\left(\mathbf{H}\right)=0,$ (14)

and $\mathbf{X}$ is called a Lie point symmetry, where
$\mathbf{X}^{\left[1\right]}$ is the first extension of the vector field
$\mathbf{X}$ in the jet space [37, 38, 39].

Therefore for the system (4)-(9) from the symmetry condition (14) we find the
Lie point symmetries

$\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle\partial_{t}~{},~{}X_{2}=\partial_{x}~{},X_{3}=t\partial_{x}+\partial_{u^{1}}~{},X_{4}=\partial_{u^{2}}~{},~{}X_{5}=\partial_{u^{3}}$
$\displaystyle X_{6}$ $\displaystyle=$ $\displaystyle
u^{3}\partial_{u^{2}}-u^{2}\partial_{u^{3}}+B^{3}\partial_{B^{2}}-B^{2}\partial_{B^{3}}~{},$
$\displaystyle X_{7}$ $\displaystyle=$ $\displaystyle
x\partial_{x}+u^{1}\partial_{u^{1}}+u^{2}\partial_{u^{2}}+u^{3}\partial_{u^{3}}-2\rho\partial_{\rho}.$

For simplicity on our presentation we have omitted the presentation of the
determining equations.

We observe that the admitted Lie point symmetries are the time and space
translation, the Galilean boost in the $x$ direction is described by $X_{3}$,
while $X_{4},~{}X_{5}$ are translation symmetries on the velocity on the
directions $y$ and $z$, the vector field $X_{6}$ is a rotation symmetry and
$X_{7}$ is a scaling symmetry.

In order to compare the symmetries with that of the MHD system, we assume
$\xi_{0}=0$ in (4)-(9) from where the symmetry vectors follows

$X_{1}~{},~{}X_{2}~{},~{}X_{3}~{},~{}X_{4}~{},~{}X_{5}~{},~{}X_{6}~{},~{}X_{7}\text{
and }X_{8}=t\partial_{t}+x\partial_{x}\text{.}$

Hence, we observe that in the presence of the Hall parameter the scaling
symmetry $X_{8}$ is omitted

The existence of Lie point symmetries for the system (4)-(9) it is essential
for the determination of similarity solutions and of conservation laws. In
this work we are interested on the determination of similarity solutions which
follow by the application of the Lie invariants [38]. The application of a Lie
point symmetry for the reduction of the system of partial differential
equations (4)-(9) lead to a system of ordinary differential equations. In
order to determine all the unique similarity transformations we should
calculate Adjoint representation of the admitted seven-dimensional Lie algebra
an after find the one-dimensional optimal system.

In order to understand this consider the two vector fields

$\mathbf{Z}=\sum\limits_{A=1}^{n}a_{A}X_{A}~{},~{}\mathbf{W}=\sum\limits_{i=1}^{n}b_{A}X_{A}~{},~{}\text{\
}a_{A},~{}b_{A}\text{ are constants and }A=1,2,...,7.$ (15)

The vector fields $Z,~{}W$ are equivalent if an only

$\mathbf{W}=Ad\left(\exp\left(\varepsilon_{A}X_{A}\right)\right)\mathbf{Z~{},~{}}$
(16)

or $W=cZ~{},~{}c=const.~{}$Operator
$Ad\left(\exp\left(\varepsilon_{A}X_{A}\right)\right)$ is called the Adjoint
representation [38]

$Ad\left(\exp\left(\varepsilon
X_{A}\right)\right)X_{B}=X_{A}-\varepsilon\left[X_{A},X_{B}\right]+\frac{1}{2}\varepsilon^{2}\left[X_{A},\left[X_{A},X_{B}\right]\right]+...~{},$
(17)

where $\left[X_{A},X_{B}\right]$ is the commutator operator. In Tables 1 and 2
we present the commutators and the Adjoint representation for the admitted
seven-dimensional Lie algebra by the system 1D HMHD system (4)-(9).

Table 1: Commutators of the admitted Lie point symmetries for the 1D HMHD system $\left[~{},~{}\right]$ | $\mathbf{X}_{1}$ | $\mathbf{X}_{2}$ | $\mathbf{X}_{3}$ | $\mathbf{X}_{4}$ | $\mathbf{X}_{5}$ | $\mathbf{X}_{6}$ | $\mathbf{X}_{7}$
---|---|---|---|---|---|---|---
$\mathbf{X}_{1}$ | $0$ | $0$ | $X_{2}$ | $0$ | $0$ | $0$ | $0$
$\mathbf{X}_{2}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $X_{2}$
$\mathbf{X}_{3}$ | $-X_{2}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $X_{3}$
$\mathbf{X}_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-X_{5}$ | $X_{4}$
$\mathbf{X}_{5}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $X_{4}$ | $X_{5}$
$\mathbf{X}_{6}$ | $0$ | $0$ | $0$ | $X_{5}$ | $-X_{4}$ | $0$ | $0$
$\mathbf{X}_{7}$ | $0$ | $-X_{2}$ | $-X_{3}$ | $-X_{4}$ | $-X_{5}$ | $0$ | $0$
Table 2: Adjoint representation for the admitted Lie point symmetries of the 1D HMHD system $Ad\left(\exp\left(\varepsilon X_{A}\right)\right)X_{B}$ | $\mathbf{X}_{1}$ | $\mathbf{X}_{2}$ | $\mathbf{X}_{3}$ | $\mathbf{X}_{4}$ | $\mathbf{X}_{5}$ | $\mathbf{X}_{6}$ | $\mathbf{X}_{7}$
---|---|---|---|---|---|---|---
$\mathbf{X}_{1}$ | $X_{1}$ | $X_{2}$ | $X_{3}-\varepsilon X_{2}$ | $X_{4}$ | $X_{5}$ | $X_{6}$ | $X_{7}$
$\mathbf{X}_{2}$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}$ | $X_{7}-\varepsilon X_{2}$
$\mathbf{X}_{3}$ | $X_{1}+\varepsilon X_{2}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}$ | $X_{7}-\varepsilon X_{3}$
$\mathbf{X}_{4}$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}+\varepsilon X_{5}$ | $X_{7}-\varepsilon X_{4}$
$\mathbf{X}_{5}$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}-\varepsilon X_{4}$ | $X_{7}-\varepsilon X_{5}$
$\mathbf{X}_{6}$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}\cos\varepsilon-X_{5}\sin\varepsilon$ | $X_{4}\sin\varepsilon+X_{4}\cos\varepsilon$ | $X_{6}$ | $X_{7}$
$\mathbf{X}_{7}$ | $X_{1}$ | $e^{\varepsilon}X_{2}$ | $e^{\varepsilon}X_{3}$ | $e^{\varepsilon}X_{4}$ | $e^{\varepsilon}X_{5}$ | $X_{6}$ | $X_{7}$

###  2.1 One-dimensional optimal system

The determination of the one-dimensional optimal system is essential in order
to perform a complete classification of the similarity transformations
according to the definition presented in [37]. As a first step the invariants
$\phi\left(a_{a}\right)~{}$of the Adjoint action should be derived. They are
given by the system

$\Delta_{A}\left(\phi\right)=C_{AB}^{C}a^{B}\frac{\partial}{\partial
a^{A}}\phi\equiv 0$ (18)

where $C_{AB}^{C}$ are the structure constants of the Lie algebra, as they are
presented in Table 1.

Therefore, we end with the system

$a_{3}\frac{\partial\phi}{\partial
a_{2}}=0~{},~{}-a_{1}\frac{\partial\phi}{\partial
a_{2}}+a_{7}\frac{\partial\phi}{\partial
a_{3}}=0~{},~{}-a_{6}\frac{\partial\phi}{\partial
a_{5}}+a_{7}\frac{\partial\phi}{\partial a_{4}}=0$ (19)
$-a_{6}\frac{\partial\phi}{\partial a_{4}}+a_{7}\frac{\partial\phi}{\partial
a_{5}}=0~{},~{}a_{4}\frac{\partial\phi}{\partial
a_{5}}-a_{5}\frac{\partial\phi}{\partial a_{4}}=0~{},$ (20)
$a_{2}\frac{\partial\phi}{\partial a_{2}}-a_{3}\frac{\partial\phi}{\partial
a_{3}}-a_{4}\frac{\partial\phi}{\partial
a_{4}}-a_{5}\frac{\partial\phi}{\partial a_{5}}=0.$ (21)

which gives $\phi=\phi\left(a_{1},a_{6},a_{7}\right)$, that is, the invariants
are the $a_{1},~{}a_{6},~{}$ and $a_{7}$.

Consider now the generic symmetry vector

$Y=a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}+a_{4}X_{4}+a_{5}X_{5}+a_{6}X_{6}+a_{7}X_{7},$
(22)

for the case where $a_{1}a_{6}a_{7}\neq 0$, it follows

$\bar{Y}=Ad\left(e^{\varepsilon_{2}X_{2}}\right)Ad\left(e^{\varepsilon_{3}X_{3}}\right)Ad\left(e^{\varepsilon_{4}X_{4}}\right)Ad\left(e^{\varepsilon_{5}X_{5}}\right)Y,$
(23)

where for specific values of the free parameters
$\varepsilon_{2},~{}\varepsilon_{3},~{}\varepsilon_{4}$ and $\varepsilon_{5}$,
the vector field $\bar{Y}$ takes the form

$\bar{Y}=\bar{a}_{1}X_{1}+\bar{a}_{6}X_{6}+\bar{a}_{7}X_{7}.$ (24)

which is the equivalent symmetry vector to the generic field $Y.$

For $a_{1}a_{6}\neq 0$ , $a_{7}=0$, with the same approach we find the
invariants $\phi\left(a_{1},a_{3},a_{6}\right)$, from where we get the
equivalent vector field of $Y$ to be

$Y^{\prime}=a_{1}^{\prime}X_{1}+a_{3}^{\prime}X_{3}+a_{6}^{\prime}X_{6}.$ (25)

In a similar way we continue and for the rest of the invariants. Therefore, we
find that the one-dimensional optimal system for the 1D HMHD system (4)-(9) is
consisted by the symmetry vector fields

$\displaystyle\left\\{X_{1}\right\\}~{},~{}\left\\{X_{2}\right\\}~{},~{}\left\\{X_{3}\right\\}~{},~{}\left\\{X_{4}\right\\}~{},~{}\left\\{X_{5}\right\\}~{},~{}\left\\{X_{6}\right\\}~{},~{}\left\\{X_{7}\right\\}$
$\displaystyle\left\\{X_{1}+\alpha X_{2}\right\\}~{},~{}\left\\{X_{1}+\alpha
X_{3}\right\\}~{},~{}\left\\{X_{1}+\alpha
X_{4}\right\\}~{},~{}\left\\{X_{1}+\alpha X_{5}\right\\}~{},~{}$
$\displaystyle\left\\{X_{1}+\alpha X_{7}\right\\}~{},~{}\left\\{X_{1}+\alpha
X_{3}+\beta X_{4}\right\\},~{}\left\\{X_{1}+\alpha X_{3}+\beta
X_{5}\right\\}~{},$ $\displaystyle\left\\{X_{1}+\alpha X_{3}+\beta
X_{6}\right\\}~{},~{}\left\\{X_{1}+\alpha X_{6}+\beta
X_{7}\right\\}~{}~{},~{}\left\\{X_{2}+\alpha
X_{4}\right\\}~{},~{}\left\\{X_{2}+\alpha X_{5}\right\\}~{},$
$\displaystyle\left\\{X_{2}+\alpha X_{6}\right\\}~{},~{}\left\\{X_{2}+\alpha
X_{3}+\beta X_{4}\right\\}~{},~{}\left\\{X_{2}+\alpha X_{3}+\beta
X_{5}\right\\}~{},\left\\{X_{2}+\alpha X_{3}+\beta X_{6}\right\\}~{},$
$\displaystyle\left\\{X_{2}+\alpha X_{3}+\beta X_{4}+\gamma
X_{5}\right\\}~{},~{}\left\\{X_{2}+\alpha X_{3}+\beta X_{4}+\gamma
X_{6}\right\\}~{},~{}\left\\{X_{2}+\alpha X_{3}+\beta X_{5}+\gamma
X_{6}\right\\}~{},$ $\displaystyle\left\\{X_{3}+\alpha
X_{4}\right\\}~{},~{}\left\\{X_{3}+\alpha
X_{5}\right\\}~{},\left\\{X_{3}+\alpha
X_{6}\right\\}~{},\left\\{X_{3}+aX_{4}+\beta
X_{5}\right\\}~{},~{}\left\\{X_{3}+aX_{4}+\beta X_{6}\right\\}~{},$
$\displaystyle\left\\{X_{3}+aX_{5}+\beta
X_{6}\right\\}~{},~{}\left\\{X_{4}+\alpha
X_{5}\right\\}~{},~{}\left\\{X_{4}+\alpha
X_{6}\right\\}~{},~{}\left\\{X_{5}+\alpha
X_{6}\right\\}~{},~{}\left\\{X_{6}+\alpha X_{7}\right\\}\text{.}$

The Lie invariants which define the similarity transformations for the one-
dimensional optimal system are presented in Tables 3, 4 and 5. In this tables
the $\mathcal{R}\left(\theta\right)$ denotes the rotation matrix defined as
follows

$\mathcal{R}\left(\theta\right)=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\text{.}$

We proceed our analysis with the application of the Lie invariants for the
determination of similarity solutions for the 1D HMHD system.

Table 3: Invariant functions of the one-dimensional optimal system for the 1D HMHD system (1/3) Symmetry | Invariants
---|---
$X_{1}$ | $x,\mathbf{u,}B^{2},B^{3},\rho$
$X_{2}$ | $t,\mathbf{u,}B^{2},B^{3},\rho$
$X_{3}$ | $t,\frac{x}{t}-u^{1},u^{2},u^{3},B^{2},B^{3},\rho$
$X_{4}$ | $t,x,u^{1},u^{3},B^{2},B^{3},\rho$
$X_{5}$ | $t,x,u^{1},u^{2},B^{2},B^{3},\rho$
$X_{6}$ | $t,x,u^{1}~{},~{}\left(u^{2}\right)+\left(u^{3}\right)^{2},\left(B^{2}\right)+\left(B^{3}\right)^{2},\arctan\left(\frac{u^{3}}{u^{2}}\right)-\arctan\left(\frac{B^{3}}{B^{2}}\right),~{}\rho$
$X_{7}$ | $t,\mathbf{u,}B^{2},B^{3},x^{2}\rho$
$X_{1}+\alpha X_{2}$ | $x-\alpha t,\mathbf{u,}B^{2},B^{3},\rho$
$X_{1}+\alpha X_{3}$ | $x-\frac{\alpha}{2}t^{2},t-u^{1},u^{2},u^{3},B^{2},B^{3},\rho$
$X_{1}+\alpha X_{4}$ | $x,u^{1},\alpha t-u^{2},u^{3},B^{2},B^{3},\rho$
$X_{1}+\alpha X_{5}$ | $x,u^{1},u^{2},\alpha t-u^{3},B^{2},B^{3},\rho$
Table 4: Invariant functions of the one-dimensional optimal system for the 1D HMHD system (2/3) Symmetry | Invariants
---|---
$X_{1}+\alpha X_{7}$ | $xe^{-\alpha t},e^{-at}\mathbf{u,}B^{2},B^{3},e^{-2at}\rho$
$X_{2}+\alpha X_{4}$ | $t,u^{1},ax-u^{2},u^{3},B^{2},B^{3},\rho$
$X_{2}+\alpha X_{5}$ | $t,u^{1},u^{2},ax-u^{3},B^{2},B^{3},\rho$
$X_{2}+\alpha X_{6}$ | $t,u^{1},\mathcal{R}\left(\alpha x\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix},\mathcal{R}\left(\alpha x\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}\rho$
$X_{3}+\alpha X_{4}$ | $t,\frac{x}{t}-u^{1},\alpha\frac{x}{t}-u^{2},u^{3},B^{2},B^{3},\rho$
$X_{3}+\alpha X_{5}$ | $t,\frac{x}{t}-u^{1},u^{2},\alpha\frac{x}{t}-u^{3},B^{2},B^{3},\rho$
$X_{3}+\alpha X_{6}$ | $t,\frac{x}{t}-u^{1},\mathcal{R}\left(\alpha\frac{x}{t}\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix},\mathcal{R}\left(\alpha\frac{x}{t}\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}\rho$
$X_{4}+\alpha X_{5}$ | $t,x,u^{1},u^{2}-\alpha u^{3},B^{2},B^{3},\rho$
$X_{4}+\alpha X_{6}$ | $t,x,u^{1},\left(u^{2}\right)^{2}+\left(u^{3}\right)^{2}+\frac{2}{\alpha}u^{3},~{}\left(B^{2}\right)+\left(B^{3}\right)^{2}~{},~{}\arctan\left(\frac{B^{3}}{B^{2}}\right)-\arctan\left(\frac{a}{1+au^{3}}\right)~{},~{}\rho$
$X_{5}+\alpha X_{6}$ | $t,x,u^{1},\left(u^{2}\right)^{2}+\left(u^{3}\right)^{2}+\frac{2}{\alpha}u^{2},~{}\left(B^{2}\right)+\left(B^{3}\right)^{2}~{},~{}\arctan\left(\frac{B^{3}}{B^{2}}\right)-\arctan\left(\frac{a}{1+au^{2}}\right)~{},~{}\rho$
$X_{6}+\alpha X_{7}$ | $t,\frac{u^{1}}{x},\frac{i}{2}\left(u^{2}-iu^{3}\right)x^{-\frac{\alpha+i}{\alpha}},~{}-\frac{1}{2}\left(iu^{2}-u^{3}\right)x^{-\frac{\alpha+i}{\alpha}},\mathcal{R}\left(\frac{\ln x}{a}\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}x^{2}\rho$
Table 5: Invariant functions of the one-dimensional optimal system for the 1D HMHD system (3/3) Symmetry | Invariants
---|---
$X_{1}+\alpha X_{3}+\beta X_{4}$ | $x-\frac{\alpha}{2}t^{2},\alpha t-u^{1},\beta t-u^{2},u^{3},B^{2},B^{3},\rho$
$X_{1}+\alpha X_{3}+\beta X_{5}$ | $x-\frac{\alpha}{2}t^{2},\alpha t-u^{1},u^{2},\beta t-u^{3},B^{2},B^{3},\rho$
$X_{1}+\alpha X_{3}+\beta X_{6}$ | $x-\frac{\alpha}{2}t^{2},\alpha t-u^{1},\mathcal{R}\left(\beta t\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix},\mathcal{R}\left(\beta t\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}\rho$
$X_{1}+\alpha X_{6}+\beta X_{7}$ | $xe^{-\beta t},e^{-\beta t}u^{1},e^{-\beta t}\mathcal{R}\left(at\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix},\mathcal{R}\left(at\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}e^{2\beta t}\rho$
$X_{2}+\alpha X_{3}+\beta X_{4}$ | $t,\frac{\alpha x}{1+\alpha t}-u^{1},\frac{\alpha x}{1+\alpha t}-u^{2},u^{3},B^{2},B^{3},\rho$
$X_{2}+\alpha X_{3}+\beta X_{5}$ | $t,\frac{\alpha x}{1+\alpha t}-u^{1},u^{2},\frac{\alpha x}{1+\alpha t}-u^{3},B^{2},B^{3},\rho$
$X_{2}+\alpha X_{3}+\beta X_{6}$ | $t,\frac{\alpha x}{1+\alpha t}-u^{1},\mathcal{R}\left(\frac{\beta x}{1+\alpha t}\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix},\mathcal{R}\left(\frac{\beta x}{1+\alpha t}\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}\rho$
$X_{3}+aX_{4}+\beta X_{5}$ | $t,\frac{x}{t}-u^{1},\frac{x}{t}-u^{2},\frac{x}{t}-u^{3},B^{2},B^{3},\rho$
$X_{3}+aX_{4}+\beta X_{6}$ | $t,\frac{x}{t}-u^{1},\mathcal{R}\left(\frac{\beta x}{t}\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix}-\begin{pmatrix}\frac{a}{\beta}\cos\frac{\beta x}{t}\\\ -\frac{a}{\beta}\sin\frac{\beta x}{t}\end{pmatrix},\mathcal{R}\left(\frac{\beta x}{t}\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}\rho$
$X_{3}+aX_{5}+\beta X_{6}$ | $t,\frac{x}{t}-u^{1},\mathcal{R}\left(\frac{\beta x}{t}\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix}-\begin{pmatrix}\frac{a}{\beta}\sin\frac{\beta x}{t}\\\ -\frac{a}{\beta}\cos\frac{\beta x}{t}\end{pmatrix},\mathcal{R}\left(\frac{\beta x}{t}\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}\rho$
$X_{2}+\alpha X_{3}+\beta X_{4}+\gamma X_{5}$ | $t,\frac{\alpha x}{1+\alpha t}-u^{1},\frac{\beta x}{1+\alpha t}-u^{2},\frac{\gamma x}{1+\alpha t}-u^{3},B^{2},B^{3},\rho$
$X_{2}+\alpha X_{3}+\beta X_{4}+\gamma X_{6}$ | $t,\frac{\alpha x}{1+\alpha t}-u^{1},\mathcal{R}\left(\frac{\gamma x}{1+\alpha t}\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix}-\begin{pmatrix}\frac{\beta}{\gamma}\cos\frac{\gamma x}{1+\alpha t}\\\ -\frac{\beta}{\gamma}\sin\frac{\gamma x}{1+\alpha t}\end{pmatrix},\mathcal{R}\left(\frac{\gamma x}{1+\alpha t}\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}\rho$
$X_{2}+\alpha X_{3}+\beta X_{5}+\gamma X_{6}$ | $t,\frac{x}{t}-u^{1},\mathcal{R}\left(\frac{\gamma x}{1+\alpha t}\right)\begin{pmatrix}u^{2}\\\ u^{3}\end{pmatrix}-\begin{pmatrix}\frac{\beta}{\gamma}\sin\frac{\gamma x}{1+\alpha t}\\\ -\frac{\beta}{\gamma}\cos\frac{\gamma x}{1+\alpha t}\end{pmatrix},\mathcal{R}\left(\frac{\gamma x}{1+\alpha t}\right)\begin{pmatrix}B^{2}\\\ B^{3}\end{pmatrix},~{}\rho$

##  3 Similarity transformations

Before we proceed with the application of similarity transformations for the
determination of exact solutions it is important to mention that the
application of Lie point symmetries for partial differential equations
reducing the number of the independent variables until the reduced system to
be consisted by ordinary differential equations. Thus, not all the Lie
symmetries can play role in the reduction of the system (4)-(9) into a system
of ordinary differential equations.

In the following we continue by demonstrate with some applications how the Lie
symmetries can be applied for the determination of exact solutions.

###  3.1 Symmetry vector $X_{1}-X_{2}$

The application of the Lie invariants which follows from the symmetry vector
$\left\\{X_{1}-X_{2}\right\\}$ in the 1D HMHD system (4)-(9) provides the
following system of ordinary differential equations

$\displaystyle\left(\rho+\rho u^{1}\right)_{,w}$ $\displaystyle=$
$\displaystyle 0,$ (26)
$\displaystyle\left(u^{1}\rho+\rho\left(u^{1}\right)^{2}+\frac{1}{2}\left(\left(B^{2}\right)^{2}+\left(B^{3}\right)^{2}\right)\right)_{,w}$
$\displaystyle=$ $\displaystyle 0,$ (27) $\displaystyle\left(\rho u^{2}+\rho
u^{1}u^{2}-B^{1}B^{2}\right)_{,w}$ $\displaystyle=$ $\displaystyle 0,$ (28)
$\displaystyle\left(\rho u^{3}\right)_{,t}+\left(\rho u^{3}+\rho
u^{1}u^{3}-B^{1}B^{3}\right)_{,w}$ $\displaystyle=$ $\displaystyle 0,$ (29)
$\displaystyle\left(B^{2}-\left(u^{1}B^{2}-u^{2}B_{0}^{1}\right)+\xi_{0}\frac{B_{0}^{1}}{\rho}B_{,x}^{3}\right)_{,w}$
$\displaystyle=$ $\displaystyle 0,$ (30)
$\displaystyle\left(B_{,t}^{3}-\left(u^{3}B_{0}^{1}-u^{1}B^{3}\right)+\xi_{0}\frac{B_{0}^{1}}{\rho}B_{,x}^{2}\right)_{,w}$
$\displaystyle=$ $\displaystyle 0.$ (31)

where $w=t+x$ and the dependent variables are functions of $w$. From equations
(26)-(29) it follows

$\displaystyle\rho\left(w\right)$ $\displaystyle=$ $\displaystyle
I_{1}^{2}\left(\left(I_{0}+I_{1}\right)-\frac{1}{2}\left(\left(B^{2}\right)^{2}+\left(B^{3}\right)^{2}\right)\right)^{-1}~{},$
(32) $\displaystyle u^{1}\left(w\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2I_{1}}\left(2I_{2}-\left(\left(B^{2}\right)^{2}+\left(B^{3}\right)^{2}\right)\right)~{},$
(33) $\displaystyle u^{2}\left(w\right)$ $\displaystyle=$
$\displaystyle\frac{B_{0}^{1}B^{2}+I_{3}}{I_{1}}~{},~{}u^{3}\left(w\right)=\frac{B_{0}^{1}B^{3}+I_{3}}{I_{1}}~{},$
(34)

in which $I_{1},I_{2},I_{3}$ and $I_{4}$ are integration constants. Hence, for
$B^{2}\left(w\right)$ and $B^{3}\left(w\right)$ we find the first-order
ordinary differential equations

$\displaystyle B_{,w}^{2}$ $\displaystyle=$
$\displaystyle-\frac{I_{1}}{B_{0}^{1}\xi_{0}}B^{3}+\frac{I_{1}\left(B_{0}^{1}B^{3}+2B_{0}^{1}I_{4}+2I_{5}I_{1}\right)}{B_{0}^{1}\xi_{0}\left(2\left(I_{1}+I_{2}\right)-\left(\left(B_{2}\right)^{2}+\left(B_{2}^{3}\right)\right)\right)}~{},$
(35) $\displaystyle B_{,w}^{3}$ $\displaystyle=$
$\displaystyle+\frac{I_{1}}{B_{0}^{1}\xi_{0}}B^{2}-\frac{I_{1}\left(B_{0}^{1}B^{2}+2B_{0}^{1}I_{3}+2I_{6}I_{1}\right)}{B_{0}^{1}\xi_{0}\left(2\left(I_{1}+I_{2}\right)-\left(\left(B_{2}\right)^{2}+\left(B_{2}^{3}\right)\right)\right)}~{},$
(36)

with $I_{5},I_{6}$ to be integration constants. The phase portrait of the
dynamical system (35), (36) is presented in Fig. 1 where we observe the
existence of oscillating trajectories [29].

Figure 1: Phase space portrait for the system (35), (36) for different values
of the free parameters. We observe that there are periodic solutions which
indicates the existence of travel-wave solutions for the system.

###  3.2 Symmetry vector $X_{1}+X_{3}$

From the point symmetry vector $X_{1}+X_{3}$ we find

$I_{1}=\rho
u^{1}~{},~{}u^{2}=\frac{B_{0}^{1}B^{2}+I_{2}}{I_{1}}~{},~{}u^{3}=\frac{B_{0}^{1}B^{3}+I_{3}}{I_{1}}~{},$
(37)

where
$\rho=\rho\left(\sigma\right),~{}B^{2}=B^{2}\left(\sigma\right)~{},~{}B^{3}=B^{3}\left(\sigma\right),~{}$
$\sigma=x+\frac{t^{2}}{2}$ satisfy the following system of first-order
ordinary differential equations

$\displaystyle\rho_{,\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\xi_{0}B_{0}^{1}I_{1}^{2}}\rho^{3}\left(\xi_{0}B_{0}^{1}+I_{5}B^{2}-I_{6}B^{3}\right)~{},~{}$
(38) $\displaystyle B_{,\sigma}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\xi_{0}B_{0}^{1}I_{1}}\left(I_{5}I_{1}\rho+\left(B_{0}^{1}\rho-
I_{1}^{2}\right)B^{3}\right)~{},~{}$ (39) $\displaystyle B_{,\sigma}^{3}$
$\displaystyle=$
$\displaystyle-\frac{1}{\xi_{0}B_{0}^{1}I_{1}}\left(I_{6}I_{1}\rho+\left(B_{0}^{1}\rho-
I_{1}^{2}\right)B^{2}\right)~{}.$ (40)

In analytic solution of the latter system can be written in terms of the
Laurent expansion

$\displaystyle\rho\left(\sigma\right)$ $\displaystyle=$
$\displaystyle\rho_{0}\sigma^{-\frac{1}{2}}+\rho_{1}\sigma^{-1}+\rho_{2}\sigma^{-\frac{3}{2}}+...~{},$
(41) $\displaystyle B^{2}\left(\sigma\right)$ $\displaystyle=$ $\displaystyle
B_{0}^{2}\sigma^{\frac{1}{2}}+B_{1}^{2}+B_{2}^{2}\sigma^{-\frac{1}{2}}+...~{},$
(42) $\displaystyle B^{3}\left(\sigma\right)$ $\displaystyle=$ $\displaystyle
B_{0}^{3}\sigma^{\frac{1}{2}}+B_{1}^{3}+B_{2}^{3}\sigma^{-\frac{1}{2}}+...~{},$
(43)

in which
$\rho_{0}=\frac{iI_{1}}{\sqrt{2}}~{},~{}B_{0}^{2}=\frac{i\sqrt{2}I_{1}}{B_{0}^{1}\xi_{0}}I_{5},~{}B_{0}^{3}=\frac{i\sqrt{2}I_{1}}{B_{0}^{1}\xi_{0}}I_{6},...$
.

The qualitative evolution of the dynamical variables $\rho,~{}B^{2}$ and
$B^{3}$ are presented for various sets of initial conditions in Fig. 2. From
the figure we observe a periodic behaviour on the dynamical variables.

Figure 2: Qualitative evolution of the dynamical variables
$\rho\left(\sigma\right)$ (red line), $B^{2}\left(\sigma\right)$ (blue line)
and $B^{3}\left(\sigma\right)$ (orange line) for different sets of the free
variables. From the plots we observe a periodic solution, this is a new wave
solution.

###  3.3 Symmetry vector $X_{7}$

Application of the symmetry vector $X_{7}$ in the system (4)-(9) provides the
closed-form solution

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}t+\rho_{1}}{x^{2}}~{},~{}u^{1}=\frac{\rho_{0}x}{\rho_{0}t+\rho_{1}}~{},~{}$
(44) $\displaystyle u^{2}$ $\displaystyle=$
$\displaystyle\frac{u_{0}^{2}x}{\rho_{0}t+\rho_{1}}~{},~{}u^{3}=\frac{u_{0}^{3}x}{\rho_{0}t+\rho_{1}}~{},~{}$
(45) $\displaystyle B^{2}$ $\displaystyle=$
$\displaystyle\frac{B_{0}^{1}u_{0}^{2}t+B_{0}^{2}}{\rho_{0}t+\rho_{1}}~{},~{}B^{2}=\frac{B_{0}^{1}u_{0}^{3}t+B_{0}^{3}}{\rho_{0}t+\rho_{1}}~{}.$
(46)

###  3.4 Symmetry vector $X_{1}+aX_{7}$

From the vector field $X_{1}+aX_{7}$ we find $z=xe^{-\alpha
t},~{}v=e^{-at}\mathbf{u~{},~{}}B^{2},B^{3},\mu=e^{-2at}\rho$, hence by
replacing in (4)-(9) it follows

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle\rho
v^{1}\left(v^{1}-az\right)+\frac{1}{2}\left(\left(B^{2}\right)+\left(B^{3}\right)^{2}\right)^{2}~{},$
(47) $\displaystyle I_{2}$ $\displaystyle=$ $\displaystyle\rho
v^{2}\left(v^{1}-az\right)-B_{0}^{1}B^{2}~{},$ (48) $\displaystyle I_{3}$
$\displaystyle=$ $\displaystyle\rho
v^{3}\left(v^{1}-az\right)-B_{0}^{1}B^{3}~{},$ (49)

where $\rho\left(z\right),~{}B^{2}\left(z\right)$ and $B^{3}\left(z\right)$
satisfy the following system of ordinary differential equation

$\displaystyle\left(\rho v^{1}-az\rho\right)_{,z}-az\rho$ $\displaystyle=$
$\displaystyle 0,$ (50)
$\displaystyle\left(B^{2}v^{1}-B_{0}^{1}v^{2}-\frac{\xi_{0}B_{0}^{1}}{\rho}B_{,z}^{2}\right)_{,z}-azB_{,z}^{2}$
$\displaystyle=$ $\displaystyle 0,$ (51)
$\displaystyle\left(B^{3}v^{1}-B_{0}^{1}v^{3}+\frac{\xi_{0}B_{0}^{1}}{\rho}B_{,z}^{3}\right)_{,z}-azB_{,z}^{3}$
$\displaystyle=$ $\displaystyle 0.$ (52)

The qualitative evolution of the variables
$\rho\left(z\right),~{}B^{2}\left(z\right)$ and $B^{3}\left(z\right)$ as it is
given by the numerical simulation of the system (50)-(52) is presented in Fig.
3.

Figure 3: Numerical simulation and evolution of the dynamical variables
$\rho\left(z\right)~{}$(red line)$,~{}B^{2}\left(z\right)$ (blue line) and
$B^{3}\left(z\right)$ (orange line) as it is given by the system (50)-(52).
The plots are for different values of the free parameter. For the initial
conditions we considerered
$\rho\left(0\right)=0.1$,$~{}B^{2}\left(0\right)=0.1$,
$B^{3}\left(0\right)=-0.01$, $B_{,z}^{2}\left(0\right)=0.01,$
$B_{z}^{3}\left(0\right)=0.01,~{}v^{1}\left(0\right)=0.1,~{}v^{2}\left(0\right)=-0.02,~{}v^{3}\left(0\right)=0.2$.

###  3.5 Symmetry vector $X_{3}+aX_{6}$

From the Lie symmetry vector $X_{3}+X_{6}$ it follows

$\displaystyle B^{2}\left(t,x\right)$ $\displaystyle=$
$\displaystyle-b^{2}\left(t\right)\cos\left(\frac{x}{t}\right)+b^{3}\left(t\right)\sin\left(\frac{x}{t}\right),$
$\displaystyle B^{3}\left(t,x\right)$ $\displaystyle=$ $\displaystyle
b^{2}\left(t\right)\sin\left(\frac{x}{t}\right)+b^{3}\left(t\right)\cos\left(\frac{x}{t}\right),$
$\displaystyle u^{1}\left(t,x\right)$ $\displaystyle=$
$\displaystyle\frac{x}{t}+v^{1}\left(t\right),$ $\displaystyle
u^{2}\left(t,x\right)$ $\displaystyle=$ $\displaystyle
v^{2}\left(t\right)\sin\left(\frac{x}{t}\right)+v^{3}\cos\left(\frac{x}{t}\right)~{},$
$\displaystyle u^{3}\left(t,x\right)$ $\displaystyle=$ $\displaystyle
v^{2}\left(t\right)\cos\left(\frac{x}{t}\right)-v^{3}\sin\left(\frac{x}{t}\right)~{},$
$\displaystyle\rho\left(t,x\right)$ $\displaystyle=$
$\displaystyle\rho\left(t\right),$

while the 1D HMHD system (4)-(9) provides

$\rho\left(t\right)=\rho_{0}t^{-1},~{}v^{1}\left(t\right)=v_{0}^{1}t^{-1}~{},$
(53)

and

${}_{0}v_{,t}^{2}-_{0}v_{0}^{1}t^{-2}v^{3}-B_{0}^{1}b^{2}$ $\displaystyle=$
$\displaystyle 0,$ (54)
$\displaystyle\rho_{0}v_{,t}^{3}+_{0}v_{0}^{1}t^{-2}v^{2}-B_{0}^{1}b^{3}$
$\displaystyle=$ $\displaystyle 0,$ (55) $\displaystyle
tb_{,t}^{2}-\frac{\xi_{0}B_{0}^{1}}{\rho_{0}}b^{3}+\left(B_{0}^{1}v^{2}+b^{2}-b^{3}v_{0}^{1}t\right)$
$\displaystyle=$ $\displaystyle 0,$ (56) $\displaystyle
tb_{,t}^{3}+\frac{\xi_{0}B_{0}^{2}}{\rho_{0}}b^{2}+\left(B_{0}^{1}v^{3}+b^{3}+b^{2}v_{0}^{1}t\right)$
$\displaystyle=$ $\displaystyle 0.$ (57)

Consider the special case where $v_{0}^{1}=0$, then from the two first
equation of the latter system it follows
$b^{2}=\frac{{}_{0}}{B_{0}^{1}}v_{,t}^{2}$ and
$b^{3}=\frac{{}_{0}}{B_{0}^{3}}v_{,t}^{3}$. Thus by replacing in the rest two
equations we find

$\displaystyle
t\frac{{}_{0}}{B_{0}^{1}}v_{,tt}^{2}-\left(\xi_{0}-\frac{{}_{0}}{B_{0}^{1}}\right)v_{,t}^{2}+B_{0}^{1}v^{2}$
$\displaystyle=$ $\displaystyle 0~{},$ (58) $\displaystyle
t\frac{{}_{0}}{B_{0}^{1}}v_{,tt}^{3}+\left(\xi_{0}+\frac{{}_{0}}{B_{0}^{1}}\right)v_{,t}^{3}+B_{0}^{1}v^{3}$
$\displaystyle=$ $\displaystyle 0~{},$ (59)

with solution which can expressed in terms of the Bessel functions
$J_{\alpha}\left(t\right)$ and $Y_{\alpha}\left(t\right)$. Note that for
$v_{0}^{1}\neq 0$ system (54)-(57) can be written into the form of an
equivalent system of two linear second-order ordinary differential equations
which means that the system is integrable.

In Fig. 4 we present the numerical solution of the dynamical system (54)-(57)
while some phase space portraits are given in Fig. 5. It is clear that the
behaviour is of the dynamical variables has oscillations for $v_{0}^{1}$ and
$B_{0}^{1}$ positive.

Figure 4: Numerical simulation and evolution of the dynamical variables
$v^{2}\left(t\right)~{}$(red line)$,~{}v^{3}\left(t\right)~{}$(blue line)
$b^{2}\left(z\right)$ (orange line) and $b^{3}\left(z\right)$ (purple line) as
it is given by the system (54)-(57). The plots are for different values of the
free parameter. Figure 5: Phase space portraits on the variables
$\left\\{v^{2}-v^{3}\right\\}$ and $\left\\{b^{2}-b^{3}\right\\}$ for the
oscillating solutions of Fig. 4

##  4 Conclusions

In this work we studied the group properties the nonlinear one-dimensional
system of MHD included the Hall term. The dynamical system is consisted by six
1+1 partial differential equations. For this system, we applied the theory of
Sophus Lie and we determined all the possible one-parameter point
transformations in which the HMHD equations are invariant. We found that the
admitted Lie point symmetries form a seven dimensional Lie algebra. The
admitted Lie algebra has common element with that of the MHD system without
the Hall term, thus it admits a smaller number of Lie symmetries for the
equivalent system without the Hall term.

For the admitted Lie symmetries we calculated the commutators and the Adjoint
representations as also the adjoint invariants. These results were applied in
order to determine all the one-dimensional Lie algebras which consisted the
optimal system. We found that the one-dimensional system is consisted by
thirty five independent vector fields. For the latter vector fields the
invariant functions which define the similarity transformations were
determined which are applied for the definition of the similarity
transformations.

Furthermore, we applied the Lie point symmetries to reduce the partial
differential equations into a system of ordinary differential equations and to
study the behaviour of the dynamical variables. Travel-wave and scaling
solutions were found.

These results contribute to the subject of the application of the Lie point
symmetries to fluid dynamics and specifically on MHD.

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# Observation of $\pi$ solitons in oscillating waveguide arrays

Antonina A. Arkhipova  Yiqi Zhang  Yaroslav V. Kartashov
<EMAIL_ADDRESS>Sergei A. Zhuravitskii  NNikolay N. Skryabin  Ivan
V. Dyakonov  Alexander A. Kalinkin  Sergei P. Kulik  Victor O. Kompanets
Sergey V. Chekalin  Victor N. Zadkov  Institute of Spectroscopy, Russian
Academy of Sciences, Troitsk, Moscow, 108840, Russia  Faculty of Physics,
Higher School of Economics, Moscow, 105066, Russia  Key Laboratory for
Physical Electronics and Devices of the Ministry of Education & Shaanxi Key
Lab of Information Photonic Technique, School of Electronic and Information
Engineering, Xi’an Jiaotong University, Xi’an, 710049, China  Quantum
Technology Centre, Faculty of Physics, M. V. Lomonosov Moscow State
University, Moscow, 119991, Russia

###### Abstract

Floquet systems with periodically varying in time parameters enable
realization of unconventional topological phases that do not exist in static
systems with constant parameters and that are frequently accompanied by
appearance of novel types of the topological states. Among such Floquet
systems are the Su-Schrieffer-Heeger lattices with periodically-modulated
couplings that can support at their edges anomalous $\pi$ modes of topological
origin despite the fact that the lattice spends only half of the evolution
period in topologically nontrivial phase, while during other half-period it is
topologically trivial. Here, using Su-Schrieffer-Heeger arrays composed from
periodically oscillating waveguides inscribed in transparent nonlinear optical
medium, we report experimental observation of photonic anomalous $\pi$ modes
residing at the edge or in the corner of the one- or two-dimensional arrays,
respectively, and demonstrate a new class of topological $\pi$ solitons
bifurcating from such modes in the topological gap of the Floquet spectrum at
high powers. $\pi$ solitons reported here are strongly oscillating nonlinear
Floquet states exactly reproducing their profiles after each longitudinal
period of the structure. They can be dynamically stable in both one- and two-
dimensional oscillating waveguide arrays, the latter ones representing the
first realization of the Floquet photonic higher-order topological insulator,
while localization properties of such $\pi$ solitons are determined by their
power.

###### keywords:

Floquet topological insulators, $\pi$ states, edge solitons, SSH model

††journal: Science Bulletin

##  1 Introduction

Photonic topological insulators [1, 2] are unique materials hosting localized
topologically protected states at their edges by analogy with edge modes in
electronic topological insulators, first predicted in solid-state physics [3,
4]. Various mechanisms of formation of the photonic topological edge states
were discovered, most of which are associated with breakup of certain
symmetries of the underlying system possessing specific degeneracies in the
linear spectrum. The most representative feature of topological edge states is
their remarkable robustness with respect to deformations of the structure,
disorder, and their persistence for different geometries of the edge between
topologically distinct materials. Their formation and robustness has been
predicted and demonstrated for various photonic systems with broken time-
reversal symmetry, for valley-Hall systems with broken inversion symmetry, and
in higher-order topological insulators [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20]. Particularly nontrivial situation is realized when
topological phase is induced by periodic modulations of system parameters in
the evolution variable [21], for example in the direction of light
propagation. Characterization of such driven Floquet systems usually requires
special topological invariants, as shown in [22, 23, 24]. Among the most
intriguing manifestations of topological effects in Floquet systems is the
formation of unidirectional edge states as proposed in [8] and observed at
optical frequencies in [9], observation of anomalous topological states [25,
26, 27], and of so-called anomalous $\pi$ modes associated with nonzero $\pi$
gap invariant and studied in [28, 29, 30, 31, 32]. Recent surge of interest to
topological pumping in near-solitonic regime should be mentioned too [33, 34,
35].

$\pi$ modes are unique topological states that may appear in a quasi-energy
spectrum of the Floquet system that due to modulation of its parameters spends
half of the evolution period in “instantaneous” nontopological phase, while on
the other half of the period it is topologically nontrivial. In tight-binding
models describing Floquet systems, $\pi$ modes usually appear at the edges of
the “longitudinal” Brillouin zone with quasi-energies equal to $\pm\pi/T$
(where $T$ is period of the Floquet system), in contrast to conventional
“zero-energy” edge states in static topological systems, hence the notion of
$\pi$ modes that we also use in this work for convenience. So far, the
photonic $\pi$ modes have been observed only in linear regime in one-
dimensional modulated Su-Schrieffer-Heeger (SSH) arrays in a microwave range
[36] and at optical frequencies in non-Hermitian or plasmonic SSH arrays [37,
38, 39] with high refractive index contrast, where, however, considerable
losses limit propagation distances to hundreds of micrometers. $\pi$ modes may
have applications in the design of systems supporting high-quality cavity
modes [40, 41], for realization of low-threshold lasers [42, 43, 44, 45], in
strongly correlated electron–photon systems [46], and other areas. They have
been also encountered beyond the realm of optics, e.g., in acoustics [47, 48].
Nevertheless, to date the photonic $\pi$ modes remain unobserved in higher-
dimensional conservative systems and their nonlinear analogs were never
reported experimentally. At the same time, photonic Floquet systems offer
unique testbed for the exploration of nonlinear effects in specifically
designed low-loss topological guiding structures, where observation of so far
elusive class of $\pi$ solitons is possible.

It should also be mentioned that nonlinearity is playing an increasingly
important role in all-optical control of topological systems, see recent
reviews [49, 2]. In particular, nonlinearity may stimulate modulational
instability of the nonlinear edge states [50, 51, 52], it leads to rich
bistability effects for edge states in pumped dissipative resonator structures
[53, 54, 55], it may cause power-controlled topological transitions [56], and
enables the formation of topological solitons both in the bulk of the
insulator [57, 58] and at its edges [51, 59, 60, 61, 62, 63, 64, 65, 66, 67,
68, 69, 70]. The important property of such solitons is that they remain
localized due to nonlinearity, and at the same time they inherit topological
protection from linear edge modes, from which they usually bifurcate. Corner
solitons in higher-order topological insulators have been reported too [71,
72, 73]. Very recently it was theoretically predicted [74] that Floquet
topological systems may support a new class of topological solitons,
qualitatively different from previously observed unidirectional states [67] –
namely, $\pi$ soliton – that represents dynamically oscillating nonlinear
Floquet state with a quasi-energy in the topological bandgap that exactly
reproduces its intensity distribution after each longitudinal period of the
structure. Even strongly localized $\pi$ solitons are practically free from
radiative losses that usually restrict propagation distances for
unidirectional edge solitons in Floquet waveguiding systems [64, 67].

In this work, we report on the experimental observation of $\pi$ solitons in
one- and two-dimensional Floquet waveguide arrays, where nontrivial
topological properties arise due to $z$-periodic oscillations of waveguide
centers in each unit cell of the structure. The arrays considered here are
inscribed in a transparent nonlinear dielectric medium (fused silica) using
the technique of direct femtosecond laser writing [75, 76, 77] and represent
SSH-like structures, which, however, are not static, but spend half of the
$z$-period in “instantaneous” topological phase, while during other half of
the period they are “instantaneously” non-topological, as defined by
periodically varying intra- and inter-cell coupling strengths. In two
dimensions such arrays represent the realization of the photonic Floquet
higher-order insulator. Floquet spectrum of such arrays is characterized by
the presence of in-gap topological $\pi$ modes, from which robust $\pi$
solitons can bifurcate in the nonlinear regime. We observe such solitons using
single-site excitations, study their periodic evolution with distance, and
dependence of their localization properties on the amplitude of the waveguide
oscillations and power.

Figure 1: (Color online) (a) Schematic image of the 1D oscillating waveguide
array (three longitudinal $z$-periods) containing 7 unit cells. (b)
Microphotographs of the fs‐laser written oscillating waveguide array at
different distances in topological phase $(z=0)$ , uniform phase $(z=Z/4)$,
and trivial phase $(z=Z/2)$. (c) Quasi‐propagation constants of the Floquet
modes of the oscillating array versus amplitude of the waveguide oscillations
$r$ within three longitudinal Brillouin zones. (d) and (e) show intensity
distributions of the linear $\pi$ modes at different distances $z$ for two
selected oscillation amplitudes $r$ corresponding to the red dots in (c). In
all cases $Z=33\,\textrm{mm}$.

##  2 Results and discussions

We consider paraxial propagation of a light beam along the $z$ axis of the
medium with focusing cubic nonlinearity and shallow transverse modulation of
the refractive index that can be described by the nonlinear Schrödinger-like
equation for the dimensionless light field amplitude $\psi$:

$i\frac{\partial\psi}{\partial
z}=-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)\psi-\mathcal{R}(x,y,z)\psi-|\psi|^{2}\psi.$ (1)

Here $x,y$ are the scaled transverse coordinates, $z$ is the propagation
distance that plays in Eq. (1) the same role as time in the Schrödinger
equation describing a quantum particle in a potential, and the function
${\mathcal{R}}(x,y,z)$ describes array of periodically oscillating waveguides.
For details of normalization of Eq. (1) see Section S1 in the Supplementary
materials .

###  2.1 1D $\pi$ solitons

First of all, for observation of 1D $\pi$ solitons we consider the SSH-like
arrays of oscillating waveguides containing 7 unit cells. Refractive index
distribution in such arrays can be described by the following function:

${\mathcal{R}}=p\sum_{m}[e^{-(x^{2}_{1m}/a_{x}^{2}+y^{2}/a_{y}^{2})}+e^{-(x^{2}_{2m}/a_{x}^{2}+y^{2}/a_{y}^{2})}],$

where $x_{1m}=x_{m}+d/2+r\cos(\omega z)$ and $x_{2m}=x_{m}-d/2-r\cos(\omega
z)$ are the $x$-coordinates of the waveguide centers in each unit cell
containing two waveguides, $\omega=2\pi/Z$ is the spatial frequency of
oscillations of the waveguide centers, $Z$ is the longitudinal period of the
array, $x_{m}=x-2md$, $m$ is the integer index of the cell, $r$ is the
amplitude of the waveguide oscillations, which was varied from $1$ to
$11\,\mu\textrm{m}$, $d=30\,\mu\textrm{m}$ is the spacing between waveguides
at $r=0$ (i.e. unit cell size is $2d$), $a_{x}=2.5~{}\mu\textrm{m}$ and
$a_{y}=7.5~{}\mu\textrm{m}$ are the widths of waveguides that are elliptical
due to writing process, and $p$ is the array depth proportional to the
refractive index contrast $\delta n$ in the structure (see Section S1 in the
Supplementary materials ). Schematic illustration of such array is presented
in Fig. 1a. As one can see, the separation $d-2r\cos(\omega z)$ between two
waveguides in the unit cell (intracell separation) of this structure varies
dynamically, leading to periodic transformation between “instantaneously”
topological (inter-cell coupling exceeds intracell one) and non-topological
(inter-cell coupling is weaker than intra-cell one) SSH configurations.
Microphotographs of such 1D fs-laser written arrays in fused silica at
different distances within the sample are presented in Fig. 1b. The array
period $Z=33\,\textrm{mm}$ was selected such that our samples contained three
full $z$-periods of this Floquet structure (see Section S2 in the
Supplementary materials ).

Nontrivial topological properties in this system arise due to longitudinal
variations of the structure (oscillations of the waveguides). Its modes are
the Floquet states $\psi=u(x,y,z)e^{ibz}$, where $b$ is a quasi-propagation
constant [for first Brillouin zone $b\in[-\omega/2,+\omega/2)$], and
$u(x,y,z)=u(x,y,z+Z)$ is the $Z$-periodic complex field that satisfy the
equation

$bu=\frac{1}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)u+\mathcal{R}u+i\frac{\partial u}{\partial z}+|u|^{2}u.$ (2)

Neglecting nonlinear term in Eq. (2) we calculate linear spectrum of 1D array
using the method proposed in [27] (see Section S5 in the Supplementary
materials ). The transformation of linear spectrum with increase of the
amplitude $r$ of waveguide oscillations is shown in Fig. 1c. Quasi-propagation
constant $b$ is defined modulo $\omega$ and in Fig. 1c we show the spectrum
within three longitudinal Brillouin zones. Gray lines correspond to the
delocalized bulk modes, while the red lines correspond to the linear
topological $\pi$ modes [28, 29, 30, 31, 32]. Notice that they emerge around
the points, where Floquet replicas of the bands spectrally overlap, and
longitudinal modulation hybridizes states at the band edges lifting their
degeneracy and opening the topological gap. Because our structure is
symmetric, $\pi$ modes appear near both edges of the array. Quasi-propagation
constants of $\pi$ modes are located in the forbidden gap in the Floquet
spectrum that guarantees absence of coupling with bulk states. Their
localization near a given edge increases with increase of the gap width, cf
Fig. 1d and e. Such modes show strong shape transformations within
longitudinal period, but exactly reproduce their shape after each period $Z$.
Remarkably, Fig. 1e clearly shows that global intensity maximum of the $\pi$
mode is not always located in the edge waveguide. For instance, at $z=Z/2$,
where the array is in instantaneous nontopological phase, the intensity
maximum switches into next to edge waveguide, while at $z=Z$, exactly after
one oscillation period, where structure returns into instantaneous topological
configuration, the light also switches back to the edge waveguide. Already for
$r=7\,\mu\textrm{m}$ the $\pi$ mode contracts practically to single waveguide
in some points within evolution period that enables its efficient excitation
in the experiment [this determined our choice of the initial “phase” of the
waveguide oscillations in Fig. 1a; linear spectrum clearly does not depend on
this phase]. Very similar results and Floquet spectrum were obtained also for
arrays with odd number of waveguides (where one of the unit cells is
incomplete), because even in this case due to waveguide oscillations both
edges periodically pass through stages when truncation becomes topological or
nontopological and therefore support $\pi$ modes.

The appearance of topological $\pi$ modes in the spectrum of this Floquet
system is associated with nonzero value of the $\pi$ gap invariant $w_{\pi}$
(see Section S4 in the Supplementary materials for details of its calculation
and literature [30, 36]). One can observe the formation of $\pi$ modes in the
spectrum of truncated array when $w_{\pi}=1$ (for sufficiently large
oscillation amplitudes $r$), while these modes are absent when $w_{\pi}=0$
(e.g., at $r\to 0$).

Inspecting spectrum in Fig. 1c one can see that while longitudinal modulation
with frequency $\omega$ creates Floquet replicas of bands and localized $\pi$
modes, by itself it does not induce parametric resonances between localized
and bulk modes, which nevertheless can occur, if additional weak modulation of
optical potential at frequencies $2\omega$, $3\omega$, $\cdots$ is added that
keeps $Z$-periodicity of array.

The $\pi$ solitons are the topological nonlinear Floquet states bifurcating
from the linear $\pi$ modes. To find their profiles we iteratively solve Eq.
(2) with the last nonlinear term included (see Section S6 in the Supplementary
materials ), by varying soliton power $U=\iint|\psi|^{2}dxdy$ and calculating
for each $U$ corresponding $Z$-periodic soliton profile $u(x,y,z)$, quasi-
propagation constant $b$, and averaged amplitude
$A=Z^{-1}\int_{z}^{z+Z}\textrm{max}|\psi|dz$. The $\pi$ solitons bifurcate
from linear $\pi$ modes, as evident from the representative $b(U)$ dependence
in Fig. 2a, where quasi-propagation constant of linear mode is shown by the
dashed line. Their amplitude $A$ increases with the power (Fig. 2b).
Importantly, nonlinearity changes the location of $b$ inside the topological
gap, gradually shifting it towards the bulk band (gray region). This is
accompanied by changes in soliton localization in the $(x,y)$ plane (it may
first increase and then decrease depending on the value of $r$), especially
when $b$ shifts into the band, where coupling with the bulk modes occurs.
Periodic transformation of soliton intensity distribution with $z$ is
illustrated in Fig. 2c. It should be stressed that for our parameters even for
the amplitude of oscillations $r\sim 9~{}\mu\textrm{m}$ the $\pi$ solitons
obtained here are robust objects that practically do not radiate and survive
over hundreds of $Z$ periods that is beneficial in comparison with the
previously observed unidirectional edge solitons [67].

We tested stability of such states by adding broadband small noise (typically,
$5\%$ in amplitude) into input field distributions and propagating such
perturbed $\pi$ solitons over distances $\sim 500Z$ that allows to detect the
presence of even very weak instabilities. Such stability analysis has shown
that for our parameters and for amplitudes of oscillations
$r>5~{}\mu\textrm{m}$ 1D solitons belonging to forbidden gap are stable, while
they become unstable, when they shift into band. Notice that stabilization of
such states that have powers well below power of Townes soliton in uniform
cubic medium is consistent with arguments of Ref. [78].

Figure 2: (Color online) Quasi‐propagation constant (a) and $z$‐averaged peak
amplitude (b) of the $\pi$ soliton versus its power $U$. Gray region in (a)
corresponds to the bulk band, while white region corresponds to the forbidden
gap in Floquet spectrum. Horizontal dashed line shows quasi-propagation
constant $b$ of linear $\pi$ mode. (c) Intensity distributions in the $\pi$
soliton with power $U=0.5$ at different distances within one oscillation
period $Z$. The state is shown near left edge only. Here $r=6\,\mu\textrm{m}$,
$Z=33~{}\textrm{mm}$. The dependencies $b(U)$ and $A(U)$ for other values of
$3\,\mu\textrm{m}\leq r\leq 9\,\mu\textrm{m}$ are qualitatively similar.
Figure 3: (Color online) (a) Formation of the $\pi$ modes with increase of the
amplitude of waveguide oscillations $r$ in linear regime. The impact of
nonlinearity on such states and formation of $\pi$ solitons is illustrated in
(b) for $r=7\,\mu\textrm{m}$ and in (c) for $r=9\,\mu\textrm{m}$ at different
propagation distances. In all panels red lines show experimental 1D intensity
cross-sections at $y=0$, blue inset shows 2D experimental intensity
distributions for a given pulse energy $E$, and black-red insets show
corresponding theoretical 2D intensity distributions for different input
powers $U$.

To demonstrate 1D $\pi$ solitons experimentally we inscribed in a fused silica
sample the series of SSH-like arrays with the different amplitudes $r$ of
waveguide oscillations ranging from $1$ to $11~{}\mu\textrm{m}$, with a step
in $r$ of $2~{}\mu\textrm{m}$ using fs-laser writing technique (see Fig. 1b
with exemplary microphotographs of the array and Section S2 in the
Supplementary materials for the details of inscription). While full sample
length contains three $Z$-periods of the structure, to demonstrate dynamics in
the internal points of the last period, we additionally inscribed arrays with
fractional lengths $2.25Z$, $2.50Z$, $2.75Z$ with the same parameters (see
Section S2 in the Supplementary materials for the details).

Figure 4: (Color online) (a) Schematic illustration of the 2D oscillating
waveguide array (for illustrative purposes we show only $3\times 3$ unit
cells). (b) Microphotographs of laser‐written 2D array with $5\times 5$ unit
cells at different distances. (c) Quasi‐propagation constants of the Floquet
modes of the 2D array with oscillating waveguides versus amplitude of
waveguide oscillations $r$ within three longitudinal Brillouin zones. (d)
Intensity distribution in linear 2D $\pi$ mode at different distances $z$ for
$r=6\,\mu\textrm{m}$. In all cases $Z=49.5\,\textrm{mm}$.

In experiments we excited the waveguide at the left edge using the fs-laser
pulses of variable energy $E$ (for correspondence between pulse energy $E$ and
input peak power in the waveguide see Section S3 in the Supplementary
materials ). Output intensity cross-sections at $y=0$ (red lines) and 2D
distributions (blue insets) are compared in Fig. 3 with the results of
theoretical simulations of the single-site excitation with different input
powers $U$ in the frames of Eq. (1) (black-red insets). In all cases
theoretical results well agree with the experimental observations. In Fig. 3a
and top right image in Fig. 3c we show how output beam localization
progressively increases in linear regime (low pulse energies
$E=20\,\textrm{nJ}$) with increase of the amplitude $r$ of the waveguide
oscillations. Efficient excitation of well-localized linear $\pi$ modes is
evident for $r\geq 7\,\mu\textrm{m}$, while for $r=5\,\mu\textrm{m}$ the
excitation efficiency is lower and some fraction of power penetrates into the
bulk of the array. First rows in Fig. 3b and c illustrate that $\pi$ mode
undergoes strong oscillations on one $Z$ period, main intensity maximum
switches into second waveguide at $z=2.50Z$ (consistently with the dynamics of
exact state in Fig. 1e), but it returns to the edge one at $z=3.00Z$
illustrating periodic evolution.

By increasing pulse energy we observe the formation of the 1D $\pi$ solitons.
As mentioned above, they are in-gap topological states bifurcating from $\pi$
modes under the action of nonlinearity. Nonlinearity leads to soliton
reshaping (in particular, for $r=7-9\,\mu\textrm{m}$ it slightly broadens with
increase of $U$), but when its quasi-propagation constant shifts into the
band, strong radiation into the bulk occurs. This is most clearly visible for
$r=7\,\mu\textrm{m}$ (Fig. 3b), where solitons were observed well-localized
near the edge for the pulse energies $E<350\,\textrm{nJ}$, but radiating
around $E\sim 430\,\textrm{nJ}$ (notice that at this energy the level of
radiation becomes visible only after three $Z$ periods). Further increase of
the pulse energy leads to stronger radiation. At $r=9\,\mu\textrm{m}$ the
range of the pulse energies, where formation of robust $\pi$ solitons is
observed substantially increases (Fig. 3c). Well localized $\pi$ solitons
performing $Z$-periodic oscillations are observed for the pulse energies
$E<900\,\textrm{nJ}$ (rows 1-3) and only at $E\sim 1000\,\textrm{nJ}$ small
radiation due to the coupling with the bulk modes appears (row 4). Simulations
over much larger distances $(z>100Z)$ confirm robustness of such dynamically
excited nonlinear Floquet states with in-gap quasi-propagation constants. It
should be stressed that excitation of the waveguide in the bulk of the above
arrays does not yield localization for considered pulse energies.

###  2.2 2D $\pi$ solitons

For observation of 2D $\pi$ solitons we utilize 2D generalization of the SSH
array with oscillating waveguides. Unit cell of such an array (quadrimer)
contains four waveguides, whose centers oscillate with period $Z$ along the
diagonals of the unit cell. We consider sufficiently large structure
containing $5\times 5$ unit cells. Refractive index distribution in this
Floquet structure is described by the function

$\begin{split}{\mathcal{R}}=p\sum_{m,n}[&e^{-(x^{2}_{1m}/a_{x}^{2}+y^{2}_{1n}/a_{y}^{2})}+e^{-(x^{2}_{2m}/a_{x}^{2}+y^{2}_{1n}/a_{y}^{2})}+\\\
&e^{-(x^{2}_{1m}/a_{x}^{2}+y^{2}_{2n}/a_{y}^{2})}+e^{-(x^{2}_{2m}/a_{x}^{2}+y^{2}_{2n}/a_{y}^{2})}],\end{split}$

where $x_{1m,2m}=x_{m}\pm d/2\pm r\cos(\omega z)$ and ${y_{1n,2n}=y_{n}}\pm
d/2\pm r\cos(\omega z)$ are the coordinates of centers of four waveguides in
the unit cell with $x_{m}=x-2md$ and $y_{n}=y-2nd$, $m,n$ are the integers. In
2D case the oscillation period was taken as $Z=49.5\,\textrm{mm}$, so that
sample contained two full longitudinal array periods. Spacing between
waveguides at $r=0~{}\mu\textrm{m}$ was set to $d=32\,\mu\textrm{m}$, and to
achieve more uniform coupling between elliptic waveguides, their longer axes
were oriented along the diagonal of the array (see schematics in Fig. 4a and
microphotographs of inscribed structure at different distances in Fig. 4b). As
one can see, such structure realizes the photonic Floquet higher-order
insulator periodically switching between “instantaneous” topological and non-
topological phases.

Figure 5: (Color online) Quasi-propagation constant (a) and $z$‐averaged peak
amplitude (b) versus power $U$ for the $\pi$ solitons in 2D array. (c)
Intensity distributions at different distances for a soliton with $U=0.2$. In
all cases $r=7\,\mu\textrm{m}$, $Z=49.5~{}\textrm{mm}$.

Dependence of quasi-propagation constants $b$ of the Floquet eigenmodes of the
2D array, obtained from linear version of Eq. (2), on amplitude of waveguide
oscillations $r$ shown in Fig. 4c reveals the formation of 2D $\pi$ modes (red
lines) that reside in the corners of the structure, but in comparison with 1D
case they appear in sufficiently narrow range of oscillation amplitudes. This
is a consequence of substantially more complex spectrum of static 2D SSH
structures [73] featuring four bands in topological phase (in contrast to only
two bands in 1D SSH arrays), that in our case experience folding due to
longitudinal array modulation, resulting in a very complex Floquet spectrum.
For instance, quasi-propagation constants of 2D $\pi$ modes may overlap with
the band, as it also happens with eigenvalues of usual corner modes in static
higher-order insulators [73]. To obtain such spectrum, where topological gap
can be opened by longitudinal array modulation, we had to select not too small
depth of potential $p=5$ to ensure that the width of the bulk bands is
comparable with the width of the longitudinal Brillouin zone and no too strong
band folding occurs. Example of the $\pi$ mode performing periodic
oscillations (only one corner is shown) is depicted in Fig. 4d. The properties
of 2D $\pi$ solitons, whose family was obtained from nonlinear Eq. (2) using
iterative method, are summarized in Fig. 5. As in the 1D case, quasi-
propagation constant of 2D solitons crosses the gap with increase of power $U$
and enters into the band (Fig. 5a). For the selected amplitude
$r=7\,\mu\textrm{m}$ the soliton exists practically in the entire gap, because
$b$ of linear $\pi$ mode from which it bifurcates is located near the lower
gap edge (we have checked that this linear $\pi$ mode indeed falls into
forbidden gap of bulk system by calculating quasienergy spectrum of periodic,
i.e. infinite in the transverse direction, Floquet array). The average
amplitude $A$ increases with $U$ (Fig. 5b). The intensity distributions at
different distances illustrating periodic $\pi$ soliton evolution in $z$ are
presented in Fig. 5c. Despite the fact that this state is 2D and oscillates
strongly, the collapse is suppressed and one observes very robust propagation
for all powers, when soliton resides in the gap. This conclusion was also
supported by the results of propagation of weakly perturbed 2D $\pi$ solitons
over large distances. For $r=7\,\mu\textrm{m}$ all such states in the gap were
found stable.

Figure 6: (Color online) Excitation of the $\pi$ solitons in 2D oscillating
waveguide arrays for different amplitudes of the waveguide oscillations $r$ at
$z=2Z$. Top rows (blue background) show experimentally measured intensity
distributions for different pulse energies $E$, while bottom rows (black
background) show theoretically calculated output patterns for different powers
$U$.

To observe 2D $\pi$ solitons we inscribed the series of 2D oscillating arrays
with various amplitudes of waveguide oscillations up to $r=9~{}\mu\textrm{m}$.
Excitations in the right corner of the array were used, but it should be
stressed that excitations of other corners yields nearly identical results due
to high symmetry and uniformity of the array. At low pulse energies
$E=10\,\textrm{nJ}$ one observes strong diffraction into the bulk at
$r=3\,\mu\textrm{m}$ (Fig. 6a), while efficient excitation of linear $\pi$
modes takes place at amplitudes $r\geq 5\,\mu\textrm{m}$ (Fig. 6b and c). To
the best of our knowledge, this constitutes the first observation of 2D $\pi$
modes in photonics. Increasing pulse energy at low $r\sim 3\,\mu\textrm{m}$
results first in concentration of light in the bulk of the sample and then its
gradual displacement toward the corner (Fig. 6a). By contrast, for $r\geq
5\,\mu\textrm{m}$ one observes the formation of $\pi$ solitons, whose range of
existence in terms of input power grows with increase of $r$. Thus, at
$r=5\,\mu\textrm{m}$ the well-localized solitons form at pulse energies
$E<300~{}\textrm{nJ}$, while at $E\sim 400~{}\textrm{nJ}$ strong radiation
into the bulk occurs (Fig. 6b) due to nonlinearity-induced shift into the
allowed band. At $r=7\,\mu\textrm{m}$ one observes the formation of the $\pi$
solitons even at $E\sim 600~{}\textrm{nJ}$ (Fig. 6c) with tendency for slight
increase of secondary intensity maxima in soliton profile at highest power
levels that is observed also in exact soliton solution of Eq. (2). Excitations
in other corners of the array (e.g., top one) yield similar results confirming
the $\pi$ soliton formation, while excitations in the bulk strongly diffract
at these pulse energies.

##  3 Conclusion

We presented experimental observation of a new type of $\pi$ solitons in
nonlinear Floquet system, where nontrivial topology arises from periodic
modulation of the underlying photonic structure in evolution variable (along
the light propagation path). Such solitons exist both in 1D and 2D geometries
and they show exceptionally robust evolution due to practically absent
radiative losses at considered periods and amplitudes of oscillations. The
results obtained here may be used in the design of a class of topological
Floquet lasers based on $\pi$ modes, for the control and enhancement of
parametric processes, such as generation of new harmonics assisted by topology
of the Floquet system, and for design of new types of on-chip all-optically
controlled topological devices.

## Conflict of interest

The authors declare that they have no conflict of interest.

## Acknowledgments

This research is funded by the research project FFUU-2021-0003 of the
Institute of Spectroscopy of the Russian Academy of Sciences and partially by
the RSF grant 21-12-00096. Y. Z. acknowledges funding by the National Natural
Science Foundation of China (Grant Nos.: 12074308). S. Z. acknowledges support
by the Foundation for the Advancement of Theoretical Physics and Mathematics
“BASIS” (Grant No.: 22-2-2-26-1).

## Author contributions

Yiqi Zhang and Yaroslav V. Kartashov formulated the problem. Sergei A.
Zhuravitskii, Nikolay N. Skryabin, Ivan V. Dyakonov, and Alexander A. Kalinkin
fabricated the samples. Antonina A. Arkhipova, Victor O. Kompanets, and Sergey
V. Chekalin performed experiments. Yiqi Zhang performed numerical modeling.
Yaroslav V. Kartashov, Sergei P. Kulik, and Victor N. Zadkov supervised the
work. All co-authors took part in discussion of results and writing of
manuscript.

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