#  $\sqrt{T\overline{T}}$-deformed oscillator inspired by ModMax

J. Antonio García1 and R. Abraham Sánchez-Isidro2
Departamento de Física de Altas Energías, Instituto de Ciencias Nucleares
Universidad Nacional Autónoma de México,
Apartado Postal 70-543, Ciudad de México, 04510, México
1<EMAIL_ADDRESS>2<EMAIL_ADDRESS>
###### Abstract

Inspired by a recently proposed Duality and Conformal invariant modification
of Maxwell theory (ModMax), we construct a one-parameter family of two-
dimensional dynamical system in classical mechanics that share many features
with the ModMax theory. It consists of a couple of
$\sqrt{T\overline{T}}$-deformed oscillators that nevertheless preserves
duality $(q\rightarrow p,p\rightarrow-q)$ and depends on a continuous
parameter $\gamma$, as in the ModMax case. Despite its non-linear features,
the system is integrable. Remarkably can be interpreted as a pair of two
coupled oscillators whose frequencies depend on some basic invariants that
correspond to the duality symmetry and rotational symmetry. Based on the
properties of the model, we can construct a non-linear map dependent on
$\gamma$ that maps the oscillator in 2D to the nonlinear one, but with
parameter $2\gamma$. The dynamics also shows the phenomenon of energy transfer
and we calculate a Hannay angle associated to geometric phases and holonomies.

##  1 Introduction

Consistent deformation of field theories is a powerful tool to explore in what
sense or to what extent a given theory can be deformed by deforming its gauge
symmetries and/or couplings in such a way the resulting theory is also
consistent (removes the gauge degree of freedom in a consistent way). It is
very useful to explore questions about if the theory is unique and/or how it
can be extended to new field theories that could be interesting in physics.
The deformations are regulated by symmetries and other constraints that we
impose according to what questions we are addressing when deforming. All
deformations preserve the field content of the given theory. A well known
example is a deformation of N free Maxwell fields where the gauge symmetry and
the interaction, preserving global Poincaré symmetry, can be deformed to
construct the Yang-Mills SU(N) theory. This deformation is also unique under
the restrictions imposed [1].

We can also deform a free action in many other ways by imposing different
restrictions. Maxwell’s theory can be deformed in different ways by preserving
duality and conformal symmetries in 4D. Some celebrated examples are the Born-
Infeld [2], Plebansky [3], and Biankii-Birula deformations [4, 5]. Recently a
new deformation of Maxwell theory preserving conformal and duality invariance
has been constructed [6, 7]111The question about the more general Hamiltonian
preserving duality and Lorentz invariance was also addressed in [8].. The
result is quite interesting. It reduces to previously known deformations and
gives us a new non-linear electromagnetic theory (ModMax). Interestingly the
theory preserves duality symmetry. As is well known duality invariance is
difficult to implement in interacting theories even if we know that the
corresponding free theory is duality invariant. Another example of a theory
with a duality invariance is the Fierz-Pauli action[9]. Still we do not know
how to implement this duality in the full General Relativity. The duality
symmetry plays a very important role in the formulation of string theory and
many investigations have been developed to understand its implications.
Understanding this fascinating symmetry is the focus of many studies in string
theories.

The ModMax theory is a one-parameter family of non-linear electrodynamics
theories that was constructed in [6]. The Lagrangian derivation was also
presented in [7]. The Legendre transformation between the Lagrangian and
Hamiltonian theories is non trivial. ModMax is constructed from the two basic
Lorentz invariants of Maxwell theory $S=-(1/4)F^{\mu\nu}F_{\mu\nu}$ and
$P=-(1/4)F_{\mu\nu}\tilde{F}^{\mu\nu}$, where $\tilde{F}^{\mu\nu}$ is the
Hodge dual of $F_{\mu\nu}$ the usual Maxwell strength field. The conformal
symmetry implies that the trace of the energy-momentum tensor is zero and this
in turn implies that the Lagrangian is a homogeneous function of degree one in
$S$ and $P$. ModMax theory is the unique conformal $U(1)$ gauge theory with a
non-linear constitutive relations, which preserves electric/magnetic duality
invariance. In the phase space this means that ModMax is invariant under the
$S$-duality transformation
$(\mathbf{D}\rightarrow-\mathbf{B},\mathbf{B}\rightarrow\mathbf{D})$. It is
important to stress that, the duality symmetry is only manifest in Hamiltonian
formalism. In Lagrangian formalism this invariance is not manifest [10]. The
standard definition of $\mathbf{D}$ is $\mathbf{D}=\frac{\partial
L}{\partial\mathbf{E}}$ and in ModMax the vector $\mathbf{D}$ depends on
$\mathbf{E}$ and $\mathbf{B}$ as in a generic non-linear electrodynamics
theory [11]. We review some of the basic properties of ModMax theory in
appendices C, and D.

Although, there may be other Lorentz and duality invariant theories of
electrodynamics corresponding to other deformations, there exist just two
theories that are conformal invariant, Bialynicki-Birula electrodynamics and
ModMax [6]. ModMax theory has been scrutinized and extended in recent works
[15, 20, 16, 21, 12, 22, 17, 18, 19, 14, 23, 13].

A very interesting question is if ModMax theory can be constructed from a
deformation of Maxwell theory. Fortunately the answer to this question is
affirmative and the deformation is know as a $T\overline{T}$ deformation. In
2D CFT a $T\overline{T}$ operator is an irrelevant operator where $T$ is the
energy-momentum tensor. In holomorphic coordinates $z,\bar{z}$ the components
are denoted by $T=T_{zz}$ and $\overline{T}=T_{\bar{z}\bar{z}}$ [26, 27].
Among the interesting properties of this irrelevant operator is that starting
from an integrable QFT, the deformed theory preserve integrability [28, 29]
and complete $1+1$ gravity in the UV [30]. An example is the $T\overline{T}$
deformation of the massless scalar field action that leads to Nambu-Goto
action [27]. Other examples and extensions can be found in [33, 31, 32, 25,
34]. Another interesting property of this deformation is that relates
non/ultra-relativistic limits of the string sigma model [35, 36].

Denoting the deformed action as $S_{QFT}=S_{CFT}+S_{\gamma}$ where
$S_{\gamma}=\gamma\int d^{2}xT\overline{T}$ and $\gamma$ is the deformation
parameter, the deformed action is a solution of the flow equation
$\frac{dS_{QFT}}{d\gamma}=\int d^{2}xT\overline{T}$ for finite $\gamma$.

By applying these technics, we can prove that ModMax is a deformation of
Maxwell action [24, 25].

The role played by $\sqrt{T\overline{T}}$-deformations could have a relation
with the realization of dualities in non linear theories [37].

Identifying
$\mathbf{E}\rightarrow\mathbf{\dot{q}},\mathbf{D}\rightarrow\mathbf{p},\mathbf{B}\rightarrow\mathbf{q}$,
we present here a $T\bar{T}$ deformation of a classical harmonic oscillator in
2D that shares many properties already present in ModMax theory. Duality
symmetry is replaced by the transformation that rotates configuration
variables with its conjugate momentum variables in the sense that
$(\mathbf{p}\rightarrow-\mathbf{q},\mathbf{q}\rightarrow\mathbf{p})$. Lorentz
symmetry is just represented by SO(2) space rotations and conformal symmetry
restrict the form of the possible Hamiltonians (or Lagrangians) in a precise
way that we will reveal later. The solution to these restrictions results in a
dynamical system with a well defined Hamiltonian (or Lagrangian), which also
is parametrized by $\gamma$ in the same way as in the ModMax theory.

Either in phase space or configuration space, the equations of motion and
couplings are quite complicated and seem to be very difficult to find an
analytical solution to the system (see appendice A and B). Nevertheless, a
numerical study reveals that the solution space is quite symmetric, developing
beautiful curves, that encourage us to try to find an analytical solution.
Moreover, as we know that the $T\overline{T}$-deformations preserve
integrability we worked out the analytical solution.

Figure 1: This solution is numerical and plotted using $\gamma=1$ and the
initial conditions $q^{1}(0)=0.7$, $p^{1}(0)=0.2$, $q^{2}(0)=1.0$,
$p^{2}(0)=0.5$, with time $t\in\\{0,100\\}$.

The Legendre transformation is also non-trivial as in the ModMax theory. In
the classical mechanical system presented here the identification of two
conserved quantities allows us to identify a relation between conjugate
momenta and the configuration space variables $q,\dot{q}$ that is invertible
resolving the explicit Legendre transformation.

In phase space, the generators of the infinitesimal duality rotations and the
corresponding generator of space rotations are given respectively by

$s=\frac{1}{2}(p^{2}+q^{2}),\hskip 28.45274ptj=\epsilon^{ij}p^{i}q^{j},$

and are constants of motion. Using them we can parametrize the space of
solutions in terms of two functions $A(s,j)$ and $B(s,j)$. Notice that we are
not reducing the degrees of freedom but just parametrizing, in a different
way, the space of solutions. This was a key observation that allowed us to
find the analytical solution. The oscillator frequencies emerge under this
parametrization as the functions $A$ and $B$.

Another interesting finding of this investigation is that we can construct a
map (depending on $q,p,\gamma$) of the 2D harmonic oscillator to the full non-
linear oscillator at $2\gamma$.

A reformulation of the problem in configuration space starting from the ModMax
Lagrangian presented in [7] can be constructed in terms of two constants of
motion $C_{1}(\sigma,\rho)$, and $C_{2}(\sigma,\rho)$ where the variables
$\sigma,\rho$ are defined by

$\sigma=\frac{1}{2}(\dot{q}^{2}+q^{2}),\hskip
28.45274pt\rho=\epsilon^{ij}\dot{q}^{i}q^{j},$

The analytical results suggests an interesting physical interpretation. The
couplings between oscillators corresponds to a oscillator mounted in a non-
inertial oscillating frame. The frequencies of the oscillators are
parametrized by the constants of motion $C_{1}$ and $C_{2}$.

We reveal also a phenomenon of beats related to the transfer of energy between
the coupled oscillators and we calculate the Hannay angle (that depends on the
initial conditions through the numerical values of $C_{1}$ and $C_{2}$
evaluated on the given set of initial conditions).

All these interesting properties are rooted in the fact that we have a
structure given by the duality symmetry implemented in the non-linear problem.

The the paper is organized as follows. In the section 2 we construct the non-
linear model in the Lagrangian and Hamiltonian formalism. Later, in section 3,
we integrate the Hamiltonian system, additionally we introduce the necessary
notation to implement Legendre’s transformation and integrate the Lagrangian
system in the sections 4 and 5 respectively. In section 6, we show how to
construct the explicit map that transformation the 2D harmonic oscillator to
the full non linear problem parametrized by $2\gamma$ and in the section 7, we
show that our model present a energy transfer phenomena between the
oscillators and compute the Hannay angle. Finally, in 8, we give our
conclusions and the appendices A and B have complementary notes.

##  2 Non-linear oscillator from a $\sqrt{T\overline{T}}$-deformation

Here we will construct the model using a $\sqrt{T\overline{T}}$-deformation.
The model can be also constructed from scaling invariance (conformal
invariance), see the appendices A and B. It consists of taking the homogeneous
harmonic oscillator in 2D and deforming it as ModMax. Even though this is the
simpler model, we consider that its theoretical details are relevant to the
understanding the non-linear systems that preserve duality and moreover
improve the understanding of $T\overline{T}$-deformations.

###  2.1 $\sqrt{T\overline{T}}$-deformations in Lagrangian formalism

We perform a $\sqrt{T\overline{T}}$-deformations for the harmonic oscillator.
First, we consider the Lagrangian of the harmonic oscillator in $2D$ with
masses $m_{i}=1$ and frequencies $\omega_{i}=1$ for $i=\\{1,2\\}$

$L_{0}=\frac{1}{2}(\mathbf{\dot{q}}^{2}-\mathbf{q}^{2}),$ (1)

where $\mathbf{\dot{q}}^{2}=\dot{q}_{1}^{2}+\dot{q}_{2}^{2}$ and
$\mathbf{q}^{2}=q_{1}^{2}+q_{2}^{2}$.

Due to the rotational and time translations invariance of the action, there
exist two Noether conserved charges defined by

$\delta_{E}q^{i}=\dot{q}^{i},\hskip
28.45274ptE_{0}=\frac{1}{2}(\mathbf{\dot{q}}^{2}+\mathbf{q}^{2}),$ (2)
$\delta_{J}q^{i}=\epsilon^{ij}q^{j},\hskip
28.45274ptJ_{0}=\epsilon^{ij}\dot{q}^{i}q^{j}.$ (3)

In order to preserve these symmetries at any order in $\gamma$, the parameter
of the deformation, we define the conserved quantities $E_{n}$ and $J_{n}$
whose are the energy and angular momentum of the deformed action at order $n$.
Then we define the $T\overline{T}$-like operator of order $n$

$O^{\gamma}_{n}=\sqrt{E_{n}^{2}-J_{n}^{2}}.$ (4)

This is the analog to the $T\overline{T}$-like operator used in [24, 25, 31]
to deform the Maxwell into ModMax theory.

The deformed Lagrangian of order $n+1$ is defined by flow equation

$L_{n+1}=L_{0}+\int d\gamma\ O_{n}^{\gamma},$ (5)

then the first order deformation of the Lagrangian is

$L_{1}=L_{0}+\gamma\sqrt{E_{0}^{2}-J_{0}^{2}},$ (6)

which is invariant under $\delta_{E}q^{i}$ and $\delta_{J}q^{i}$ symmetries.
Then the deformed conserved quantities are

$E_{1}=E_{0}\bigg{(}1+\frac{L_{0}\gamma}{\sqrt{E_{0}^{2}-J_{0}^{2}}}\bigg{)},$
(7)
$J_{1}=J_{0}\bigg{(}1+\frac{L_{0}\gamma}{\sqrt{E_{0}^{2}-J_{0}^{2}}}\bigg{)}.$
(8)

Now, we are able to compute the operator $O^{\gamma}_{1}$ as

$O^{\gamma}_{1}=\sqrt{E_{1}^{2}-J_{1}^{2}}=\sqrt{E_{0}^{2}-J_{0}^{2}}+\gamma
L_{0},$ (9)

thus the second order Lagrangian is

$L_{2}=L_{0}+\gamma\sqrt{E_{0}^{2}-J_{0}^{2}}+\frac{1}{2}\gamma^{2}L_{0}.$
(10)

Because of $\delta_{E}q^{i}$ and $\delta_{J}q^{i}$ are still symmetries of the
action, we obtain the deformed second order conserved quantities

$E_{2}=E_{0}\bigg{(}1+\frac{L_{0}\gamma}{\sqrt{E_{0}^{2}-J_{0}^{2}}}+\frac{1}{2}\gamma^{2}\bigg{)},$
(11)
$J_{2}=J_{0}\bigg{(}1+\frac{L_{0}\gamma}{\sqrt{E_{0}^{2}-J_{0}^{2}}}+\frac{1}{2}\gamma^{2}\bigg{)}.$
(12)

and the second order $T\overline{T}$-like operator is

$O_{2}^{\gamma}=\sqrt{E_{0}^{2}-J_{0}^{2}}+\gamma
L_{0},+\frac{1}{2}\gamma^{2}\sqrt{E_{0}^{2}-J_{0}^{2}},$ (13)

then the third order Lagrangian is

$L_{3}=L_{0}+\gamma\sqrt{E_{0}^{2}-J_{0}^{2}}+\frac{1}{2}\gamma^{2}L_{0},+\frac{1}{3!}\gamma^{3}\sqrt{E_{0}^{2}-J_{0}^{2}}.$
(14)

By induction, it is straightforward to prove that the $n$ order deformation to
the Lagrangian function is

$L=\sum_{n=0}^{\infty}\bigg{[}\bigg{(}\frac{\gamma^{2n}}{(2n)!}\bigg{)}L_{0}+\bigg{(}\frac{\gamma^{2n+1}}{(2n+1)!}\bigg{)}\sqrt{E_{0}^{2}-J_{0}^{2}}\bigg{]}=L_{0}\cosh\gamma+\sinh\gamma\sqrt{E_{0}^{2}-J_{0}^{2}},$
(15)

the $\delta_{E}q^{i}$ and $\delta_{J}q^{i}$ symmetries generate the conserved
quantities 222Compare this result with the Energy-momentum tensor of the
ModMax theory
$T^{\mu\nu}=(F^{\mu\sigma}F^{\nu}{}_{\sigma}+S\eta^{\mu\nu})\Big{(}\cosh\gamma+\sinh\gamma\frac{S}{\sqrt{S^{2}+P^{2}}}\Big{)}.$

$E=E_{0}\sum_{n=0}^{\infty}\bigg{[}\bigg{(}\frac{\gamma^{2n}}{(2n)!}\bigg{)}+\bigg{(}\frac{\gamma^{2n+1}}{(2n+1)!}\bigg{)}\frac{L_{0}}{\sqrt{E_{0}^{2}-J_{0}^{2}}}\bigg{]}=E_{0}\bigg{(}\cosh\gamma+\frac{L_{0}\sinh\gamma}{\sqrt{E_{0}^{2}-J_{0}^{2}}}\bigg{)},$
(16)
$J=J_{0}\sum_{n=0}^{\infty}\bigg{[}\bigg{(}\frac{\gamma^{2n}}{(2n)!}\bigg{)}+\bigg{(}\frac{\gamma^{2n+1}}{(2n+1)!}\bigg{)}\frac{L_{0}}{\sqrt{E_{0}^{2}-J_{0}^{2}}}\bigg{]}=J_{0}\bigg{(}\cosh\gamma+\frac{L_{0}\sinh\gamma}{\sqrt{E_{0}^{2}-J_{0}^{2}}}\bigg{)},$
(17)

it is important to stress that, the energy and the angular momentum of the
theory are $E$ and $J$ respectively. The Hamiltonian function is NOT the
energy of the dynamical system as we will see later.

To all orders $T\overline{T}$-like operator is

$O^{\gamma}=\sum_{k=0}^{\infty}\bigg{[}\bigg{(}\frac{\gamma^{2n+1}}{(2n+1)!}\bigg{)}L_{0}+\bigg{(}\frac{\gamma^{2n}}{(2n)!}\bigg{)}\sqrt{E_{0}^{2}-J_{0}^{2}}\bigg{]}=L_{0}\sinh\gamma+\cosh\gamma\sqrt{E_{0}^{2}-J_{0}^{2}},$
(18)

which is solution of the $T\overline{T}$ flow equation

$\frac{dS_{NL}}{d\gamma}=\int dt\ O^{\gamma},$ (19)

with $S_{NL}=\int dt\ L$.

If we use the definitions

$S=\frac{1}{2}(\mathbf{\dot{q}}^{2}-\mathbf{q}^{2}),\hskip
28.45274ptP=\mathbf{\dot{q}}\cdot\mathbf{q},$ (20)

we note that

$S=L_{0},\hskip 28.45274ptE_{0}^{2}-J_{0}^{2}=S^{2}+P^{2}.$ (21)

Thus the deformed Lagrangian function is

$L=S\cosh\gamma+\sinh\gamma\sqrt{S^{2}+P^{2}}.$ (22)

In terms of $E_{0}$ and $J_{0}$ we can write down the Lagrangian function as

$L=E_{0}\cosh\gamma+\sinh\gamma\sqrt{E_{0}^{2}-J_{0}^{2}}-\mathbf{q}^{2}\cosh\gamma.$
(23)

This form of the Lagrangian will be useful later when we implement the
Legendre transformation to construct the Hamiltonian formalism.

###  2.2 $\sqrt{T\overline{T}}$-deformations in Hamiltonian formalism

Now we will develop the $\sqrt{T\overline{T}}$-deformations in the Hamiltonian
formalism. To implement the deformation in Hamiltonian formalism we start with
conserved quantities and then compute compute the Noether symmetries, so we
use the deformed conserved quantities just constructed. Notice that the
deformed energy and angular momentum are (16) and (17). In order to construct
the deformations order by order in $\gamma$ as in the Lagrangian case, we
start with the harmonic oscillator and its conserved quantities,

$H_{0}=\frac{1}{2}(\mathbf{p}^{2}+\mathbf{q}^{2}),$ (24)
$s_{0}=\frac{1}{2}(\mathbf{p}^{2}+\mathbf{q}^{2}),\hskip
28.45274ptj_{0}=\epsilon^{ij}p^{i}q^{j},$ (25)

where $\mathbf{p}^{2}=p_{1}^{2}+p_{2}^{2}$ and
$\mathbf{q}^{2}=q_{1}^{2}+q_{2}^{2}$ and $i=\\{1,2\\}$. The conserved
quantities $s_{0}$ and $j_{0}$ are just the Hamiltonian generators of the
duality and rotational symmetries,

$\delta_{s_{0}}q^{i}=[q^{i},s_{0}]=p^{i},\hskip
28.45274pt\delta_{s_{0}}p^{i}=[p^{i},s_{0}]=-q^{i},$ (26)
$\delta_{j_{0}}q^{i}=[q^{i},j_{0}]=\epsilon^{ij}q^{j},\hskip
28.45274pt\delta_{j_{0}}p^{i}=[p^{i},j_{0}]=-\epsilon^{ij}p^{j}.$ (27)

It is important to notice that $s_{0}$ coincide with the energy of the system
at zero order in $\gamma$. Nevertheless at order $n>0$, $s_{0}$ is not the
energy but is still the conserved quantity associated with the duality
symmetry.

In order to keep $s_{0}$ and $j_{0}$ as Hamiltonian generators of the duality
and rotational symmetry, we define the $T\overline{T}$-like Hamiltonian
operator of order $n$ as

$O^{\gamma}_{n}=\sqrt{s_{n}^{2}-j_{n}^{2}}.$ (28)

The deformed Hamiltonian is defined as (flow equation)

$H_{n+1}=s_{0}-\int d\gamma\ O^{\gamma}_{n}.$ (29)

The first order deformation of the Hamiltonian function is then

$H_{1}=s_{0}-\gamma\sqrt{s_{0}^{2}-j_{0}^{2}},$ (30)

As a second step we propose $s_{1}$ and $j_{1}$ in terms of $s_{0}$ and
$j_{0}$ in the following way

$s_{1}=s_{0}\bigg{(}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{)},$
(31)
$j_{1}=j_{0}\bigg{(}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{)}.$
(32)

$s_{1}$ and $j_{1}$ are conserved quantities of the system due to $s_{0}$ and
$j_{0}$ are conserved. Using the Noether theorem, we compute the deformed
symmetries that are generated by $s_{1}$ and $j_{1}$

$\delta_{s_{1}}q^{i}=\frac{\partial}{\partial
p^{i}}\bigg{(}s_{0}\bigg{[}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{]}\bigg{)},\hskip
28.45274pt\delta_{s_{1}}p^{i}=-\frac{\partial}{\partial
q^{i}}\bigg{(}s_{0}\bigg{[}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{]}\bigg{)},$
(33) $\delta_{j_{1}}q^{i}=\frac{\partial}{\partial
p^{i}}\bigg{(}j_{0}\bigg{[}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{]}\bigg{)},\hskip
28.45274pt\delta_{j_{1}}p^{i}=-\frac{\partial}{\partial
q^{i}}\bigg{(}j_{0}\bigg{[}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{]}\bigg{)},$
(34)

that correspond to duality and rotation deformed symmetries.

Now we construct the first order $T\overline{T}$ operator

$O_{1}^{\gamma}=\sqrt{s_{0}^{2}-j_{0}^{2}}-\gamma s_{0},$ (35)

then the second order Hamiltonian function is

$H_{2}=s_{0}-\gamma\sqrt{s_{0}^{2}-j_{0}^{2}}+\frac{1}{2}\gamma^{2}s_{0}.$
(36)

In order to construct the second order $T\overline{T}$-like operator, we
propose $s_{2}$ and $j_{2}$ conserved quantities in terms of $s_{0}$ and
$j_{0}$ as folows

$s_{2}=s_{0}\bigg{(}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}+\frac{1}{2}\gamma^{2}\bigg{)},$
(37)
$j_{2}=j_{0}\bigg{(}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}+\frac{1}{2}\gamma^{2}\bigg{)},$
(38)

these conserved quantities generate the deformed symmetries

$\delta_{s_{1}}q^{i}=\frac{\partial}{\partial
p^{i}}\bigg{(}s_{0}\bigg{[}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}+\frac{1}{2}\gamma^{2}\bigg{]}\bigg{)},\hskip
28.45274pt\delta_{s_{1}}p^{i}=-\frac{\partial}{\partial
q^{i}}\bigg{(}s_{0}\bigg{[}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}+\frac{1}{2}\gamma^{2}\bigg{]}\bigg{)},$
(39) $\delta_{j_{1}}q^{i}=\frac{\partial}{\partial
p^{i}}\bigg{(}j_{0}\bigg{[}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}+\frac{1}{2}\gamma^{2}\bigg{]}\bigg{)},\hskip
28.45274pt\delta_{j_{1}}p^{i}=-\frac{\partial}{\partial
q^{i}}\bigg{(}j_{0}\bigg{[}1-\frac{s_{0}\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}+\frac{1}{2}\gamma^{2}\bigg{]}\bigg{)}.$
(40)

which in turn are the deformed duality and rotation symmetries.

The second order $T\overline{T}$-like operator is

$O_{2}^{\gamma}=\sqrt{s_{0}^{2}-j_{0}^{2}}-\gamma
s_{0}+\frac{1}{2}\gamma^{2}\sqrt{s_{0}^{2}-j_{0}^{2}},$ (41)

Continuing with the same steps, the third order Hamiltonian function is

$H_{3}=s_{0}-\gamma\sqrt{s_{0}^{2}-j_{0}^{2}}+\frac{1}{2}\gamma^{2}s_{0}-\frac{1}{3!}\gamma^{3}\sqrt{s_{0}^{2}-j_{0}^{2}}.$
(42)

By induction we prove that the Hamiltonian function at all orders in $\gamma$
is

$H=\sum_{n=0}^{\infty}\bigg{[}\bigg{(}\frac{\gamma^{2n}}{(2n)!}\bigg{)}s_{0}-\bigg{(}\frac{\gamma^{2n+1}}{(2n+1)!}\bigg{)}\sqrt{s_{0}^{2}-j_{0}^{2}}\bigg{]}=s_{0}\cosh\gamma-\sinh\gamma\sqrt{s_{0}^{2}-j_{0}^{2}},$
(43)

which was constructed using the deformed energy and angular momentum $s$ and
$j$

$s=s_{0}\bigg{(}\cosh\gamma-\frac{s_{0}\sinh\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{)},$
(44)
$j=j_{0}\bigg{(}\cosh\gamma-\frac{s_{0}\sinh\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{)}.$
(45)

As a consequence of the underlaying scaling symmetry we found

$\frac{s}{j}=\frac{s_{0}}{j_{0}}.$ (46)

These conserved quantities generate the deformed symmetries

$\delta_{s}q^{i}=\frac{\partial}{\partial
p^{i}}\bigg{(}s_{0}\bigg{[}\cosh\gamma-\frac{s_{0}\sinh\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{]}\bigg{)},\hskip
28.45274pt\delta_{s}p^{i}=-\frac{\partial}{\partial
q^{i}}\bigg{(}s_{0}\bigg{[}\cosh\gamma-\frac{s_{0}\sinh\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{]}\bigg{)},$
(47) $\delta_{j}q^{i}=\frac{\partial}{\partial
p^{i}}\bigg{(}j_{0}\bigg{[}\cosh\gamma-\frac{s_{0}\sinh\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{]}\bigg{)},\hskip
28.45274pt\delta_{j}p^{i}=-\frac{\partial}{\partial
q^{i}}\bigg{(}j_{0}\bigg{[}\cosh\gamma-\frac{s_{0}\sinh\gamma}{\sqrt{s_{0}^{2}-j_{0}^{2}}}\bigg{]}\bigg{)}.$
(48)

The conserved quantities $s$ and $j$ can be used to construct
$T\overline{T}$-like operator that deforms the harmonic oscillators to the
non-linear oscillators using the flow equation

$H=-\int d\gamma\ O^{\gamma}$

where

$O^{\gamma}=-s_{0}\sinh\gamma+\cosh\gamma\sqrt{s_{0}^{2}-j_{0}^{2}},$ (49)

In the remaining of the article we will use instead of $s_{0},j_{0}$ just
$s,j$ so the Hamiltonian function is

$H=s\cosh\gamma-\sqrt{s^{2}-j^{2}}\sinh\gamma.$ (50)

Notice that this procedure generate the correct deformed Hamiltonian that can
be constructed in same was as the ModMax Hamiltonian as presented in appendix
C. In contrast with the Lagrangian case here the symmetries and the constants
of motion are deformed.

##  3 Integration of the Hamiltonian system

Now we discuss how to use the conserved quantities $s$ and $j$ to integrate
the Hamiltonian system. As is well known when we have $n$ degrees of freedom
the Hamiltonian system needs $n$ conserved quantities in involution to be
integrable [38]. So from the knowledge of $s$ and $j$ we were able to
integrate the system.

It is easy to prove that our non-linear oscillators has two conserved
quantities in involution that can be used to find an explicit analytic
integration of the Hamiltonian equations of motion.

First notice that the Hamilton equations are

$\begin{split}\dot{p}^{i}&=-\frac{\partial H}{\partial
q^{i}}=-\bigg{(}\cosh(\gamma)-\frac{\sinh(\gamma)s}{\sqrt{s^{2}-j^{2}}}\bigg{)}q^{i}+\frac{\sinh(\gamma)j}{\sqrt{s^{2}-j^{2}}}\epsilon^{ij}p^{j},\\\
\dot{q}^{i}&=\frac{\partial H}{\partial
p^{i}}=\bigg{(}\cosh(\gamma)-\frac{\sinh(\gamma)s}{\sqrt{s^{2}-j^{2}}}\bigg{)}p^{i}+\frac{\sinh(\gamma)j}{\sqrt{s^{2}-j^{2}}}\epsilon^{ij}q^{j}.\end{split}$
(51)

It is straightforward to prove that $s$ and $j$ are conserved quantities in
involution, i.e

$[s,j]=[s,H]=[j,H]=0.$ (52)

where $[F,G]$ denotes the Poisson bracket.

Using this fact we can define two conserved quantities

$A\equiv\cosh(\gamma)-\frac{\sinh(\gamma)s}{\sqrt{s^{2}-j^{2}}},\hskip
28.45274ptB\equiv\frac{\sinh(\gamma)j}{\sqrt{s^{2}-j^{2}}},$ (53)

that are also in involution

$[A,B]=[A,H]=[B,H]=0.$ (54)

In terms of these quatities the Hamilton equations can be rewritten as

$\begin{split}\dot{p}^{i}&=-Aq^{i}+B\epsilon^{ij}p^{j},\\\
\dot{q}^{i}&=Ap^{i}+B\epsilon^{ij}q^{j},\end{split}$ (55)

Therefore, over the surface defined by $A$ and $B$ constants, the non-linear
Hamiltonian equations are just linear equations! Moreover this reduction is
consistent with the variational principle as we show in the appendices A and
B.

We can rewritte the differential equations in a matrix notation as

$\begin{pmatrix}\dot{p}^{i}\\\
\dot{q}^{i}\end{pmatrix}=\begin{pmatrix}B\epsilon^{ij}&-A\delta^{ij}\\\
A\delta^{ij}&B\epsilon^{ij}\end{pmatrix}\begin{pmatrix}p^{j}\\\
q^{j}\end{pmatrix},\hskip
28.45274ptC^{ab}\equiv\begin{pmatrix}B\epsilon^{ij}&-A\delta^{ij}\\\
A\delta^{ij}&B\epsilon^{ij}\end{pmatrix},\hskip
28.45274ptz^{a}\equiv\begin{pmatrix}p^{j}\\\ q^{j}\end{pmatrix},$ (56)

with $a,b=\\{1,2,3,4\\}$. Over the surface defined by $A$ and $B$ constants,
the solutions of the system are obtained by taking the exponential of the
matrix $C$

$\dot{z}^{a}=D^{ab}z_{0}^{b},\hskip
28.45274ptD^{ab}=\big{(}e^{Ct}\big{)}^{ab}$ (57)

where $z_{0}^{b}$ are the initial conditions.

A remakable property of this dynamics is the fact the the matrix $C^{ab}$
admits a decomposition in term of two matrices $C_{A}$ and $C_{B}$
respectively

$C^{ab}=\begin{pmatrix}B\epsilon^{ij}&-A\delta^{ij}\\\
A\delta^{ij}&B\epsilon^{ij}\end{pmatrix}=\begin{pmatrix}0&-A\delta^{ij}\\\
A\delta^{ij}&0\end{pmatrix}+\begin{pmatrix}B\epsilon^{ij}&0\\\
0&B\epsilon^{ij}\end{pmatrix},$ (58)

These matrices commute between themselves and the exponential of $C$ is the
exponential of $C_{A}$ times the exponential of $C_{B}$,

$D^{ab}=\big{(}e^{C_{A}t}e^{C_{B}t}\big{)}^{ab}=\big{(}e^{C_{B}t}e^{C_{A}t}\big{)}^{ab},\hskip
28.45274ptD_{A}\equiv e^{C_{A}t},\hskip 28.45274ptD_{B}\equiv e^{C_{B}t}.$
(59)

Then we observe that the dynamics of our system is the same as two coupled
oscillators with frequencies $A$ and $B$ (that now we fix as numbers
corresponding with the initial data). The explicit construction is then

$D^{ab}_{A}z_{0}^{b}=\begin{pmatrix}p^{i}_{0}\cos(At)-q^{i}_{0}\sin(At)\\\
q^{i}_{0}\cos(At)+p^{i}_{0}\sin(At)\end{pmatrix},$ (60)

and

$D^{ab}_{B}z_{0}^{b}=\begin{pmatrix}p^{i}_{0}\cos(Bt)+\epsilon^{ij}p^{j}_{0}\sin(Bt)\\\
q^{i}_{0}\cos(Bt)+\epsilon^{ij}q^{j}_{0}\sin(Bt)\end{pmatrix},$ (61)

Notice that the oscillations are not the same in phase-space.

The solutions of the system are

$\begin{split}p^{i}(t)=&\cos(At)(p_{0}^{i}\cos(Bt)+\epsilon^{ij}p_{0}^{j}\sin(Bt))-\sin(At)(q_{0}^{i}\cos(Bt)+\epsilon^{ij}q_{0}^{j}\sin(Bt)),\\\
q^{i}(t)=&\sin(At)(p_{0}^{i}\cos(Bt)+\epsilon^{ij}p_{0}^{j}\sin(Bt))+\cos(At)(q_{0}^{i}\cos(Bt)+\epsilon^{ij}q_{0}^{j}\sin(Bt)),\end{split}$
(62)

where $p^{i}_{0}$ and $q^{i}_{0}$ are the initial conditions. These relations
are the general solution of the system (51) and correspond to two coupled
oscillators with frequencies $A$ and $B$ respectively.

##  4 Legendre’s transform

As has been shown in [6], Legendre’s transformation in ModMax theory must be
implemented carefully. Due to the non-linear behavior of the theory, the task
of inverting the velocities in terms of momentum variables is cumbersome. A
more easy task is to implement Legendre’s transformation over the surface $A$
and $B$ constants because there the system is linear (cf. eq. (55)). Moreover
we know that a deduction of the corresponding Lagrangian [7] can be
implemented from first principles (appendix D).

First, we consider the Hamiltonian function,

$H(q,p)=\frac{1}{2}A(q,p)(\mathbf{p}^{2}+\mathbf{q}^{2})+B(q,p)\epsilon^{ij}p^{i}q^{j},$
(63)

and we take $A$ and $B$ as constants. The Hamiltonian equations are

$\begin{split}\dot{p}^{i}&=-Aq^{i}+B\epsilon^{ij}p^{j},\\\
\dot{q}^{i}&=Ap^{i}+B\epsilon^{ij}q^{j},\end{split}$ (64)

by definition the substitution of $A$ and $B$ in terms of phase space
variables in (63) leads to the non-linear Hamiltonian function (50). A
remarkable property of this Hamiltonian is that we can recover from it the
complete equations of motion. Now we can invert the momentum variables in
terms of the configuration space

$p^{i}=\frac{1}{A}\dot{q}^{i}-\frac{B}{A}\epsilon^{ij}q^{j},$ (65)

and over the surface defined by $A$ and $B$ constants, compute the Lagrangian
function

$L=\frac{1}{2A}(\dot{\mathbf{q}}^{2}-(A^{2}-B^{2})\mathbf{q}^{2})-\frac{B}{A}\epsilon^{ij}\dot{q}^{i}q^{j},$
(66)

of course, this Lagrangian functions is consistent with the momentum
definition (65).

Now the question is how to write the corresponding expressions for $A$ and $B$
in terms of the configuration space. With the notation

$\sigma\equiv\frac{1}{2}(\mathbf{\dot{q}}^{2}+\mathbf{q}^{2})=E_{0},\hskip
28.45274pt\rho\equiv\epsilon^{ij}\dot{q}^{i}q^{j}=J_{0},$ (67)

where $E_{0}$ and $J_{0}$ are in terms of $q,\dot{q}$ (eqs. (2),(3)), we can
define the two functions $C_{1}$ and $C_{2}$

$C_{1}\equiv\cosh\gamma+\frac{\sigma\sinh\gamma}{\sqrt{\sigma^{2}-\rho^{2}}},\hskip
28.45274ptC_{2}\equiv-\frac{\rho\sinh\gamma}{\sqrt{\sigma^{2}-\rho^{2}}},$
(68)

Now the Lagrangian function (22) is

$L(\dot{\mathbf{q}},\mathbf{q})=C_{1}(\dot{\mathbf{q}},\mathbf{q})\sigma(\dot{\mathbf{q}},\mathbf{q})+C_{2}(\dot{\mathbf{q}},\mathbf{q})\rho(\dot{\mathbf{q}},\mathbf{q})-\mathbf{q}^{2}\cosh\gamma,$
(69)

Now we define the momentum by

$p^{i}=C_{1}(\dot{\mathbf{q}},\mathbf{q})\dot{q}^{i}+C_{2}(\dot{\mathbf{q}},\mathbf{q})\epsilon^{ij}q^{j},$
(70)

This definition of the momentum is consistent with the starting Lagrangian in
terms of $(q,\dot{q})$. This property comes from the fact that the Lagrangian
is a homogeneus function of degree one in $\sigma$ and $\rho$ (see appendix A
for details). Demanding the consistency between the Hamiltonian momentum (65)
and the Lagrangian momentum (70) we conclude that the implicit Legendre’s
transformation is

$C_{1}(\dot{\mathbf{q}},\mathbf{q})=\frac{1}{A(\mathbf{p},\mathbf{q})},\hskip
28.45274ptC_{2}(\dot{\mathbf{q}},\mathbf{q})=-\frac{B(\mathbf{p},\mathbf{q})}{A(\mathbf{p},\mathbf{q})},$
(71)

and using these identities, we can prove that the Lagrangian function (66)
matches the Lagrangian function (69), the complete non-linear Lagrangian
function.

Therefore we have implemented Legendre’s transformation over the surface
defined by $A$ and $B$ constants. Due to $A$ and $B$ being constants of motion
in phase space, $C_{1}$ and $C_{2}$ are also conserved in configuration space.
This last statment can also be proved using the Lagrangian formalism. We want
to stress that although $C_{1}$ and $C_{2}$ are conserved, $\sigma$ and $\rho$
are NOT conserved.

##  5 Integration of the Lagrangian system

Now we have all the necessary tools to show the complete integration of the
Lagrangian equations of motion. As in Hamiltonian formalism, we use the
conserved quantities at the level of the equation of motion to integrate the
system.

The Lagrangian equations of motion are

$\ddot{q}^{i}=\frac{C_{2}^{2}-1}{C_{1}^{2}}q^{i}-2\frac{C_{2}}{C_{1}}\epsilon^{ij}\dot{q}^{j},$
(72)

where we are using that $C_{1}$ and $C_{2}$ are constants of motion.

The substitution of the expressions (68) leads to the non-linear equation of
motion of the Lagrangian function (22). More details about how the equations
of motion can be reduced to this form are given in the appendix A. In
particular, this differential equations are linear over the surface defined by
$C_{1}$ and $C_{2}$ as constants, and the solution is

$\begin{split}q^{i}(t)&=\cos\bigg{(}\frac{1}{C_{1}}t\bigg{)}\bigg{[}q_{0}^{i}\cos\bigg{(}\frac{C2}{C1}t\bigg{)}-\epsilon^{ij}q_{0}^{j}\sin\bigg{(}\frac{C2}{C1}t\bigg{)}\bigg{]}\\\
&+\sin\bigg{(}\frac{1}{C_{1}}t\bigg{)}\bigg{[}\big{(}C_{1}\dot{q}_{0}^{i}+C_{2}\epsilon^{ij}q_{0}^{j}\big{)}\cos\bigg{(}\frac{C2}{C1}t\bigg{)}-\epsilon^{ij}\big{(}C_{1}\dot{q}^{j}_{0}+C_{2}\epsilon^{jk}q_{0}^{k}\big{)}\sin\bigg{(}\frac{C2}{C1}t\bigg{)}\bigg{]}.\end{split}$
(73)

We notice that using the momentum’s definition (65), the Hamiltonian solutions
(62) are consistent with (73). This fact is another check that our Legendre’s
transformation over the surface defined by $A$ and $B$ constants, or
equivalently $C_{1}$ and $C_{2}$ as constants, is consistent.

Interestingly the functional form of the Hamiltonian and Lagrangian functions
in terms of $(s,j)$ and $(S,P)$ respectively are the same as the Hamiltonian
and Lagrangian densities of ModMax [6, 7].

##  6 A deformation map

In this section we will show how to construct a map from the usual oscillator
in 2D to the non-linear coupled oscillator presented here.

Starting from a 2D oscillator with mass and frequency equal to one

$H_{0}=\frac{1}{2}(\widehat{p}^{2}+\widehat{q}^{2})$ (74)

we define the new variables $\widehat{p}_{i},\widehat{q}^{i}$ , using the
matrix notation

$\begin{pmatrix}\widehat{p}_{i}\\\
\widehat{q}^{i}\end{pmatrix}={\mathbb{M}}(\gamma)\begin{pmatrix}p^{j}\\\
q^{j}\end{pmatrix},$ (75)

where

${\mathbb{M}}(\gamma)=\begin{pmatrix}B(q,p,\gamma)\epsilon^{ij}&-A(q,p,\gamma)\delta^{ij}\\\
A(q,p,\gamma)\delta^{ij}&B(q,p,\gamma)\epsilon^{ij}\end{pmatrix}.$ (76)

As $A$ and $B$ are functions of $q,p,\gamma$ the map is quite nonlinear. But
on the surface $A$ and $B$ constants the map looks linear. This map is very
powerful because, in just one step, implements the deformation of the harmonic
oscillator (74) into the full non-linear oscillator in $2\gamma$

$H(q,p,2\gamma)=\frac{1}{2}\Big{(}A^{2}(q,p,\gamma)+B^{2}(q,p,\gamma)\Big{)}(\mathbf{p}^{2}+\mathbf{q}^{2})+2A(q,p,\gamma)B(q,p,\gamma)\epsilon^{ij}p^{i}q^{j}.$
(77)

Doing the explicit calculation taking into account the functional dependence
of $A$ and $B$ in terms of $,q,p,\gamma$ we find

$H(q,p,2\gamma)=\frac{1}{2}A(q,p,2\gamma)(\mathbf{p}^{2}+\mathbf{q}^{2})+B(q,p,2\gamma)\epsilon^{ij}p^{i}q^{j},$
(78)

thus this map do the job of the $T\bar{T}$ deformation in just one step. But
notice that here the deformation is obtained at $2\gamma$.

Analogous properties can be found for ModMax theory333ModMax satisfies a
remarkable indentity that for the case of the oscillator considered here can
be written in the form
$H(\mathbf{p},\mathbf{q};2\gamma)=\frac{1}{2}\bigg{[}\bigg{(}\frac{\partial
H(\mathbf{p},\mathbf{q};\gamma)}{\partial
p^{i}}\bigg{)}^{2}+\bigg{(}\frac{H(\mathbf{p},\mathbf{q};\gamma)}{\partial
q^{i}}\bigg{)}^{2}\bigg{]}.$ (79) The corresponding Lagranian identity is
$L(\mathbf{\dot{q}},\mathbf{q};2\gamma)=-\frac{1}{2}\bigg{[}\bigg{(}\frac{\partial
L(\mathbf{\dot{q}},\mathbf{q};\gamma)}{\partial\dot{q}^{i}}\bigg{)}^{2}-\bigg{(}\frac{L(\mathbf{\dot{q}},\mathbf{q};\gamma)}{\partial
q^{i}}\bigg{)}^{2}\bigg{]}.$ (80) but here we will restrict ourselves to the
Hamiltonian analysis.. See [6, 7] for details.

In Lagrangian formalism the same idea can be implemented as follows. First we
notice that the Lagrangian (66) in terms of $C_{1}$ and $C_{2}$ can be written
as

$L=\frac{1}{2}C_{1}\bigg{(}\mathbf{\dot{q}}^{2}+\frac{C_{2}^{2}-1}{C_{1}^{2}}\mathbf{q}^{2}\bigg{)}+C_{2}\epsilon^{ij}\dot{q}^{i}q^{j}.$
(81)

Using the matrix notation

$L=\frac{1}{2}\begin{pmatrix}\dot{q}^{i}\\\
q^{i}\end{pmatrix}^{T}\begin{pmatrix}C_{1}\delta^{ij}&C_{2}\epsilon^{ij}\\\
-C_{2}\epsilon^{ij}&\frac{C_{2}^{2}-1}{C_{1}}\delta^{ij}\end{pmatrix}\begin{pmatrix}\dot{q}^{j}\\\
q^{j}\end{pmatrix}.$ (82)

Now, we define the matrices and the vector

$M^{ab}=\begin{pmatrix}C_{1}\delta^{ij}&C_{2}\epsilon^{ij}\\\
C_{2}\epsilon^{ij}&-\frac{C_{2}^{2}-1}{C_{1}}\delta^{ij}\end{pmatrix},\hskip
28.45274ptg^{ab}=\begin{pmatrix}\delta^{ij}&0\\\
0&-\delta^{ij}\end{pmatrix},\hskip
28.45274ptv^{a}=\begin{pmatrix}\dot{q}^{i}\\\ q^{i}\end{pmatrix},$ (83)

$i={1,2}$ and $a,b={1,2,3,4}$. Using these definitions we have

$L[v;2\gamma]=\frac{1}{2}(v^{T})^{a}(M^{T}(\gamma))^{ab}g^{bc}M^{cd}(\gamma)v^{d},$
(84)

where $M(\gamma)$ is the matrix $M$ evaluated on $\gamma$. Then we identify a
new set of variables

$\hat{v}^{a}=M(\gamma)^{ab}v^{b},$ (85)

Using this new variables (85) the Lagrangian (84) is written as

$L[\hat{v};2\gamma]=\frac{1}{2}(\hat{v}^{T})^{a}g^{ab}\hat{v}^{b},$ (86)

This last result is the Lagrangian of the harmonic oscillator in 2D. So we
started with the full non linear Lagrangian and obtain through the definition
(85) the Lagrangian of the harmonic oscillator 2D.

##  7 Some dynamical properties of the system

In this section, we present two properties of the dynamical system. First, we
show how these two coupled oscillators have an energy transfer phenomena
between oscillators. Then we compute the Hannay angle, which is the classical
analogue of the Berry phase for the corresponding quantum system. So the
Hannay angle captures geometrical information (holonomy, parallel transport)
in the space of solutions.

###  7.1 Energy transfer

A well known property of two coupled systems that oscillate and conserve
energy, is the energy transfer phenomenon [39]. It consists in that the
amplitudes of the two oscillators also oscillate in such a way that when one
of the systems oscillates with a large amplitude, the second one oscillates
with a small amplitude and viceversa.

If we consider the amplitudes

$\alpha^{i}=q_{0}^{i}\cos\bigg{(}\frac{C_{2}}{C_{1}}t\bigg{)}-\epsilon^{ij}q_{0}^{j}\sin\bigg{(}\frac{C_{2}}{C_{1}}t\bigg{)},$
(87)

and

$\beta^{i}=\big{(}C_{1}\dot{q}_{0}^{i}+C_{2}\epsilon^{ij}q_{0}^{j}\big{)}\cos\bigg{(}\frac{C_{2}}{C_{1}}t\bigg{)}-\epsilon^{ij}\big{(}C_{1}\dot{q}^{j}_{0}+C_{2}\epsilon^{jk}q_{0}^{k}\big{)}\sin\bigg{(}\frac{C_{2}}{C_{1}}t\bigg{)},$
(88)

the solutions of the Lagrangian (73) system have the form

$q(t)^{i}=\alpha^{i}\cos\bigg{(}\frac{1}{C_{1}}t\bigg{)}+\beta^{i}\sin\bigg{(}\frac{1}{C_{1}}t\bigg{)}$
(89)

which are oscillation functions that have amplitudes that also oscillate.

We can observe the transfer phenomenon if we plot some solutions of the
system.

Figure 2: This solutions are plotted using $\gamma=1$ and the initial
conditions $q^{1}(0)=1.5$, $\dot{q}^{1}(0)=-5.1$, $q^{2}(0)=0.3$,
$\dot{q}^{2}(0)=1$.

###  7.2 Hannay angle

As we have observed, our system could be interpreted as an oscillator with
frequency $A$ mounted in a non-inertial reference frame that also oscillates
but with frequency $B$. The Hannay angle [40] could be interpreted as a phase
shift between the two oscillators.

We compute the Hannay angle considering the shift $2\pi+Bt^{\prime}$, where
$t^{\prime}$ is the time that takes one period of the first oscillator with
frequency $A$. Then the Hannay angle in our system is

$\Theta_{H}=2\pi+Bt^{\prime}=2\pi\big{(}1+\frac{B}{A}\big{)},\hskip
28.45274ptt^{\prime}=\frac{2\pi}{A},$ (90)

Using the second equation in (71), we can write the Hannay angle as

$\Theta_{H}=2\pi\bigg{(}1-\frac{\rho\sinh\gamma}{\sqrt{\sigma^{2}-\rho^{2}}}\bigg{)}.$
(91)

Of course, when $\gamma=0$, then $\Theta_{H}=2\pi$, it means that the two
oscillators are in phase. This geometrical angle depends on the initial
conditions through the definitions $A$ and $B$ evaluated at the corresponding
initial condition vector. It also is the consequence of a coupling that can be
interpreted as a covariant derivative

$Dv^{i}=\frac{dv^{i}}{dt}+\frac{C_{2}}{C_{1}}\epsilon^{ij}q^{j},$ (92)

then we can write the Lagrangian function (66) as

$L=C_{1}Dq^{i}Dq^{i}-\frac{1}{C_{1}}q^{2}.$ (93)

##  8 Conclusions

We have constructed a non-linear classical system, which consists of two
coupled oscillators. The construction can be performed by a
$\sqrt{T\overline{T}}$-deformation (in Lagrangian and Hamiltonian formalisms)
of the 2D homogeneous free harmonic oscillators. The system has a duality
symmetry and conformal (scaling) symmetries. It could be interesting to study
our system in the context of $T$-duality for point particles as [41].

The system is integrable, it has two conserved charges in involution $s$ and
$j$, which are associated with duality and rotational symmetry respectively.
It is very interesting that considering the constants of motion $A$ and $B$
(defined in terms of $s$ and $j$), it is straightforward to integrate the
equation of motion as a linear system. Moreover, $A$ and $B$ are the
frequencies of the two coupled oscillators respectively. Because we can use
the conserved quantities in the action, we found how to perform the Legendre
transformation in a simply way. It could be interesting to investigate the
possibility of thinking of our model as a non-relativistic limit of some
action, this research area has been addressed by [35, 36, 42] in the context
of $T\bar{T}$ deformations. Many of the dynamical properties of the system
studied here can be translated to ModMax theory. This step is a work in
progress that we will publish elsewhere.

We construct a map of the 2D harmonic oscillator with mass and frequency equal
to 1 to the nonlinear system at $2\gamma$. This map performs, in just one
step, the complete deformation of the harmonic oscillator produced by the
$T\bar{T}$ mechanism.

We studied the mechanical properties of our system and found that the system
present energy transfer phenomenon. Finally, we compute the Hannay angle,
which is a geometrical property of the system and space of solutions.

We leave for a future work the quantum analysis of the non linear oscillator
presented here. Other applications and extensions like the supersymmetric case
and the relativistic version are also worth to study in a future development
of the ideas presented here.

## Acknowledgements

RA was partially supported by a PhD. CONACyT fellowship number 744575. The
work of JAG was partially supported by CONACyT grant A1-S-22886 and DGAPA-UNAM
grant IN107520.

##  Appendix A Remarks on the Lagrangian formalism

In this appendix, we construct, from first principles our dynamical system
showing the role played by the conserved quantities $C_{1}$ and $C_{2}$.

Starting from the Lagrangian

$L=\sigma\cosh\gamma+\sqrt{\sigma^{2}-\rho^{2}}\sinh\gamma-\mathbf{q}^{2}\cosh\gamma,$
(94)

using $\sigma$ and $\rho$ as in (67), we define

$\bar{L}=\sigma\cosh\gamma+\sqrt{\sigma^{2}-\rho^{2}}\sinh\gamma,$ (95)

and observe that $\bar{L}$ is an homogeneous function of degree $1$ in
$(\sigma,\rho)$, son we have the relation444Notice that this relation is the
expression of scaling (conformal) symmetry as in the ModMax theory.

$\bar{L}(\sigma,\rho)=\frac{\partial\bar{L}}{\partial\sigma}\sigma+\frac{\partial\bar{L}}{\partial\rho}\rho.$
(96)

The complete Lagrangian function is

$L=\frac{\partial\bar{L}}{\partial\sigma}\sigma+\frac{\partial\bar{L}}{\partial\rho}\rho-\mathbf{q}^{2}\cosh\gamma.$
(97)

From the definition of conjugate momenta

$p^{i}=\frac{\partial^{2}L}{\partial\dot{q}^{i}\partial\sigma}\sigma+\frac{\partial^{2}L}{\partial\dot{q}^{i}\partial\rho}\rho+\frac{\partial
L}{\partial\sigma}\frac{\partial\sigma}{\partial\dot{q}^{i}}+\frac{\partial
L}{\partial\rho}\frac{\partial\rho}{\partial\dot{q}^{i}},$ (98)

we can prove that

$\frac{\partial^{2}L}{\partial\dot{q}^{i}\partial\sigma}\sigma+\frac{\partial^{2}L}{\partial\dot{q}^{i}\partial\rho}\rho=0.$
(99)

Therefore, if we define

$C_{1}=\frac{\partial\bar{L}}{\partial\sigma},\hskip
28.45274ptC_{2}=\frac{\partial\bar{L}}{\partial\rho},$ (100)

we obtain the expressions for the Lagrangian and the momenta respectively

$L=C_{1}(\mathbf{\dot{q}},\mathbf{q})\sigma(\mathbf{\dot{q}},\mathbf{q})+C_{2}(\mathbf{\dot{q}},\mathbf{q})\rho(\mathbf{\dot{q}},\mathbf{q})-\mathbf{q}^{2}\cosh\gamma,$
(101)
$p^{i}=C_{1}(\mathbf{\dot{q}},\mathbf{q})\dot{q}^{i}+C_{2}(\mathbf{\dot{q}},\mathbf{q})\epsilon^{ij}q^{j}$
(102)

where it seems like if we can derive respect to $\dot{q}^{i}$ without taking
into account the dependence of $C_{1}$ and $C_{2}$ on $\dot{q}^{i}$.

The equations of motions are

$\dot{C_{1}}\dot{q}^{i}+\dot{C_{2}}\epsilon^{ij}q^{j}+C_{1}\ddot{q}^{i}+C_{2}\epsilon^{ij}\dot{q}^{j}=\frac{\partial
C_{1}}{\partial q^{i}}\sigma+\frac{\partial C_{2}}{\partial
q^{i}}\rho+\frac{C_{2}^{2}-1}{C_{1}}q^{i}-C_{2}\epsilon^{ij}\dot{q}^{j},$
(103)

Taking in to account the time derivative of $C_{1}$ and $C_{2}$, we can
collect the terms with $\ddot{q}^{i}$

$\bigg{(}C_{1}\delta^{ik}+\frac{\partial
C_{1}}{\partial\dot{q}^{k}}\dot{q}^{i}+\frac{\partial
C_{2}}{\partial\dot{q}^{k}}\epsilon^{ij}q^{j}\bigg{)}\ddot{q}^{k}=-\frac{\partial
C_{1}}{\partial q^{k}}\dot{q}^{k}\dot{q}^{i}-\frac{\partial C_{2}}{\partial
q^{k}}\dot{q}^{k}\epsilon^{ij}q^{j}+\frac{\partial C_{1}}{\partial
q^{i}}\sigma+\frac{\partial C_{2}}{\partial
q^{i}}\rho+\frac{C_{2}^{2}-1}{C_{1}}q^{i}-2C_{2}\epsilon^{ij}\dot{q}^{j},$
(104)

and identify

$W^{ik}=C_{1}\delta^{ik}+\frac{\partial
C_{1}}{\partial\dot{q}^{k}}\dot{q}^{i}+\frac{\partial
C_{2}}{\partial\dot{q}^{k}}\epsilon^{ij}q^{j}.$ (105)

It is straightforward to prove that

$\frac{\partial C_{1}}{\partial q^{i}}\sigma+\frac{\partial C_{2}}{\partial
q^{i}}\rho=0,$ (106)

then the equations of motion are

$W^{ik}\ddot{q}^{k}=-\frac{\partial C_{1}}{\partial
q^{k}}\dot{q}^{k}\dot{q}^{i}-\frac{\partial C_{2}}{\partial
q^{k}}\dot{q}^{k}\epsilon^{ij}q^{j}+\frac{C_{2}^{2}-1}{C_{1}}q^{i}-2C_{2}\epsilon^{ij}\dot{q}^{j},$
(107)

This equations can be simplified drastically over the surface over the surface
$C_{1},C_{2}$ constants

$\ddot{q}^{i}=\frac{C_{2}^{2}-1}{C_{1}^{2}}q^{i}-2\frac{C_{2}}{C_{1}}\epsilon^{ij}\dot{q}^{j}.$
(108)

##  Appendix B Remarks on the Hamiltonian formalism

In the Hamiltonian formalism some aspects can be simplified. The conformal
condition, as in ModMax, implies that the Hamiltonian function (50) is a
homogeneous function of degree $1$ in terms of $(s,j)$,

$H(s,j)=\frac{\partial H}{\partial s}s+\frac{\partial H}{\partial j}j.$ (109)

Then the Hamiltonian equations are

$\dot{q}^{i}=\frac{\partial^{2}H}{\partial p^{i}\partial
s}s+\frac{\partial^{2}H}{\partial p^{i}\partial j}j+\frac{\partial H}{\partial
s}\frac{\partial s}{\partial p^{i}}+\frac{\partial H}{\partial
j}\frac{\partial j}{\partial p^{i}},$ (110)
$\dot{p}_{i}=-\bigg{[}\frac{\partial^{2}H}{\partial q^{i}\partial
s}s+\frac{\partial^{2}H}{\partial q^{i}\partial j}j+\frac{\partial H}{\partial
s}\frac{\partial s}{\partial q^{i}}+\frac{\partial H}{\partial
j}\frac{\partial j}{\partial q^{i}}\bigg{]}.$ (111)

It is straightforward to show that, the first two terms in each Hamiltonian
equations satisfy

$\frac{\partial^{2}H}{\partial p^{i}\partial s}s+\frac{\partial^{2}H}{\partial
p^{i}\partial j}j=0,\hskip 28.45274pt\frac{\partial^{2}H}{\partial
q^{i}\partial s}s+\frac{\partial^{2}H}{\partial q^{i}\partial j}j=0,$ (112)

therefore, if we define

$A=\frac{\partial H(s,j)}{\partial s}\hskip 28.45274ptB=\frac{\partial
H(s,j)}{\partial j},$ (113)

then we obtain

$H(\mathbf{q},\mathbf{p})=A(\mathbf{q},\mathbf{p})s(\mathbf{q},\mathbf{p})+B(\mathbf{q},\mathbf{p})j(\mathbf{q},\mathbf{p}),$
(114)

and the Hamiltonian equations are as in (55).

Of course, as we have shown in the main text, $A$ and $B$ are conserved
quantities of the system, but the interesting remark is that we can derive the
Hamiltonian equations by explicity take into account the derivatives of $A$
and $B$ as functions of $(\mathbf{q},\mathbf{p})$, or we can obtain the
Hamiltonian equations just by taking $A$ and $B$ as constants. The two
procedures gives the same result as in the equations (53).

##  Appendix C Hamiltonian deduction of ModMax

We will summarize some relevant results from the Hamiltonian deduction of
ModMax. The Hamiltonian density $\mathcal{H}$ for a generic sourceless
electromagnetic theory in the vacuum must be dependent on magnetic induction
3-vector B and the displacement current 3-vector D. The equations of motion
are

$\begin{split}\dot{\textbf{B}}&=-\nabla\times\textbf{E},\hskip
28.45274pt\nabla\cdot\textbf{B}=0,\\\
\dot{\textbf{D}}&=\nabla\times\textbf{H},\hskip
35.56593pt\nabla\cdot\textbf{D}=0,\end{split}$ (115)

taken together with the constitutive relations

$\textbf{E}=\frac{\partial\mathcal{H}}{\partial\textbf{D}},\hskip
28.45274pt\textbf{H}=\frac{\partial\mathcal{H}}{\partial\textbf{B}}.$ (116)

The equations of motion are invariant under time and spatial translations
implying that, the integrals over space of

$\dot{\mathcal{H}}=-\nabla\cdot(\textbf{E}\times\textbf{H}),\hskip
28.45274pt\mathcal{P}_{i}=-\partial_{j}T^{j}_{\ i},$ (117)

are conserved charges, where $\mathcal{P}_{i}$, $i=\\{1,2,3\\}$, are the
components of the density momentum $\mathbfcal{P}=\textbf{D}\times\textbf{B}$
and $T^{j}_{\ i}$ are the components of $3\times 3$ energy-momentum tensor,

$T^{i}_{\ j}=\delta^{i}_{\
j}(\textbf{D}\cdot\textbf{E}+\textbf{H}\cdot\textbf{B}-\mathcal{H})-(D^{i}E_{j}+H^{i}B_{j}),$
(118)

taking into account the rotational invariance, these are the manifest
symmetries of the field equations. The rotational invariance implies
$\mathbf{B}\times\mathbf{H}+\mathbf{D}\times\mathbf{E}=0$.

There are other symmetries that are not manifest in the field equations, such
as Lorentz boost invariance. In a Lorentz invariant theory is possible to
write the equations (117) as the 4-vector continuity equation for a symmetric
energy momentum tensor, if only if,

$\textbf{E}\times\textbf{H}=\textbf{D}\times\textbf{B},$ (119)

which is therefore the condition for the equations (115) to be Lorentz
invariant.

The trace of the energy-momentum tensor is

$T^{i}_{\
i}-\mathcal{H}=2[\textbf{D}\cdot\textbf{E}+\textbf{H}\cdot\textbf{B}-2\mathcal{H}],$
(120)

thus the conditions of conformal invariance are (119) and

$\textbf{D}\cdot\textbf{E}+\textbf{H}\cdot\textbf{B}=2\mathcal{H},$ (121)

this last equation, in terms of the constitutive equations (116), means that
the Hamiltonian density is a homogeneous function of degree $2$ in terms of
$(\mathbf{D},\mathbf{B})$.

Finally, the condition for invariance under the $SO(2)$ electromagnetic
duality, which acts as rotations between the 3-vectors D and B, is

$\textbf{E}\cdot\textbf{B}=\textbf{D}\cdot\textbf{H}.$ (122)

Although, the are 3 independent rotation scalars, there are at most two
duality invariants

$s=\frac{1}{2}(\textbf{D}^{2}+\textbf{B}^{2}),\hskip
28.45274ptp=|\textbf{D}\times\textbf{B}|.$ (123)

If $\mathcal{H}$ is a duality invariant, it must be a function of $s$ and $p$.
The Lorentz invariant condition (119) implies, using (116), that

$\mathcal{H}_{s}^{2}+\frac{2s}{p}\mathcal{H}_{s}\mathcal{H}_{p}+\mathcal{H}_{p}^{2}=1.$
(124)

An alternative basis for the duality invariant rotation scalars is

$u=\frac{1}{2}\bigg{(}s+\sqrt{s^{2}-p^{2}}\bigg{)},\hskip
28.45274ptv=\frac{1}{2}\bigg{(}s-\sqrt{s^{2}-p^{2}}\bigg{)}.$ (125)

These variables are well-defined since,

$s^{2}-p^{2}=\xi^{2}+\eta^{2}\geq 0,$ (126)

where $(\xi,\eta)$ are rotation invariants,

$\xi=\frac{1}{2}(\mathbf{D}^{2}-\mathbf{B}^{2}),\hskip
28.45274pt\eta=\mathbf{D}\cdot\mathbf{B},$ (127)

notice that $(\xi,\eta)$ are not duality invariants.

For solution of the form $\mathcal{H}=\sqrt{K}+\text{constant}$, the equation
(124) in terms of $(u,v)$ variables is $K_{u}K_{v}=4K$. The solution for
quadratic and non-negative $K(u,v)$ and zero vacuum energy depend on one
parameter $T$ with dimensions of energy density and a dimensionless parameter
$\gamma$,

$\mathcal{H}=\sqrt{T^{2}+2T(e^{-\gamma}u+e^{\gamma}v)+4uv}-T.$ (128)

The Lagrangian version was constructed in [43]. This Hamiltonian density is
the Born-Infeld electrodynamics [2], when $\gamma=0$. The strong field limit,
$T\rightarrow 0$, yields the duality and conformal invariant Hamiltonian
density $\mathcal{H}=p$ of Bialynicki-Birula electrodynamics [4, 5], for all
$\gamma$. The attempt to find a Lagrangian density fails, since
$\mathbf{D}\cdot\mathbf{E}-\mathcal{H}\equiv 0$. The week field limit,
$T\rightarrow\infty$ yields the Hamiltonian density

$\mathcal{H}=(\text{cosh}\gamma)s-(\text{sinh}\gamma)\sqrt{s^{2}-p^{2}},$
(129)

an equivalent expression is,

$\mathcal{H}=\frac{1}{2}(\text{cosh}\gamma)(\mathbf{D}^{2}+\mathbf{B}^{2})-\frac{1}{2}(\text{sinh}\gamma)\sqrt{(\mathbf{D}^{2}-\mathbf{B}^{2})^{2}+4(\mathbf{D}\cdot\mathbf{B})^{2}},$
(130)

and

$\mathcal{H}=\frac{1}{2}(\text{cosh}\gamma)(\mathbf{D}^{2}+\mathbf{B}^{2})-\frac{1}{2}(\text{sinh}\gamma)\sqrt{(\mathbf{D}^{2}+\mathbf{B}^{2})^{2}-4(\mathbf{D}\times\mathbf{B})^{2}}.$
(131)

This is the one-parameter extension of Maxwell electrodynamics. For any value
of $\gamma$, this Hamiltonian density satisfies (121) required for conformal
invariance. Although, there may be other non-linear Lorentz and duality
invariant theories of electrodynamics corresponding to other solutions of
(124), there exist just two theories that are also conformal invariant,
Bialynicki-Birula electrodynamics and the family of ModMax theories as [6] has
proven.

##  Appendix D Lagrangian deduction of ModMax

The existence of the ModMax Lagrangian density can be derived in a simple way
[7].

Considering the Bessel-Hagen criterion for conformal invariance is [44]

$\Theta^{\mu}_{\ \mu}=0,$ (132)

where $\Theta^{\mu}_{\ \mu}$ is the symmetric $4\times 4$ energy-momentum
tensor.

The Lagrangian density must be a functions of $(S,P)$, in this case the
equation (132) can be cast as

$\mathcal{L}_{S}S+\mathcal{L}_{P}P=\mathcal{L},$ (133)

where $S=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$,
$P=-\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}$ are the Lorentz invariants,
together with the definitions $\mathcal{L}_{S}S=\partial\mathcal{L}/\partial
S$ and $\mathcal{L}_{P}P=\partial\mathcal{L}/\partial P$. Again, this last
equation means that the Lagrangian density is a homogeneous function of degree
$1$ in terms of $(S,P)$.

The equations of motion are

$\partial_{\mu}E^{\mu\nu}=0,$ (134)

where

$E_{\mu\nu}=\frac{\partial\mathcal{L}}{\partial
F^{\mu\nu}}=-(\mathcal{L}_{S}F_{\mu\nu}+\mathcal{L}_{P}\tilde{F}_{\mu\nu}),$
(135)

with the Bianchi identity

$\partial_{\mu}\tilde{F}^{\mu\nu}=0,$ (136)

which can be used to solve $F_{\mu\nu}$ in terms of the 1-form gauge potential
$A_{\mu}$.

In this Lagrangian approach, the constitutive equations are defined through
(135). In general, they are not duality invariant. The Gaillard-Zumino
criterion [45] for invariance under duality-rotation transformations is

$\tilde{E}_{\mu\nu}E^{\mu\nu}=\tilde{F}_{\mu\nu}F^{\mu\nu},$ (137)

which is equivalent to equation (122).

Using (135) and the identity
$\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}=-F_{\mu\nu}F^{\mu\nu}$, the equation
(137) can be rewrite as follows

$4\bigg{(}\mathcal{L}_{S}^{2}-\mathcal{L}_{P}^{2}\bigg{)}P-8\mathcal{L}_{S}\mathcal{L}_{P}S=P.$
(138)

multiplying both sides of the equation (138) and using (133), it result in

$4\bigg{(}\sqrt{S^{2}+P^{2}}\mathcal{L}_{S}-\mathcal{L}\bigg{)}\bigg{(}\sqrt{S^{2}+P^{2}}\mathcal{L}_{S}+\mathcal{L}\bigg{)}=P^{2}.$
(139)

To solve this non-linear partial differential equation, it is convenient to
define,

$x=\sqrt{S^{2}+P^{2}},\hskip 28.45274pty=S,$ (140)

this $x$ and $y$ are independent variables everywhere except for the point
$P=0$ which is the only singular point of the Gaillard-Zumino condition (137).

Since equation (133) implies that $\mathcal{L}$ is homogeneus of degree 1 in
$S$ and $P$, Searching for solutions of (139) in the form,

$\mathcal{L}=\alpha x+\beta y,$ (141)

with $\alpha$ and $\beta$ to be determined and substituting this proposal in
equation (139), the values for $x$ and $y$ can be obtained,

$\alpha=\pm\sinh\gamma,\hskip 28.45274pt\beta=\cosh\gamma.$ (142)

The solution with $\alpha=-\sinh$, $\beta=\cosh\gamma$, $\gamma>0$ represent
the Lagrangian density which is unbounded from below and should be discarded.
Hence the set of Lagrangian densities, invariant under conformal group
transformation and duality rotations, is given by the one-parameter family of
functions,

$\mathcal{L}(S,P;\gamma)=(\cosh\gamma)S+(\sinh\gamma)\sqrt{S^{2}+P^{2}},$
(143)

where $\gamma\geq 0$.

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k-100-5\max\\{\tilde{c},-c\\}}\sum_{\begin{subarray}{c}\tilde{j}\leq
j+l/(1+2\alpha)\\\
\end{subarray}}2^{-(6+\delta)l-(n/2-1)l}2^{|\gamma|\tilde{j}}2^{(1+\alpha)j-(1+\alpha)\tilde{j}+\delta
m}$ $\times 2^{-m+k_{2}/3}2^{-4k_{1,+}}\epsilon_{1}^{2}\lesssim
2^{|\gamma|j-(6+\delta)l-|\gamma|l/2+|\gamma|l/(1+2\alpha)}2^{-m+\delta
m}\epsilon_{1}^{2}.$ (4.57)

Recall (4.55). From the above estimate, we have

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\ k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\leq j+l/(1+2\alpha)\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;+,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{\begin{subarray}{c}\gamma,\gamma_{1},\gamma_{2}\in\mathbb{Z}_{+}^{3}\\\
|\gamma|=20,\gamma_{1}+\gamma_{2}=\gamma\\\
\end{subarray}}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\
k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j-|\gamma|j}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\leq j+l/(1+2\alpha)\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathfrak{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim 2^{-(6+\delta)l+10(1-4\alpha)l/(2+4\alpha)}2^{-m+\delta
m}\epsilon_{1}^{2}\lesssim 2^{-m+\delta m+\delta l}\epsilon_{1}^{2}.$ (4.58)

$\bullet$ If $b=3$ or
$l\in[-100-5\max\\{\tilde{c},-c\\},\tilde{c}+10]\cap\mathbb{Z}$.

Recall the definition of the cutoff function $\phi_{\nu,3}(\xi,\eta)$ in
(4.42). Since now $(\xi_{2}-\eta_{2},\xi_{3}-\eta_{3})/|\xi-\eta|$ and
$(c_{1}^{2}\eta_{2},c_{2}^{2}\eta_{3})/\Lambda_{2}(\eta)$ are not comparable,
an improved bound as in (4.25) holds. Thanks to the improved estimate, with
minor modifications in the above argument and the argument used for the
estimate of $\mathcal{J}_{k;k_{1},k_{2};l;3}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$
in the proof of Lemma 4.2, the case $b=3$ and the case
$l\in[-100-5\max\\{\tilde{c},-c\\},\tilde{c}+10]\cap\mathbb{Z}$ can be handled
similarly. Hence finishing the proof. ∎

###  4.3. The frequencies-comparable case

In this subsection, we consider the case that the input frequencies and the
out frequency are all comparable, e.g., $\min\\{k_{1},k_{2}\\}\geq
k-100-5\max\\{\tilde{c},-c\\}$ and
$k\geq\max\\{k_{1},k_{2}\\}-100-5\max\\{\tilde{c},-c\\},$ where absolute
constants $\tilde{c}$ and $c$ are defined in (2.4).

We single out this case because, unlike previous two subsections, the
separation between the resonance case and the non-resonance case is not
entirely determined by the sign of $\mu,\nu$. Instead, we will utilize a new
partition of unity functions, see (4.60). Moreover, as one might notice when
comparing the obtained estimate (3.22) in the non-comparable case and the
obtained estimate (3.23) in the comparable case, there is also a difference.
These estimates play roles in the threshold case, i.e., $l=-m/2+\delta m$, for
which we will elaborate here.

######  Lemma 4.5.

Under the bootstrap assumption (2.9), for any
$t\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, we have

$\sup_{k\in\mathbb{Z}}\sum_{\begin{subarray}{c}a=1,2\\\
\mu,\nu\in\\{+,-\\}\end{subarray}}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}-100-5\max\\{\tilde{c},-c\\}\leq k\\\
k\leq\min\\{k_{1},k_{2}\\}+100+5\max\\{\tilde{c},-c\\}\\\
\end{subarray}}\|\int_{t_{1}}^{t_{2}}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi)\big{]}dt\|_{Z}\lesssim
2^{\delta m}\epsilon_{1}^{2}.$ (4.59)

###### Proof.

Recall (2.6). Same as what we did in (3.5), we also do space localization for
two inputs. As in the proof of Lemma 3.1, we first rule out the case
$j\geq\max\\{m,-k_{-}\\}+\delta(m+j).$ It would be sufficient to consider the
case $j\leq\max\\{m,-k_{-}\\}+\delta(m+j).$

As in (3.11), we do dyadic decomposition for
$\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$. Note that the
obtained estimates (3.12) and (3.16) are still valid. Moreover, similar to
what we did in the proof of Lemma 4.2, from the $L^{2}-L^{\infty}$ type
bilinear estimate, the estimate (2.12), and the volume of support, we can rule
out the case $k\leq-m/4-2\alpha m.$

If $l\in[-100-5\max\\{\tilde{c},-c\\},\tilde{c}+10]\cap\mathbb{Z}$, then $l$
and the relative sizes between the input frequencies and the out frequency
don’t play a role. Hence the same argument used for the estimate of
$\mathcal{J}_{k;k_{1},k_{2};l;3}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$ in the proof
of Lemma 4.2 still holds.

From now on, we focus on the case $l\in[-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}$. Recall (2.7), from Taylor
expansion, for any $\xi,\eta\in
supp(\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta)))$, we have

$\Phi_{\mu,\nu}^{a}(\xi,\eta)=|\xi_{1}|-\mu|\xi_{1}-\eta_{1}|-\nu|\eta_{1}|+\mathcal{O}(2^{k+l}).$

Based on the possible sign of $\mu,\nu,\xi_{1},\eta_{1}$, we define the
following partition of the unity functions,

$\begin{split}\forall\mu,\nu\in\\{+,-\\},\quad
1&=\varphi_{\mu,\nu}^{1}(\xi,\eta)+\varphi_{\mu,\nu}^{2}(\xi,\eta),\\\
\varphi_{+,+}^{1}(\xi,\eta):=\mathbf{1}_{[0,\infty)}((\xi_{1}-\eta_{1})\eta_{1}),\qquad&\varphi_{+,+}^{2}(\xi,\eta):=\mathbf{1}_{(-\infty,0)}((\xi_{1}-\eta_{1})\eta_{1}),\\\
\varphi_{+,-}^{1}(\xi,\eta):=\mathbf{1}_{(-\infty,0)}(\xi_{1}\eta_{1}),\qquad&\varphi_{+,+}^{2}(\xi,\eta):=\mathbf{1}_{[0,\infty)}(\xi_{1}\eta_{1}),\\\
\varphi_{-,+}^{1}(\xi,\eta):=\mathbf{1}_{(-\infty,0)}((\xi_{1}-\eta_{1})\xi_{1}),\qquad&\varphi_{+,+}^{2}(\xi,\eta):=\mathbf{1}_{[0,\infty)}((\xi_{1}-\eta_{1})\xi_{1}),\\\
\varphi_{-,-}^{1}(\xi,\eta):=0,\qquad&\varphi_{+,+}^{2}(\xi,\eta):=1.\\\
\end{split}$ (4.60)

where the cutoff function $\varphi_{\mu,\nu}^{1}(\xi,\eta)$ determines the
resonance case and the cutoff function $\varphi_{\mu,\nu}^{2}(\xi,\eta)$
determines the non-resonance case.

With the partition of unity in (4.60), we split
$\mathcal{J}_{k;k_{1},k_{2};l}^{\mu,\nu;j_{1},j_{2}}(t,\xi)$ into two parts as
follows,

$\begin{split}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)=\sum_{i^{\prime}=1,2}&\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2};i^{\prime}}(t,\xi),\quad\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2};i^{\prime}}(t,\xi):=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\\\
&\times\widehat{h_{1,k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2,k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{\mu,\nu}^{i^{\prime}}(\xi,\eta)d\eta.\end{split}$
(4.61)

$\bullet$ If $i^{\prime}=1$, i.e., the resonance case.

For any $\mu,\nu\in\\{+,-\\},a\in\\{1,2\\},$ $\xi,\eta\in
supp(\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)$
$\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{\mu,\nu}^{1}(\xi,\eta))$, we have

$\big{|}\nabla_{\xi}\big{(}\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{)}\big{|}+\big{|}\nabla_{\xi}\big{(}\Phi_{\mu,\nu}^{a}(\xi,\xi-\eta)\big{)}\big{|}\lesssim
2^{l/2}.$

Hence, as in the proof of Lemma 4.4, we can rule out the case $j\geq
m+l/2+\delta m$. Thanks to the improved upper bound of $j$. , as in the proof
of Lemma 4.4, the non-threshold case, i.e., $l\in(-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}$ follows by the same argument.

Now, we focus on the threshold case, i.e., $l=-m/2+\delta m.$ We localize
further localize further based on the size of $\xi_{2}/|\eta|,\xi_{3}/|\eta|$
as follows,

$\begin{split}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)&=\sum_{b_{1},b_{2}\in[-m/2+\delta
m,l/2+2]\cap\mathbb{Z}}\mathcal{J}_{k;k_{1},k_{2};l;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi),\\\
\mathcal{J}_{k;k_{1},k_{2};l;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)&:=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)\varphi_{\mu,\nu}^{1}(\xi,\eta)\\\
&\quad\times\psi_{b_{1},b_{2}}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))d\eta,\\\
\psi_{b_{1},b_{2}}(\xi)&:=\varphi_{b_{1};-m/2+\delta
m}(\xi_{2}/|\xi|)\varphi_{b_{2};-m/2+\delta m}(\xi_{3}/|\xi|).\\\ \end{split}$
(4.62)

Recall the approximations in (3.21). From the estimate of symbol in (3.14),
$L^{2}-L^{\infty}$-type bilinear estimate, and the volume of support of the
frequency variable of the input putted in $L^{2}$, we have

$\begin{split}&\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim
2^{l+(1+\alpha)j}2^{3k/2}\big{(}(2^{b_{1}}+2^{k+l})(2^{b_{2}}+2^{k+l})\big{)}^{1/2}2^{-m+k-5k_{+}}\epsilon_{1}^{2}.\\\
\end{split}$ (4.63)

From the above estimate, we can rule out the case $b_{1}\leq-m/4-2\alpha m$
and the case $b_{2}\leq-m/4-2\alpha m$.

Now, it remains to consider the case $b_{1}\geq-m/4-2\alpha m$ and
$b_{2}\geq-m/4-2\alpha m$, for which we have
$|\xi_{2}|,|\xi_{3}|\in[2^{-m/4-2\alpha m+k},2^{-m/4+\delta m+k}]$. From the
approximations in (3.21), due to the lower bound of $|\xi_{2}|,|\xi_{3}|$, we
have

$\big{|}1+{\mu\nu c_{1}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|^{-1}}\big{|}\gtrsim
2^{-3\alpha m},\quad\big{|}1+{\mu\nu
c_{2}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|^{-1}}\big{|}\gtrsim 2^{-3\alpha m}.$
(4.64)

Thanks to the above lower bound, the following estimate holds for the volume
of support of $\eta$ for any fixed $\xi$,

$\begin{split}&\big{|}\\{\eta:|l_{\mu,\nu}(\xi,\eta)|\leq 2^{l},|\eta|\sim
2^{k_{2}},|\xi-\eta|\sim 2^{k_{1}},|\xi|\sim 2^{k}\\}\big{|}\lesssim
2^{3k+2l+6\alpha m}.\end{split}$ (4.65)

Now, based on the size of phases $\Phi_{\mu,\nu}^{a}(\xi,\eta)$, we split
$\mathcal{J}_{k;k_{1},k_{2};l;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)$
further into two parts as follows,

$\begin{split}\mathcal{J}_{k;k_{1},k_{2};l;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)&=\sum_{a^{\prime}=1,2}\mathcal{J}_{k;k_{1},k_{2};l;a^{\prime};b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi),\\\
\mathcal{J}_{k;k_{1},k_{2};l;1;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)&:=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)\varphi_{\mu,\nu}^{1}(\xi,\eta)\\\
&\times\psi_{b_{1},b_{2}}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\psi_{\leq-3m/4+10\alpha
m}(\Phi_{\mu,\nu}^{a}(\xi,\eta))d\eta,\\\
\mathcal{J}_{k;k_{1},k_{2};l;2;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)&:=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)\varphi_{\mu,\nu}^{1}(\xi,\eta)\\\
&\times\psi_{b_{1},b_{2}}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\psi_{\geq-3m/4+10\alpha
m}(\Phi_{\mu,\nu}^{a}(\xi,\eta))d\eta.\\\ \end{split}$ (4.66)

Moreover, note that, the approximation of phases in (4.49) are still valid.
For any fixed $\xi_{1},\xi_{2},\eta\in\mathbb{R}^{3}$, thanks to the estimate
of lower bounds in (4.64), we know that $\xi_{3}$ is localized inside a much
smaller ball. More precisely,

$\begin{split}\big{|}\\{\xi_{3}:|l_{\mu,\nu}(\xi,\eta)|\sim
2^{l},|\xi_{2}|\sim 2^{b_{1}+k},&|\xi_{3}|\sim
2^{b_{2}+k},|\xi|,|\xi-\eta|,|\eta|\sim 2^{k},\\\
&|\Phi_{\mu,\nu}^{a}(\xi,\eta)|\lesssim 2^{-3m/4+10\alpha m}\\}\big{|}\lesssim
2^{-m/2+20\alpha m}.\\\ \end{split}$ (4.67)

From the above estimates of volume of support (4.65) and (4.67), the estimate
of symbol in (3.14), we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}-100-5\max\\{\tilde{c},-c\\}\leq k\\\
k\leq\min\\{k_{1},k_{2}\\}+100+5\max\\{\tilde{c},-c\\}\\\
\end{subarray}}\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l;1;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim 2^{(1+\alpha)j+l}(2^{-m/2+20\alpha
m}2^{b_{1}})^{1/2}2^{3k+2l+6\alpha m-8k_{+}}\epsilon_{1}^{2}\lesssim
2^{-m-\alpha m}\epsilon_{1}^{2}.\\\ \end{split}$ (4.68)

For
$\mathcal{J}_{k;k_{1},k_{2};l;2;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)$,
due to high oscillation in time, we do integration by parts in time once. As a
result, from (4.65), after using the volume of support of $\xi,\eta$, the
estimate (3.18) in Proposition 3.2, we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}-100-5\max\\{\tilde{c},-c\\}\leq k\\\
k\leq\min\\{k_{1},k_{2}\\}+100+5\max\\{\tilde{c},-c\\}\\\
\end{subarray}}\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;2;b_{1},b_{2}}^{a;\mu,\nu;j_{1},j_{2};1}(t,\xi)\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim 2^{(1+\alpha)j+l+3m/4-10\alpha
m}(2^{b_{1}+b_{2}})^{1/2}2^{3k+2l+6\alpha m-8k_{+}}\epsilon_{1}^{2}\lesssim
2^{-\alpha m}\epsilon_{1}^{2}.\\\ \end{split}$ (4.69)

$\bullet$ If $i^{\prime}=2$, i.e., the non-resonance case.

Recall (4.60), $\forall a\in\\{1,2\\},$ we have

$\big{|}\Phi_{\mu,\nu}^{a}(\xi,\eta)\varphi_{\mu,\nu}^{2}(\xi,\eta)\big{|}\sim
2^{k}.$

Hence, to utilize high oscillation in time, we do integration by parts in time
once. As a result, by using the same argument used in the proof of Lemma 4.3,
we have

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}-100-5\max\\{\tilde{c},-c\\}\leq k\\\
k\leq\min\\{k_{1},k_{2}\\}+100+5\max\\{\tilde{c},-c\\}\\\
\end{subarray}}\|\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2};2}(t,\xi)dt\big{]}\|_{Z}\lesssim\epsilon_{1}^{2}.$

To sum up, our desired estimate (4.59) holds. ∎

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