# On the existence and partial stability of standing waves for a nematic
liquid crystal director field equations

Paulo Amorim, Jean-Baptiste Casteras, and João-Paulo Dias  Instituto de
Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira
Ramos 149, C.P. 68530, 21941-909 Rio de Janeiro, RJ- Brasil<EMAIL_ADDRESS>CMAFcCIO & Departamento de Matemática, Faculdade de Ciências da Universidade
de Lisboa, Campo Grande, Edifício C6, 1749-016 Lisboa, Portugal
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, following the studies in [1], we consider some new aspects of
the motion of the director field of a nematic liquid crystal submitted to a
magnetic field and to a laser beam. In particular, we study the existence and
partial orbital stability of special standing waves, in the spirit of [5] and
[12] and we present some numerical simulations.

  * Keywords:

liquid crystal, nematic, magnetic field, laser beam, standing waves,
stability.

  * MSC (2020):

76D03,76W05,78-10,35L53, 35C07.

##  1\. Introduction and main results

A great number of technological applications related to data display and non-
linear optics, use thin films of nematic liquid cristals, cf. [7] for the
general theory of nematic liquid cristals. In such devices the local direction
of the optical axis of the liquid crystal is represented by a unit vector
$\mathbf{n}(x,t)$, called the director, and may be modified by the application
of an electric or magnetic field. The interaction of a light beam with the
dynamics of the director $\mathbf{n}(x,t)$, under a magnetic field, helps to
improve the device performance.

In this paper we consider the model introduced in [1] to describe the motion
of the director field of a nematic liquid crystal submitted to an external
constant strong magnetic field $\mathbf{H}$, with intensity $H\in\mathbb{R}$,
and also to a laser beam, assuming some simplifications and approximations
motivated by previous experiments and models (cf. [2], [3], [19] and [20], for
magneto-optic experiments, and [15], [23] for the simplified director field
equation).
The system under consideration reads

(1.1) $\begin{array}[]{ll}\left\\{\begin{array}[]{l}iu_{t}+u_{xx}=-\rho
u+a|u|^{2}u+H^{2}x^{2}u\vspace*{2mm}\\\
\rho_{tt}=(\sigma(v))_{x}-b\rho+|u|^{2},\end{array}\right.&x\in\mathbb{R},\;t\geq
0,\end{array}$

where $i$ is the imaginary unit, $u(x,t)$ is a complex valued function
representing the wave function associated to the laser beam under the presence
of the magnetic field $\mathbf{H}$ orthogonal to the director field,
$\rho\in\mathbb{R}$ measures the angle of the director field with de $x$
axis,$v=\rho_{x}$, $a,H\in\mathbb{R},b>0$ are given constants, with initial
data

(1.2)
$u(x,0)=u_{0}(x),\rho(x,0)=\rho_{0}(x),\rho_{t}(x,0)=\rho_{1}(x),x\in\mathbb{R},$

and where the function $\sigma(v)$ is given by

(1.3) $\sigma(v)=\alpha v+\lambda
v^{3},\quad\lambda=\frac{2}{3}\gamma(\alpha-\beta),$

where $\alpha\geq\beta>0$ are elastic constants of the liquid crystal, cf.
[15], and

(1.4) $\gamma=4(\chi_{a})^{-1}H^{-2}\beta>0,$

where $\chi_{a}>0$ is the anisotropy of the magnetic susceptibility, cf. [19].
In the quasilinear case $\alpha>\beta,\alpha\simeq\beta$, the study of the
existence of a weak global solution to the Cauchy problem for the system (1.1)
with the initial data (1.2), in suitable spaces, has been developed in [1], by
application of the compensated compactness method introduced in [21] to the
regularised system with a physical viscosity and the vanishing viscosity
method (cf. also [8] and [9] for two examples of this technique applied to
related systems of short waves-long waves).
In Section 2 we prove, in the general case ($\lambda\geq 0$), by application
of Theorem 6 in [18], a local in time existence and uniqueness theorem of a
classical solution for the Cauchy problem (1.1),(1.2). For this purpose we
need to introduce some functional spaces and point out several well known
results:
Let be the linear operator defined in $L^{2}(\mathbb{R})$ by

(1.5) $Au=u_{xx}-H^{2}x^{2}u,\,u\in D(A),H\neq{0},$

where $D(A)=\big{\\{}u\in X_{A}|Au\in L^{2}(\mathbb{R})\big{\\}}$ , with

(1.6) $X_{A}=\big{\\{}u\in H^{1}(\mathbb{R})|xu\in
L^{2}(\mathbb{R})\big{\\}}.$

It can be proved, cf. [22], that if $u\in X_{1}=\big{\\{}u|xu,u_{x}\in
L^{2}(\mathbb{R})\big{\\}}$, then $u\in L^{2}(\mathbb{R})$ with (denoting by
$\|.\|_{p}$ the norm $\|.\|_{L^{p}(\mathbb{R})}$)

(1.7) $\|u\|^{2}_{2}\leq 2^{\frac{1}{2}}\|u_{x}\|_{2}\|xu\|_{2},\forall u\in
X_{1},$

and so $X_{A}=X_{1}$, and it is not difficult to prove that the injection of
$X_{A}$ in $L^{q}(\mathbb{R}),2\leq q<+\infty$, is compact (cf. [12]).
Moreover, it may be also proved, cf. [4], lemma 9.2.1, that $A$ is self-
adjoint in $L^{2}(\mathbb{R})$, $(Au,u)\leq 0,\forall u\in D(A)$, and (cf.
[13]),

(1.8) $D(A)=(-A+1)^{-1}L^{2}(\mathbb{R})=\big{\\{}u\in
H^{2}(\mathbb{R})|x^{2}u\in L^{2}(\mathbb{R})\big{\\}}.$

We can now state the first result that will be proved in Section 2:

######  Theorem 1.

Let $(u_{0},\rho_{0},\rho_{1})\in D(A)\times H^{3}\times H^{2}$ and
$\lambda\geq 0$. Then, there exists $T^{*}=T^{*}(u_{0},\rho_{0},\rho_{1})>0$
such that, for all $T<T^{*}$, there exists an unique solution $(u,\rho)$ to
the Cauchy problem (1.1),(1.2) with $u\in C([0,T];D(A))\cap
C^{1}([0,T];L^{2})$ and $\rho\in C([0,T];H^{3})\cap C^{1}([0,T];H^{2})\cap
C^{2}([0,T];H^{1})$.

As it is well known, in the quasilinear case the local solution, in general,
blows-up in finite time.

In Section 3, by obtaining the convenient estimates, we prove the following
result in the semilinear case ($\alpha=\beta$):

######  Theorem 2.

Let $(u_{0},\rho_{0},\rho_{1})\in D(A)\times H^{3}\times H^{2}$ and
$\lambda=0$. Then, there exists an unique global in time solution $(u,\rho)$
to the Cauchy problem (1.1),(1.2), with $u\in C([0,+\infty);D(A))\cap
C^{1}([0,+\infty);L^{2})$ and $\rho\in C([0,+\infty);H^{3})\cap
C^{1}([0,+\infty);H^{2})\cap C^{2}([0,+\infty);H^{1})$.

In the special case of initial data with compact support, we will prove in
Section 4 the following result:

######  Theorem 3.

Assuming the hypothesis of Theorem 2, consider the particular case where

(1.9) $\it{supp}\big{\\{}u_{0},\rho_{0},\rho_{1}\big{\\}}\subset
D=]-\theta,\theta[,\quad\theta>0.$

Then, for each $t>0$ and $\varepsilon>0$ , there exists a
$\delta=\delta(t,\varepsilon,\|u_{0}\|_{H^{1}})>0$, such that

(1.10)
$\int_{\mathbb{R}\setminus(D+B(0,\delta))}[|u|^{2}+|\rho|^{2}+|\rho_{x}|^{2}+|\rho_{t}|^{2}](x,t)dx\leq\varepsilon,$

where $B(0,\delta)=\big{\\{}x\in\mathbb{R}||x|<\delta\big{\\}}$.

The proof of this result follows a technique introduced in [6] in the case of
the nonlinear Schrödinger equation.
In Section 5, which contains the main result in the paper, we study the
existence and possible partial orbital stability of the standing waves for the
system (1.1) with $a=-1$ (attractive case) and $\lambda\geq 0$. These
solutions are of the form

(1.11) $(e^{i\mu t}u(x),\rho(x)),\,\mu\in\mathbb{R},$

and the system (1.1) takes the aspect (we fix $\alpha=1$, without loss of
generality):

(1.12)
$\begin{array}[]{ll}\left\\{\begin{array}[]{l}u_{xx}-H^{2}x^{2}u+|u|^{2}u+\rho
u=\mu u\vspace*{2mm}\\\
-\rho_{xx}-\lambda(\rho_{x}^{3})_{x}+b\rho=|u|^{2},\end{array}\right.&x\in\mathbb{R}.\end{array}$

We can rewrite this system as a scalar equation

(1.13) $u_{xx}-H^{2}x^{2}u+|u|^{2}u+\rho(|u|^{2})u=\mu u,$

where $\rho(f)$ is the solution to
$-\rho_{xx}-\lambda(\rho_{x}^{3})_{x}+b\rho=f$. It is not difficult to prove
that if $f\in L^{2}$, there exists a unique $\rho\in H^{2}$. This allows for
instance to prove that $\rho(|u|^{2})u^{2}\in L^{1}$ provided that $u\in
X_{A}$. Now, to find nontrivial solutions of this equation belonging to
$D(A)$, the domain of the linear operator defined by (1.5), we will closely
follow the technique introduced in [12] for the case of the Gross-Pitaevskii
equation. More precisely, we consider the energy functional defined in $X_{A}$
by (with $\int.dx=\int_{\mathbb{R}}.dx$):

(1.14)
$\begin{split}\mathcal{E}(u)=\frac{1}{2}\int|u_{x}|^{2}dx+\frac{1}{2}H^{2}\int
x^{2}|u|^{2}dx\\\
-\frac{1}{4}\int|u|^{4}dx-\frac{1}{2}\int\rho(|u|^{2})|u|^{2}dx,\,u\in
X_{A},\end{split}$

and we look to solve the following constrained minimization problem for a
prescribed $c>0$:

(1.15) $\mathcal{I}_{c}=\inf\big{\\{}\mathcal{E}(u),u\in
X_{A},\,\text{real},\int|u(x)|^{2}dx=c^{2}\big{\\}}.$

We start by proving the following result which corresponds to Lemma 1.2 in
[12].

######  Theorem 4.

We have:
i) The energy functional $\mathcal{E}$ is $C^{1}$ on $X_{A}$ real.
ii) The mapping $c\rightarrow\mathcal{I}_{c}$ is continuous.
iii) Any minimizing sequence of $\mathcal{I}_{c}$ is relatively compact in
$X_{A}$ and so, if $\big{\\{}u_{n}\big{\\}}_{n\in\mathbb{N}}\subset X_{A}$ is
a corresponding minimizing sequence,then there exists $u\in X_{A}$ such that
$\|u\|^{2}_{2}=c^{2}$ and $\lim_{n\rightarrow+\infty}u_{n}=u$ in $X_{A}$.
Moreover $u(x)=u(|x|)$ is radial decreasing and satisfies (1.13) for a certain
$\mu\in\mathbb{R}$.

To prove this result we follow the ideas in [12] and introduce the real space
$\tilde{X}_{A}=\big{\\{}w=(u,v)\in X_{A}\times X_{A}\big{\\}}$, for real $u$
and $v$, with norm

(1.16) $\|w\|^{2}_{\tilde{X}_{A}}=\|u\|^{2}_{X_{A}}+\|v\|^{2}_{X_{A}},u,v\in
X_{A},$

and observe that if $u=u_{1}+iu_{2}$,with
$u_{1}=\mathcal{R}e\,u,u_{2}=\mathcal{I}m\,u$, the equation (1.13) can be
written in the system form:

(1.17)
$\begin{array}[]{ll}\left\\{\begin{array}[]{l}u_{1xx}-H^{2}x^{2}u_{1}+|u|^{2}u_{1}+\rho(|u|^{2})u_{1}=\mu
u_{1}\vspace*{2mm}\\\
u_{2xx}-H^{2}x^{2}u_{2}+|u|^{2}u_{2}+\rho(|u|^{2})u_{2}=\mu
u_{2}\end{array}\right.&x\in\mathbb{R},\end{array}$

with
$w=(u_{1},u_{2})\in\tilde{X}_{A},u_{1}=\mathcal{R}e\,u,u_{2}=\mathcal{I}m\,u$.
In the new space $\tilde{X}_{A}$, the functional defined in (1.14) takes the
form, for $w=(u,v)\in\tilde{X}_{A},|w|^{4}=(|u_{1}|^{2}+|u_{2}|^{2})^{2},$

(1.18)
$\begin{split}\mathcal{\tilde{E}}(u)=\frac{1}{2}\int|w_{x}|^{2}dx+\frac{1}{2}H^{2}\int
x^{2}|w|^{2}dx\\\
-\frac{1}{4}\int|w|^{4}dx-\frac{1}{2}\int\rho(|w|^{2})|w|^{2}d\xi,\,w\in\tilde{X}_{A},\end{split}$

and, for all $c>0$,we introduce

(1.19)
$\mathcal{\tilde{I}}_{c}=\inf\big{\\{}\mathcal{\tilde{E}}(w),w\in\tilde{X}_{A},\int|w(x)|^{2}dx=c^{2}\big{\\}},$

and the sets

$\mathcal{W}_{c}=\big{\\{}u\in
X_{A},\|u\|^{2}_{2}=c^{2},\mathcal{I}_{c}=\mathcal{E}(u),u>0\big{\\}},$
$\mathcal{Z}_{c}=\big{\\{}w\in\tilde{X}_{A},\|w\|^{2}_{2}=c^{2},\mathcal{\tilde{I}}_{c}=\mathcal{\tilde{E}}(w)\big{\\}}.$

Following [5] and [12], we introduce the following definition:

Definition: The set $\mathcal{Z}_{c}$ is said to be stable if
$\mathcal{Z}_{c}\neq\varnothing$ and for all $\varepsilon>0$, there exists
$\delta>0$ such that, for all
$w_{0}=({u_{1}}_{0},{u_{2}}_{0})\in\tilde{X}_{A}$, we have, for all $t\geq 0$,

$\inf_{w\in\mathcal{Z}_{c}}\|w_{0}-w\|_{\tilde{X}_{A}}<\delta\Longrightarrow\inf_{w\in\mathcal{Z}_{c}}\|\psi(.,t)-w\|_{\tilde{X}_{A}}<\varepsilon,$

where $\psi(x,t)=(u_{1}(x,t),u_{2}(x,t))$ corresponds to the solution
$u(x,t)=u_{1}(x,t)+iu_{2}(x,t)$ of the first equation in the Cauchy problem
(1.1),(1.2), with initial data $u_{0}(x)={u_{1}}_{0}(x)+i{u_{2}}_{0}(x)$ and
where $\rho(x,t)=\rho(|u(x,t)|^{2})(x,t)$ satisfies

$-\rho_{xx}-\lambda(\rho_{x}^{3})_{x}+b\rho=|u(.,t)|^{2}.$

This corresponds to the hypothesis $\rho_{tt}\simeq 0$, cf. [2], [3] and [19]
. The local existence and uniqueness in $X_{A}$ to the corresponding Cauchy
problem for the Schrödinger equation is a consequence of Theorem $3.5.1$ in
[4]. It is easy to get the global existence of such solution $\psi(t)$ if
their initial data is closed to $\mathcal{Z}_{c}$. Indeed, denote by $T$ the
maximal time of existence and suppose that $\mathcal{Z}_{c}$ is stable at
least up to the time $T$. So, using the stability at time $T$, we see that
$\psi(T)$ is uniformly bounded in $\tilde{X}_{A}$. Therefore, we can apply the
local existence result for initial data $\psi(T)$. This contradicts the
maximality of $T$ and yields to the global existence.
Proceeding as in the proof of Theorem 3 (see in particular (5.9)), we can show
that

$\|\rho(|\psi(t)|^{2})-\rho(|w|^{2})\|_{H^{1}}\leq
C\|\psi(t)+w\|_{L^{2}}\|\psi(t)-w\|_{\tilde{X_{A}}},$

where $C$ is a constant not depending on $t$. So, if $w$ is stable, we derive,
in the conditions of the definition,

(1.20)
$\inf_{w\in\mathcal{Z}_{c}}\|\rho(.,t)-\varphi\|_{H^{1}}<c_{1}(\|u_{0}\|_{2}+c)\varepsilon.$

We point out that, if $w=(u_{1},u_{2})\in\mathcal{Z}_{c}$, then there exists a
Lagrange multiplier $\mu\in\mathbb{R}$ such that $w$ satisfies (1.17), that is
$u=u_{1}+iu_{2}$ satisfies (1.13).
We will prove the following result which is a variant of Theorem 2.1 in [12]:

######  Theorem 5.

The functional $\tilde{\mathcal{E}}$ is $C^{1}$ in $\tilde{X}_{A}$ and we have
i) For all
$c>0,\mathcal{I}_{c}=\mathcal{\tilde{I}}_{c},\mathcal{Z}_{c}\neq\varnothing$
and $\mathcal{Z}_{c}$ is stable.
ii) For all $w\in\mathcal{Z}_{c},|w|\in\mathcal{W}_{c}$.
iii) $\mathcal{Z}_{c}=\big{\\{}e^{i\theta}u,\theta\in\mathbb{R}\big{\\}}$,
with $u$ real being a minimizer of (1.15).

The proof of this result is similar to the proof of Theorem 2.1 in [12]. We
repeat some parts of the original proof for sake of completeness. Next, in
Section 6, also following closely [12], we prove a bifurcation result
asserting in particular that all solutions of the minimisation problem (1.15)
belongs to a bifurcation branch starting from the point $(\lambda_{0},0)$ (in
the plane $(\mu,u)$) where $\lambda_{0}$ is the first eigenvalue of the
operator $-\partial_{xx}+H^{2}x^{2}$.

######  Proposition 1.

The point $(\lambda_{0},0)$ is a bifurcation point for (1.13) in the plane
$(\mu,u)$ where $-\mu\in\mathbb{R}^{+}$ and $u\in X_{A}$. The branch issued
from this point is unbounded in the $\mu$ direction (it exists for all
$-\mu>\lambda_{0}$). Moreover solutions to (1.13) belonging to this branch are
in fact minimizers of problem (1.15).

As already mentioned, the proof of this proposition follows closely the one of
[12, Theorem 3.1]. An important ingredient which has also independent interest
is the following uniqueness result.

######  Proposition 2.

There exists a unique radial positive solution to (1.13) such that
$\lim_{r\rightarrow\infty}u(r)=0$.

The proof of this proposition is strongly inspired by [14].

Finally, in Section 7 we present some numerical simulations illustrating the
behaviour of the standing waves according to the intensity of the magnetic
field $\mathbf{H}$, and also the limit as the Lagrange multiplier $-\mu$
approaches the bifurcation value $\lambda_{0}$.

##  2\. Local existence in the general case

In order to prove Theorem 1, let us introduce the Riemann invariants
associated to the second equation in the system (1.1),

(2.1) $l=w+\int_{0}^{v}\sqrt{\alpha+3\lambda\xi^{2}}d\xi\quad\textrm{and}\quad
r=w-\int_{0}^{v}\sqrt{\alpha+3\lambda\xi^{2}}d\xi,$

where $w=\rho_{t},v=\rho_{x}$. We derive

$l-r=2\int_{0}^{v}\sqrt{\alpha+3\lambda\xi^{2}}d\xi$ $=v\sqrt{\alpha+3\lambda
v^{2}}+\frac{1}{\sqrt{3\lambda}}\,\textrm{arcsinh}(\sqrt{3\lambda}v),\quad
w=\frac{l+r}{2}.$

Noticing that

$\displaystyle f(v)=v\sqrt{\alpha+3\lambda
v^{2}}+\frac{1}{\sqrt{3\lambda}}\,\textrm{arcsinh}(\sqrt{3\lambda}v)$

is one-to-one and smooth, we have $v=f^{-1}(l-r)=v(l,r)$ and, for classical
solutions, the Cauchy problem (1.1),(1.2) is equivalent to the system

(2.2) $\left\\{\begin{array}[]{lllll}iu_{t}+u_{xx}-H^{2}x^{2}u=-\rho
u+a|u|^{2}u\\\ \\\ \rho_{t}=\frac{1}{2}(l+r)\\\ \\\
l_{t}-\sqrt{\alpha+3\lambda v^{2}}l_{x}=-b\rho+|u|^{2}\\\ \\\
r_{t}+\sqrt{\alpha+3\lambda v^{2}}r_{x}=-b\rho+|u|^{2}\\\ \end{array}\right.$

with initial data (cf. (1.5),(1.8)),

(2.3) $\begin{split}u(.,0)=u_{0}\in D(A)=\big{\\{}u\in
H^{2}{\mathbb{R}}|x^{2}u\in L^{2}(\mathbb{R})\big{\\}},\\\
\rho(.,0)=\rho_{0}\in H^{3}(\mathbb{R}),l(.,0)=l_{0}\in
H^{2}(\mathbb{R}),r(.,0)=r_{0}\in H^{2}(\mathbb{R}).\end{split}$

In order to apply Kato’s theorem (cf. [18, Thm. 6]) to obtain the existence
and uniqueness of a local in time strong solution, cf. Theorem 1, for the
corresponding Cauchy problem, we need to pass to real spaces, introducing the
variables

(2.4) $u_{1}=\mathcal{R}e\,u,u_{2}=\mathcal{I}m\,u.$

Now, we can pass to the proof of Theorem 1:
With $({u_{1}}_{0},{u_{2}}_{0})=(u_{1}(.,0),u_{2}(.,0))$, let

(2.5)
$U=(u_{1},u_{2},\rho,l,r),U_{0}=({u_{1}}_{0},{u_{2}}_{0},\rho_{0},l_{0},r_{0}),$

and

$\mathcal{A}(U)=\left[\begin{array}[]{ccccccccccc}0&A&0&0&0\\\ -A&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&-\sqrt{\alpha+3\lambda v^{2}}\frac{\partial}{\partial
x}&0\\\ 0&0&0&0&\sqrt{\alpha+3\lambda v^{2}}\frac{\partial}{\partial
x}\end{array}\right],$

$g(t,U)=\left[\begin{array}[]{cccccccccc}-\rho
u_{2}+a(u_{1}^{2}+u_{2}^{2})u_{2}\\\ \rho u_{1}-a(u_{1}^{2}+u_{2}^{2})u_{1}\\\
\frac{1}{2}(l+r)\\\ -b\rho+|u|^{2}\\\ -b\rho+|u|^{2}\end{array}\right].$

The initial value problem (2.2), (2.3) can be written in the form

(2.6) $\left\\{\begin{array}[]{ccccccc}\displaystyle\frac{\partial}{\partial
t}U+\mathcal{A}(U)U=g(t,U)\\\ U(.,0)=U_{0}.\end{array}\right.$

Let us take

$U_{0}=({u_{1}}_{0},{u_{2}}_{0},\rho_{0},l_{0},r_{0})\in
Y=(D(A))^{2}\times(H^{2}(\mathbb{R}))^{3}$

(the condition $\rho_{0}\in H^{3}(\mathbb{R})$ will be used later). We now set
$X=(L^{2}(\mathbb{R}))^{2}\times(L^{2}(\mathbb{R}))^{3}$ and
$S=((1-A)I)^{2}\times((1-\Delta)I)^{3}$, which is an isomorphism
$S:Y\rightarrow X$. Furthermore, we denote by $W_{R}$ the open ball in $Y$ of
radius $R$ centered at the origin and by $G(X,1,\omega)$ the set of linear
operators $\Lambda\,:\,D(\Lambda)\subset X\to X$ such that:

  * •

$-\Lambda$ generates a $C_{0}$-semigroup
$\\{e^{-t\Lambda}\\}_{t\in\mathbb{R}_{+}}$;

  * •

for all $t\geq 0$, $\|e^{-t\Lambda}\|\leq e^{\omega t}$, where, for all $U\in
W_{R}$,

$\omega=\frac{1}{2}\sup_{x\in\mathbb{R}}\|\frac{\partial}{\partial
x}a(\rho,l,r)\|\leq c(R),\quad c:[0,+\infty[\to[0,+\infty[\textrm{ continuous,
and }$ $a(\rho,l,r)=\left[\begin{array}[]{cccccccccc}0&0&0\\\
0&-\sqrt{\alpha+3\lambda v^{2}}&0\\\ 0&0&\sqrt{\alpha+3\lambda
v^{2}}\end{array}\right].$

By the properties of the operator $A$ (cf. Section 1) and following [18,
Section 12], we derive

$\mathcal{A}\,:U=(u_{1},u_{2},\rho,l,r)\in W_{R}\to G(X,1,\omega),$

and it is easy to see that $g$ verifies, for fixed $T>0$,
$\|g(t,U(t))\|_{Y}\leq\theta_{R}$, $t\in[0,T]$, $U\in C([0,T];W_{R})$.
For $(\rho,l,r)$ in a ball $\tilde{W}$ in $(H^{2}(\mathbb{R}))^{3}$, we set
(see [18, (12.6)]), with $[.,.]$ denoting the commutator matrix operator,

$B_{0}(\rho,l,r)=[(1-\Delta),a(\rho,l,r)](1-\Delta)^{-1}\in\mathcal{L}((L^{2}(\mathbb{R}))^{3}).$

We now introduce the operator $B(U)\in\mathcal{L}(X)$,
$U=(F_{1},F_{2},\rho,l,r)\in W_{R}$, by

$B(U)=\left[\begin{array}[]{cccccccccc}0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0\\\
0&0&&B_{0}(\rho,l,r)\\\ 0&0\end{array}\right].$

In [18, Section 12], Kato proved that for $(\rho,l,r)\in\tilde{W}$ we have

$(1-\Delta)a(\rho,l,r)(1-\Delta)^{-1}=a(\rho,l,r)+B_{0}(\rho,l,r).$

Hence, we easily derive

$S\mathcal{A}(U)S^{-1}=\mathcal{A}(U)+B(U),U\in W_{R}.$

Now, it is easy to see that conditions (7.1)–(7.7) in Section 7 of [18] are
satisfied and so we can apply Theorem 6 in [18] and we obtain the result
stated in Theorem 1, with $\rho\in C([0,T];H^{2})\cap C^{1}([0,T];H^{1})\cap
C^{2}([0,T];L^{2})$.
To obtain the requested regularity for $\rho$ it is enough to remark that,
since $\rho_{x}=v,\rho_{t}=w,\rho_{0}\in H^{3},v_{0}={\rho_{0}}_{x}\in
H^{2},w_{0}=\rho_{1}\in H^{2}$, we deduce $\rho_{x}=v\in
C([0,T];H^{2}),\rho_{t}=w\in C([0,T];H^{2})$, and this achieves the proof of
Theorem 1.

##  3\. Global existence in the semilinear case

Now, we consider the semilinear case, that is when $\alpha=\beta$ and so
$\lambda=0$.
Hence we pass to the proof of Theorem 2. For the local in time unique solution
$(u,\rho)$ defined in the interval $[0,T^{*}[,T>0$, to the Cauchy problem
(1.1),(1.2), obtained in Theorem 1, we easily deduce the following
conservation laws (cf. [1]) in the case $\lambda\geq 0,\alpha>0$:

(3.1) $\int|u(x,t)|^{2}\;dx=\int|u_{0}(x)|^{2}\;dx,t\in[0,T^{*}[.$ (3.2)
$\begin{split}E(t)=\frac{1}{2}\int(\rho_{t}(x,t))^{2}\;dx+\frac{\alpha}{2}\int(\rho_{x}(x,t))^{2}\;dx+\frac{\lambda}{4}\int(\rho_{x}(x,t))^{4}\;dx\\\
+\frac{b}{2}\int(\rho(x,t))^{2}\;dx-\int\rho(x,t)|u(x,t)|^{2}\;dx+\int|u_{x}(x,t)|^{2}\;dx\\\
+\frac{a}{2}\int|u(x,t)|^{4}\;dx+H^{2}\int
x^{2}|u(x,t)|^{2}\;dx=E(0),t\in[0,T^{*}[.\end{split}$

Applying the Gagliardo-Nirenberg inequality to the term
$|\frac{a}{2}\int|u(x,t)|^{4}\;dx|$ and since $b>0$ we easily derive (cf.
[1]), for $t\in[0,T^{*}[$,

(3.3)
$\begin{split}\int(\rho_{t}(x,t))^{2}\;dx+\int(\rho_{x}(x,t))^{2}\;dx+\lambda\int(\rho_{x}(x,t))^{4}\;dx\\\
+\int(\rho(x,t))^{2}\;dx+\int|u_{x}(x,t)|^{2}\;dx+H^{2}\int
x^{2}|u(x,t)|^{2}\;dx\leq c_{1}.\end{split}$

We continue with the proof of Theorem 2, in the semilinear case, that is
$\lambda=0$. We have, for $t\in[0,T^{*}[,$

(3.4)
$\|\rho(t)\|_{2}\leq\|\rho_{0}\|_{2}+\int_{0}^{t}\|\rho_{t}(\tau)\|d\tau\leq
c_{2}(1+t).$

Next we estimate $\|Au(t)\|_{2},\|\rho_{xt}(t)\|_{2}$ and $\|\rho_{xx}\|_{2}$.
For $\lambda=0$, the system (2.2) reads

(3.5) $\left\\{\begin{array}[]{ll}iu_{t}+u_{xx}-H^{2}x^{2}u=-\rho
u+a|u|^{2}u\\\ \rho_{t}=\frac{1}{2}(l+r)\\\
l_{t}-\sqrt{\alpha}\,l_{x}=-b\rho+|u|^{2}\\\
r_{t}+\sqrt{\alpha}\,r_{x}=-b\rho+|u|^{2}\\\ \end{array}\right.$

with initial data (2.3). To simplify, we assume $\alpha=\beta=b=1$.
Recall that we have, since $\lambda=0$,

(3.6) $\left\\{\begin{array}[]{l}l=w+v=\rho_{t}+\rho_{x}\\\ \\\
r=w-v=\rho_{t}-\rho_{x}.\\\ \end{array}\right.$

From (3.5), we derive

$r_{tx}r_{x}+r_{xx}r_{x}=-\rho_{x}r_{x}+2\mathcal{R}e(\bar{u}u_{x})r_{x},$

and so

$\frac{1}{2}\frac{d}{dt}\int(r_{x})^{2}dx\leq\frac{1}{2}\int[(\rho_{x})^{2}+(r_{x})^{2}]dx+c_{3}\int(r_{x})^{2}dx+c_{3},$

and a similar estimate for $l_{x}.$ We deduce, with $c_{4}(t)$ being a
positive, increasing and continuous function,

(3.7) $\|r_{x}(t)\|^{2}_{2}+\|l_{x}(t)\|^{2}_{2}\leq c_{4}(t),t\in[0,T^{*}[.$

Moreover, we derive from (3.5), formally,

$\mathcal{R}e(u_{tt}\bar{u}_{t})+\mathcal{I}m[(u_{xxt}-H^{2}x^{2}u_{t})\bar{u}_{t}]=a\mathcal{I}m[(|u|^{2}u)_{t}\bar{u}_{t}],$
$\frac{1}{2}\frac{d}{dt}\int|u_{t}|^{2}dx-\mathcal{I}m\int
u_{xt}\bar{u}_{xt}dx=2a\mathcal{I}m\int\mathcal{R}e(u\bar{u}_{t})u\bar{u}_{t}dx$

$\leq c_{5}\int|u_{t}|^{2}dx$, and hence

(3.8) $\int|u_{t}|^{2}dx\leq c_{6}(t),t\in[0,T^{*}[.$

We deduce from (3.5),

(3.9) $\|Au(t)\|_{2}\leq c_{7}(t),t\in[0,T^{*}[.$

We have by (3.5),

$r_{txx}r_{xx}+r_{txx}r_{xx}=-\rho_{xx}r_{xx}+2\frac{d}{dx}[\mathcal{R}e(u\bar{u}_{x})]r_{xx}$

and so, formally,

(3.10)
$\begin{split}\frac{1}{2}\frac{d}{dt}\int(r_{xx})^{2}dx\leq\frac{1}{2}\int[(\rho_{xx})^{2}+(r_{xx})^{2}]dx\\\
+2\int(|u||u_{xx}|+|u_{x}|^{2})|r_{xx}|dx\\\
\leq\frac{1}{2}\int[(\rho_{xx})^{2}+(r_{xx})^{2}]dx+c_{8}(t)\int|r_{xx}|^{2}dx,\end{split}$

by (3.9) and (1.8). But, by (3.6), we derive

$\rho_{xx}=\frac{1}{2}(l_{x}-r_{x}),$

and so, by (3.7) and (3.10), we deduce

(3.11) $\frac{d}{dt}\int(r_{xx})^{2}dx\leq
c_{9}(t)\int(r_{xx})^{2}dx+c_{10}(t)$

and similarly

(3.12) $\frac{d}{dt}\int(l_{xx})^{2}dx\leq
c_{9}(t)\int(l_{xx})^{2}dx+c_{10}(t).$

We conclude that

(3.13) $\|r_{xx}\|^{2}_{2}+\|l_{xx}\|^{2}_{2}\leq c_{11}(t),t\in[0,T^{*}[,$

with $c_{11}(t)$ being a positive, increasing and continuous function of
$t\geq{0}$. This achieves the proof of Theorem 2 (the operations that we made
formally can be easily justified by a convenient smoothing procedure).

##  4\. Special case of initial data with compact support

We assume the hypothesis of Theorem 2, that is is we consider the semilinear
case ($\lambda=0$) and, without loss of generality, we take
$\alpha=\beta=b=|a|=1$. We also assume that the initial data verifies (1.9)
for a certain $d>0$. Following [6, Section 2], if we take $\phi\in
W^{1,\infty}(\mathbb{R}),$ real valued, and $u$ is the solution of the
Schrödinger equation in (1.1), we easily obtain

$\mathcal{R}e\int\phi^{2}u_{t}\bar{u}dx+\mathcal{I}m\int\phi^{2}u_{xx}\bar{u}dx=0.$

We derive

(4.1) $\|\phi u(t)\|_{2}\leq\|\phi
u_{0}\|_{2}+c_{0}t\|\phi_{x}\|_{\infty},\quad t\geq 0,$

where

$c_{0}=2\sup_{t\geq 0}\|u_{x}(t)\|_{2}.$

Moreover, from the wave equation in (1.2) with $\lambda=0$, we deduce for
$t\geq 0$,

$\phi^{2}\rho_{tt}\rho_{t}-\phi^{2}\phi_{xx}\rho_{t}=-\phi^{2}\rho\rho_{t}+\phi^{2}\rho_{t}|u|^{2},$
(4.2)
$\begin{split}\frac{d}{dt}\int(\phi\rho_{t})^{2}dx+\frac{d}{dt}\int(\phi\rho_{x})^{2}dx+\frac{d}{dt}\int(\phi\rho)^{2}dx\\\
=2\int\phi^{2}\rho_{t}|u|^{2}dx\leq 2\|\phi\rho_{t}\|_{2}\|\phi
u\|_{2}\|u_{0}\|_{2}.\end{split}$

We assume

(4.3) $0\leq\phi\leq 1.$

We have, by the Gagliardo-Nirenberg inequality and (4.1),

(4.4) $\displaystyle\|\phi u\|_{\infty}\leq\|\phi u\|_{2}^{\frac{1}{2}}\|(\phi
u)_{x}\|_{2}^{\frac{1}{2}}$ $\displaystyle\leq(\|\phi
u_{0}\|_{2}+c_{0}t\|\phi_{x}\|_{\infty}))^{\frac{1}{2}}(\|\phi_{x}\|_{\infty}\|u_{0}\|_{2}+\frac{c_{0}}{2})^{\frac{1}{2}}$
$\displaystyle=g_{0}(t).$

Now, with

(4.5) $g_{1}(t)=g_{0}(t)\|u_{0}\|_{2},$

we deduce, from (4.2), (4.4) and with

(4.6)
$f_{1}(t)=\int(\phi\rho_{t})^{2}dx+\int(\phi\rho_{x})^{2}dx+\int(\phi\rho)^{2}dx,$
$f_{1}^{\frac{1}{2}}(t)\leq
f_{1}^{\frac{1}{2}}(0)+2\int_{0}^{t}g_{1}(\tau)f_{1}^{\frac{1}{2}}(\tau)d\tau,$
(4.7) $f_{1}^{\frac{1}{2}}(t)\leq
f_{1}^{\frac{1}{2}}(0)+\int_{0}^{t}g_{1}(\tau)d\tau\leq
f_{1}^{\frac{1}{2}}(0)+t\|u_{0}\|_{2}\,g_{0}(t),t\geq 0.$

Hence, if we define

(4.8) $f(t)=f_{1}(t)+\|\phi u(t)\|_{2}^{2},\quad t\geq 0,$

we derive, by (4.7), (4.8) and (4.1),

(4.9) $\begin{split}f^{\frac{1}{2}}(t)\leq f_{1}^{\frac{1}{2}}(t)+\|\phi
u(t)\|_{2}\leq f_{1}^{\frac{1}{2}}(0)+t\|u_{0}\|_{2}g_{0}(t)+\|\phi
u_{0}\|_{2}+c_{0}t\|\phi_{x}\|_{\infty}\\\ \leq
f^{\frac{1}{2}}(0)+t\|u_{0}\|_{2}(\|\phi
u_{0}\|_{2}+c_{0}t\|\phi_{x}\|_{\infty}))^{\frac{1}{2}}(\|\phi_{x}\|_{\infty}\|u_{0}\|_{2}+\frac{c_{0}}{2})^{\frac{1}{2}}\\\
+\|\phi u_{0}\|_{2}+c_{0}t\|\phi_{x}\|_{\infty}.\end{split}$

Now, we fix $t>0$ and $\varepsilon>0$ and assume that the initial data verify
(1.9). We introduce the set $C=\mathbb{R}\setminus(D+B(0,\delta))$, $\delta$
to be chosen, and the function $\phi\in W^{1,\infty}(\mathbb{R}),$ real
valued, verifying (4.3), $\phi=0$ in $D$, $\phi=1$ in $C$ and
$\|\phi_{x}\|_{\infty}=\frac{1}{\delta}$. We have $f(0)=0$, $\phi u_{0}=0$,
and so, by (4.9), we easily obtain

(4.10) $f(t)\leq
2c_{0}\|u_{0}\|_{2}^{3}\frac{t^{3}}{{\delta}^{2}}+c_{0}^{2}\|u_{0}\|_{2}^{2}\frac{t^{3}}{\delta}+2c_{0}^{2}\frac{t^{2}}{{\delta}^{2}},$

and now we can choose $\delta$ such that (1.10) is satisfied and the Theorem 3
is proved.

##  5\. Existence and partial stability of standing waves

We will consider the system (1.1) in the attractive case $a=-1$ and without
loss of generality we assume that $\alpha=1$. We want to study the existence
and behaviour of standing waves of the system (1.1), that is solutions of the
form (1.11). As we have seen in the introduction, we can rewrite this system
as a scalar equation (1.13). Following the technique introduced in [12] fort
the Gross-Pitaevski equation, we consider the energy functional defined in
$X_{A}$ by (1.14). Recall that $X_{A}\subset L^{q}(\mathbb{R}),2\leq
q<+\infty$, with compact injection, and the norm in $X_{A}$ is equivalent to
the norm

(5.1) $\|u\|_{X_{A}}^{2}=\int|u_{x}|^{2}dx+H^{2}\int x^{2}|u|^{2}dx,H\neq
0,u\in{X_{A}}.$

We now pass to the proof of Theorem 4, which is a variant of Lemma 1.2 in
[12], whose proof we closely follow. Let $\big{\\{}u_{n}\big{\\}}$ be a
minimizing sequence of $\mathcal{E}$ defined by (1.14) in $X_{A}$ (real), that
is

$u_{n}\in
X_{A},\|u_{n}\|^{2}=c^{2},\lim_{n\rightarrow+\infty}\mathcal{E}(u_{n})=\mathcal{I}_{c}\,,$

defined by (1.15). Multiplying the equation satisfied by $\rho(|u|^{2})$ by
$u$ and integrating by parts, we find

$\int[\rho_{x}^{2}+\lambda\rho_{x}^{4}+b\rho^{2}]dx=\int|u|^{2}\rho dx.$

Using Young’s inequality, we get, for a constant $C$ depending on $b$ (we
allow this constant to change from line to line),

$\int|u|^{2}\rho dx\leq\frac{b}{2}\int\rho^{2}dx+C_{b}\int|u|^{4}dx.$

So using the two previous lines, we get that

(5.2) $\int\rho^{2}dx\leq C_{b}\int|u|^{4}dx.$

From Hölder’s inequality, we obtain that, for some constant $\tilde{C}>0$,

(5.3)
$\int\rho|u|^{2}dx\leq(\int\rho^{2}dx)^{\frac{1}{2}}(\int|u|^{4}dx)^{\frac{1}{2}}\leq\tilde{C}\int|u|^{4}dx.$

and, by Gagliardo-Nirenberg inequality,

(5.4) $\|u\|^{4}\leq C\|u_{x}\|_{2}\|u\|_{2}^{3},u\in H^{1}(\mathbb{R}).$

Hence, reasoning as in [12], (1.1) in lemma 1.2, we derive, for each
$\varepsilon>0$ and $x\in X_{A}$, such that $\|u\|_{2}^{2}=c^{2}$,

(5.5)
$\|u\|_{4}^{4}\leq\frac{\varepsilon^{2}}{2}\|u_{x}\|_{2}^{2}+\frac{1}{2\varepsilon^{2}}c^{6},$

and so, for $u\in X_{A}$ such that $\|u\|_{2}^{2}=c^{2}$, we deduce

(5.6)
$\mathcal{E}\geq(\frac{1}{2}-\frac{\varepsilon^{2}}{2}(\frac{1}{4}+\frac{\tilde{C}}{2}))\|u_{x}\|^{2}-\frac{1}{2\varepsilon^{2}}(\frac{1}{4}+\frac{\tilde{C}}{2})c^{6}-\frac{1}{2}H^{2}\int
x^{2}|u|^{2}dx,$

and we can choose $\varepsilon$ such that
$1-\epsilon^{2}(\frac{1}{4}+\frac{\tilde{C}}{2})>0$.
Hence, the minimizing sequence is bounded in $X_{A}$ and there exists a
subsequence $\big{\\{u_{n}}\big{\\}}$ such that $u_{n}\rightharpoonup u$ in
$X_{A}$ (weakly). Recalling that the injection of $X_{A}$ in
$L^{4}(\mathbb{R})$ is compact, we derive

(5.7) $u_{n}\rightarrow u\,\textit{in}\,L^{4}(\mathbb{R}).$

Moreover, by the lower semi-continuity, we deduce

(5.8)
$\int(|u_{x}|^{2}+H^{2}x^{2}|u|^{2})dx\leq\liminf_{n\to\infty}\int(|{u_{n}}_{x}|^{2}+H^{2}x^{2}|u_{n}|^{2})dx.$

On the other hand, we have, setting
$f:=\rho(|u|^{2})-\rho(|u_{n}|^{2}):=\rho-\rho_{n}$,

$-f_{xx}-\lambda(\rho_{x}^{3}-(\rho_{n})_{x}^{3})_{x}+bf=|u|^{2}-|u_{n}|^{2}.$

Noticing using Young’s inequality that

$-\int(\rho-\rho_{n})(\rho_{x}^{3}-(\rho_{n})_{x}^{3})_{x}dx=\int(\rho_{x}^{4}+(\rho_{n})_{x}^{4}-\rho_{x}^{3}(\rho_{n})_{x}-(\rho_{n})_{x}^{3}\rho_{x})dx\geq
0.$

So proceeding as in (5.2), we can show that

(5.9) $\|f\|_{L^{2}}^{2}\leq C\||u|^{2}-|u_{n}|^{2}\|_{L^{2}}^{2}.$

Using this last estimate, we deduce that

$\displaystyle|\int\rho(|u|^{2})|u|^{2}-\int\rho(|u_{n}|^{2})|u_{n}|^{2}|$
$\displaystyle\leq|\int\rho(|u|^{2})(|u|^{2}-|u_{n}|^{2})dx|+|\int(\rho(|u|^{2})-\rho(|u_{n}|^{2}))|u_{n}|^{2}dx|$
$\displaystyle\leq\|\rho(|u|^{2})\|_{L^{2}}\||u_{n}|^{2}-|u|^{2}\|_{L^{4}}^{2}+\|u_{n}\|_{L^{4}}^{2}\|\rho(|u|^{2})-\rho(|u_{n}|^{2})\|_{L^{2}}\rightarrow
0,$

since $|u_{n}|^{2}\rightarrow|u|^{2}$ in $L^{2}(\mathbb{R})$.
Hence, $u$ is a minimizer of (1.14), that is

$u\in X_{A},\|u\|_{2}^{2}=c^{2},\mathcal{E}=\mathcal{I}_{c}.$

We conclude that $\mathcal{E}(u_{n})\rightarrow\mathcal{E}(u)$ and so

(5.10) $\int|{u_{n}}_{x}|^{2}dx+H^{2}\int
x^{2}|u_{n}|^{2}dx\rightarrow\int|u_{x}|^{2}dx+H^{2}\int x^{2}|u|^{2}dx.$

We derive that $u\in X_{A}$ and, denoting by $u^{\star}$ the Schwarz
rearrangement of the real function $u$, (cf. [17] for the definition and
general properties), we know that

$\|u^{\star}_{x}\|^{2}\leq\|u_{x}\|^{2},\|u^{\star}\|^{4}=\|u\|^{4}.$

The Polya-Szego inequality asserts that, for any $f\in W^{1,p}$ with
$p\in[1,\infty]$,

$\|\nabla f\|_{L^{p}}\geq\|\nabla f^{\star}\|_{L^{p}}.$

Moreover, by [12], we have

(5.11) $\int x^{2}|u^{\star}|^{2}dx<\int
x^{2}|u|^{2}dx,\textit{unless}\,\,u=u^{\star}.$

By [10, Theorem 6.3], we know that

$\int G(v(x))dx\leq\int G(v^{\star}(x))dx,$

provided that $G(t)=\int_{0}^{t}g(s)ds$ and
$g:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is such that

$|g(s)|\leq K(s+s^{l}),$

where $K>0$, $l>1$ and $s\geq 0$. We want to apply this result for
$G(s)=\rho(s^{2})s^{2}$. So $g(s)=(\rho(s^{2}))_{s}s^{2}+2s\rho(s^{2})$.
Observe that $(\rho(s^{2}))_{s}:=f$ is the solution to
$-f_{xx}-3\lambda((\rho(s^{2}))_{x}^{2}f_{x})_{x}+bf=2s$. Using the maximum
principle, we can show that $g:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$. On
the other hand, by standard elliptic regularity theory, we have that
$|\rho(s^{2})|,|\rho(s^{2})_{s}|\leq C(s+s^{2})$, for any $s>0$. So, [10,
Theorem 6.3] yields that

$\int\rho(|u|^{2})|u|^{2}dx\leq\int\rho(|u^{\star}|^{2})|u^{\star}|^{2}dx.$

Combining all the previous inequalities, we see that
$\mathcal{E}(u^{\star})<\mathcal{E}(u)$ unless $u=u^{\star}$ a.e. and this
proves that the minimizers of (1.14) are non-negative and radial decreasing.
This completes the proof of Theorem 4.

We now pass to the proof of Theorem 5, which follows the lines of the proof of
Theorem 2.1 in [12]. For sake of completeness we repeat some parts of the
proof to make it easier to follow.
We recall that, cf. [5], to prove the orbital stability it is enough to prove
that $\mathcal{Z}\neq\varnothing$ and that any sequence
$\big{\\{}w_{n}=(u_{n},v_{n})\big{\\}}\subset\tilde{X}_{A}$ such that
$\|w_{n}\|_{2}^{2}\rightarrow c^{2}$ and
$\tilde{E}(w_{n})\rightarrow\tilde{\mathcal{I}}_{c}$, is relatively compact in
$\tilde{\mathcal{E}}$. By the computations in the proof of Theorem 4, we have
that the sequence $\big{\\{}w_{n}\big{\\}}$ is bounded in $\tilde{X}_{A}$ and
so we can assume that there exists a subsequence, still denoted by
$\big{\\{}w_{n}\big{\\}}$ and $w=(u,v)\in\tilde{X}_{A}$ such that
$w_{n}\rightharpoonup w$ weakly in $\tilde{X}_{A}$, that is
$u_{n}\rightharpoonup u,v_{n}\rightharpoonup v$ in $X_{A}$. Hence, there
exists a subsequence, still denoted by $\big{\\{}w_{n})\big{\\}}$, such that
there exists

(5.12) $\lim_{n\to\infty}\int(|{u_{n}}_{x}|^{2}+|{v_{n}}_{x}|^{2})dx.$

Now, we introduce $\varrho_{n}=|w_{n}|=(u_{n}^{2}+v_{n}^{2})^{\frac{1}{2}}$,
which belongs to ${X}_{A}$. Following the proof of [12, Theorem 2.1], we have
${\varrho_{n}}_{x}=\frac{u_{n}{u_{n}}_{x}+v_{n}{v_{n}}_{x}}{(u_{n}^{2}+v_{n}^{2})^{\frac{1}{2}}}$,
if $u_{n}^{2}+v_{n}^{2}>0$, and ${\varrho_{n}}_{x}=0$, otherwise.
We deduce

(5.13)
$\displaystyle\tilde{\mathcal{E}}(w_{n})-\mathcal{E}(\varrho_{n})=\frac{1}{2}\int_{u_{n}^{2}+v_{n}^{2}>0}\Big{(}\frac{u_{n}{u_{n}}_{x}-v_{n}{v_{n}}_{x}}{u_{n}^{2}+v_{n}^{2}}\Big{)}^{2}dx$
$\displaystyle-\frac{1}{4}\int{|(|u_{n}|^{2}+|v_{n}|^{2})|^{2}}d\xi+\frac{1}{4}\int{|\varrho_{n}|^{4}d\xi}$
$\displaystyle=\frac{1}{2}\int_{u_{n}^{2}+v_{n}^{2}>0}\Big{(}\frac{u_{n}{u_{n}}_{x}-v_{n}{v_{n}}_{x}}{u_{n}^{2}+v_{n}^{2}}\Big{)}^{2}dx.$

Hence, we derive as in [12, Theorem 2.1],

(5.14)
${\tilde{\mathcal{I}}}_{c}=\lim_{n\to\infty}\mathcal{\tilde{E}}(w_{n})\geq\limsup_{n\to\infty}\mathcal{E}(\varrho_{n})$

and

(5.15)
$\lim_{n\to\infty}\|\varrho_{n}\|_{2}^{2}=\lim_{n\to\infty}\|w_{n}\|_{2}^{2}=c^{2}.$

Applying Theorem 4 and with $c_{n}=\|\varrho_{n}\|_{2}$, we obtain

(5.16)
$\liminf_{n\to\infty}\mathcal{E}(\varrho_{n})\geq\liminf_{n\to\infty}\mathcal{I}_{c_{n}}\geq\mathcal{I}_{c}\geq\mathcal{\tilde{I}}_{c}.$

Hence, by (5.14) and (5.16), we derive

(5.17)
$\lim_{n\to\infty}\mathcal{E}(\varrho_{n})=\lim_{n\to\infty}\mathcal{\tilde{E}}(w_{n})=\mathcal{I}_{c}=\mathcal{\tilde{I}}_{c}\,,$

and so, by (5.13) and (5.17), we get

(5.18)
$\lim_{n\to\infty}\int|{u_{n}}_{x}|^{2}+|{v_{n}}_{x}|^{2}-|\partial_{x}\big{(}(u_{n}^{2}+v_{n}^{2})^{\frac{1}{2}}\big{)}|^{2}dx=0.$

We can rewrite this last line as

(5.19)
$\lim_{n\to\infty}\int(|{u_{n}}_{x}|^{2}+|{v_{n}}_{x}|)^{2}dx=\lim_{n\to\infty}\int|{\varrho_{n}}_{x}|^{2}dx.$

Now, by (5.15), (5.17) and iii) in Theorem 4, we conclude that there exists
$\varrho\in X_{A}$ such that $\varrho_{n}\rightarrow\varrho$ in $X_{A}$ and
$\|\varrho\|_{2}^{2}=c^{2},\mathcal{E}(\varrho)=\mathcal{I}_{c}.$ Moreover
$\varrho\in H^{2}(\mathbb{R})\subset C^{1}(\mathbb{R})$ is a solution of
(1.13) and $\varrho>0$. We prove that $\varrho=(u^{2}+v^{2})^{\frac{1}{2}}$
just as in the proof of Theo. 2.1 in [12, p.279].
Finally, we prove that $\|{w_{n}}_{x}\|_{2}^{2}\rightarrow\|w_{x}\|_{2}^{2}$:
By applying (5.19) we have
$\lim_{n\to\infty}\|w_{n}\|_{2}^{2}=\lim_{n\to\infty}\|{\varrho_{n}}_{x}\|_{2}^{2}$
and
$\|{\varrho_{n}}_{x}\|_{2}^{2}\rightarrow\|\varrho_{x}\|_{2}^{2}$, since
$\varrho_{n}\rightarrow\varrho$ in $X_{A}$.
Hence,$\|w_{x}\|_{2}^{2}\leq\lim_{n\to\infty}\|{w_{n}}_{x}\|_{2}^{2}=\|\varrho_{x}\|_{2}^{2}.$
But it is easy to see that

$\|w_{x}\|_{2}^{2}=\int(|u_{x}|^{2}+(|v_{x}|^{2})dx\geq\int_{u^{2}+v^{2}>0}\Big{(}\frac{(uu_{x}+vv_{x})^{2}}{u^{2}+v^{2}}\Big{)}^{2}dx=\|\varrho_{x}\|_{2}^{2},$

because $({uu_{x}+vv_{x}}^{2})\leq(u^{2}+v^{2})(|u_{x}|^{2}+(|v_{x}|^{2}).$
Hence,$\|{w_{n}}_{x}\|_{2}^{2}\rightarrow\|w_{x}\|_{2}^{2}$. We also have that
$w_{n}\rightharpoonup w$, weakly in ${\tilde{X}}_{A}$. In particular, by
compactness, $w_{n}\rightarrow w$ in
$(L^{2}(\mathbb{R}))^{2}\cap(L^{4}(\mathbb{R}))^{2}.$
Since
$\mathcal{\tilde{E}}(w_{n})\rightarrow\mathcal{\tilde{I}}_{c}=\mathcal{\tilde{E}}(c)$,
we derive that $\int x^{2}|w_{n}|^{2}dx\rightarrow\int x^{2}|w|^{2}dx$ and so
$\|w_{n}\|_{{\tilde{X}}_{A}}^{2}\rightarrow\|w\|_{{\tilde{X}}_{A}}^{2}$. We
conclude that $w_{n}\rightarrow w$ in ${\tilde{X}}_{A}$, and this achieves the
proof of Theorem 5.

######  Remark 1.

We would like to remark that in the semilinear case, namely when $\lambda=0$,
we can simplify some arguments. Indeed, by applying the Fourier transform to
(1.13), we can solve explicitly this equation and derive

(5.20)
$\rho=\mathcal{F}^{-1}\Big{(}\frac{\mathcal{F}|u|^{2}}{b+4\pi^{2}\xi^{2}}\Big{)}.$

The energy functional is then given by:

(5.21)
$\begin{split}\mathcal{E}(u)=\frac{1}{2}\int|u_{x}|^{2}dx+\frac{1}{2}H^{2}\int
x^{2}|u|^{2}dx\\\
-\frac{1}{4}\int|u|^{4}dx-\frac{1}{4}\int\frac{|\mathcal{F}|u|^{2}|^{2}}{1+4\pi^{2}\xi^{2}}d\xi,\,u\in
X_{A}.\end{split}$

We can use directly (5.20) to obtain an estimate on $\rho$. To prove the
symmetry of minimizers, we can use Proposition 3.2 in [16], noticing that
$(|u|^{2})^{\star}=|u^{\star}|^{2}$, to deduce that

(5.22)
$\int\frac{|\mathcal{F}|u|^{2}|^{2}}{1+4\pi^{2}\xi^{2}}d\xi\leq\int\frac{|\mathcal{F}(|u|^{2})^{\star}|^{2}}{1+4\pi^{2}\xi^{2}}d\xi=\int\frac{|\mathcal{F}(|u^{\star}|^{2})|^{2}}{1+4\pi^{2}\xi^{2}}d\xi.$

##  6\. Bifurcation structure

This section is devoted to the study of the bifurcation structure of solution
to the minimization problem (1.15) namely we prove Proposition 1. We begin by
showing a Pohozaev identity which is also of independent interest.

######  Lemma 1 (Pohozaev identity).

Let $u\in X_{A}$ be a solution to (1.13). Then we have

$2\|u_{x}\|_{2}^{2}-2H^{2}\|xu\|_{2}^{2}-\frac{1}{2}\|u\|_{4}^{4}+\int
u^{2}x\rho_{x}(|u|^{2})dx=0.$

###### Proof.

To simplify notation, we set $\rho:=\rho(|u|^{2})$. Multiplying the equation
(1.13) by $xu$ and integrating by parts, we get

$\|u_{x}\|_{2}^{2}-3H^{2}\|xu\|_{2}^{2}+\dfrac{1}{2}\|u\|_{4}^{4}+\int
u^{2}(\rho+x\rho_{x})dx-\mu\|u\|_{2}^{2}=0.$

On the other hand, multiplying the equation by $u$ and integrating by parts,
we get

(6.1) $\|u_{x}\|_{2}^{2}+H^{2}\|xu\|^{2}_{2}-\int\rho
u^{2}dx-\|u\|_{4}^{4}+\mu\|u\|_{2}^{2}=0.$

So combining the two previous lines, we find

$2\|u_{x}\|_{2}^{2}-2H^{2}\|xu\|_{2}^{2}-\frac{1}{2}\|u\|_{4}^{4}+\int
u^{2}x\rho_{x}dx=0.$

∎

Let us denote by $u_{c}$ a function achieving the minimum for the problem
(1.15) and by $\mu_{c}$ its lagrange multiplier. We also set $\lambda_{0}$ for
the first eigenvalue of the harmonic oscillator $-\partial_{xx}+H^{2}x^{2}$.
We will show that $\mu_{c}$ converges to $-\lambda_{0}$ when the mass $c$ goes
to $0$.

######  Proposition 3.

We have

$\lim_{c\rightarrow 0}\mu_{c}=-\lambda_{0}.$

###### Proof.

In a first time, we are going to show that $-\mu_{c}\leq\lambda_{0}$.
Multiplying the equation satisfied by $u_{c}$ by $u_{c}$ and integrating by
parts, we get

$-c^{2}\mu_{c}=\|(u_{c})_{x}\|_{2}^{2}+H^{2}\|xu_{c}\|_{2}^{2}-\|u_{c}\|_{4}^{4}-\int\rho(|u_{c}|^{2})u_{c}^{2}dx=2E(u_{c})-\dfrac{\|u_{c}\|_{4}^{4}}{2}.$

Thus, we deduce that

$-\mu_{c}\leq\dfrac{2E(u_{c})}{c^{2}}.$

Let $u_{0}$ be the eigenfunction associated to $\lambda_{0}$ namely
$\|u_{0}\|_{X_{A}}^{2}=\lambda_{0}$ and $\|u_{0}\|_{2}=1$. We set
$v_{c}=cu_{0}$. Using that $u_{c}$ is a minimiser of problem (1.15), we have
that $E(u_{c})\leq E(v_{c})$ and

$\dfrac{2E(cu_{0})}{c^{2}}=\|(u_{0})_{x}\|_{2}^{2}+H^{2}\|xu_{0}\|_{2}^{2}-\dfrac{c^{2}}{2}\int
u_{0}^{4}dx-\int\rho(|u_{0}|^{2})u_{0}^{2}dx\leq\lambda_{0}.$

This proves that $-\mu_{c}\leq\lambda_{0}$.

Using Pohozaev’s identity (see Lemma 1) and (6.1), we have

$2\|(u_{c})_{x}\|_{2}^{2}+\int
u_{c}^{2}(\frac{x\rho_{x}(|u_{c}|^{2})}{2}-\rho(|u_{c}|^{2}))dx-\frac{5}{4}\|u_{c}\|_{4}^{4}+\mu_{c}\|u_{c}\|_{2}^{2}=0.$

So, recalling that $-\mu_{c}\leq\lambda_{0}$, we have for a constant $M>0$ not
depending on $c$ that

$\|(u_{c})_{x}\|_{2}^{2}\leq Mc^{2}+M\|u_{c}\|_{4}^{4}-\int
u_{c}^{2}(\frac{x\rho_{x}(|u_{c}|^{2})}{2}-\rho(|u_{c}|^{2}))dx.$

Notice that, integrating by parts and using radial coordinates,

$\displaystyle\int u_{c}^{2}x\rho_{x}(|u_{c}|^{2})dx$ $\displaystyle=\int
u_{c}^{2}(x\rho(|u_{c}|^{2}))_{x}dx-\int u_{c}^{2}\rho(|u_{c}|^{2})dx$
$\displaystyle=-2\int u_{c}(u_{c})_{r}r\rho(|u_{c}|^{2})dr-\int
u_{c}^{2}\rho(|u_{c}|^{2})dx$ $\displaystyle\geq-\int
u_{c}^{2}\rho(|u_{c}|^{2})dx.$

In the last inequality, we used that $u_{r}\leq 0$. So, by (5.3), we obtain,
for some constant $M$ not depending on $c$,

$\|(u_{c})_{x}\|_{2}^{2}\leq Mc^{2}+M\|u_{c}\|_{4}^{4}+M\int
u_{c}^{2}\rho(|u_{c}|^{2})dx\leq Mc^{2}+M\|u_{c}\|_{4}^{4}.$

The Gagliardo-Nirenberg’s inequality (5.4) and Young’s inequality then imply
that

$\|(u_{c})_{x}\|_{2}^{2}\leq Mc^{2}.$

We have, by definition of $\lambda_{0}$,

$\displaystyle-\mu_{c}$
$\displaystyle=\dfrac{\|(u_{c})_{x}\|_{2}^{2}+H^{2}\|xu_{c}\|_{2}^{2}}{c^{2}}-\dfrac{\|u_{c}\|_{4}^{4}+\int\rho(|u_{c}|^{2})u_{c}^{2}dx}{c^{2}}$
$\displaystyle\geq\lambda_{0}-\dfrac{\|u_{c}\|_{4}^{4}+\int\rho(|u_{c}|^{2})u_{c}^{2}dx}{c^{2}}.$

Then, using (5.3) and Gagliardo-Nirenberg’s inequality (5.4), we deduce that,
for some constant $k$ not depending on $c$,

$-\mu_{c}\geq\lambda_{0}-kc^{4}.$

Taking $c\rightarrow 0$, the result follows.

∎

Adapting the proof of Proposition $6.7$ of [14], we can prove the uniqueness
of positive solution to (1.13) namely Proposition 2.

###### Proof of Proposition 2.

We denote by $u(r,\alpha_{1})$ the radial solution to (1.13) such that
$u(0,\alpha_{1})=\alpha_{1}$. Suppose that there exists two number
$0<\alpha_{1}<\tilde{\alpha}_{1}$ such that $u(r,\alpha_{1})$ and
$u(r,\tilde{\alpha}_{1})$ are two positive radial solution decaying to $0$ at
infinity. To simplify notation, we set $u(r)=u(r,\alpha_{1})$ and
$\eta(r)=u(r,\tilde{\alpha}_{1})$. Let $\psi=\eta-u$. In the following, we
denote by $u^{\prime}=\partial_{r}u$. Then $\psi$ satisfies

(6.2)
$\psi^{\prime\prime}-(\lambda+r^{2})\psi+\dfrac{|\eta|^{2}\eta-|u|^{2}u+\rho(|\eta|^{2})\eta-\rho(|u|^{2})u}{\eta-u}\psi=0.$

Multiplying the previous equation by $u$ and multiplying (1.13) by $\psi$,
taking the difference and integrating by parts, we find

$\displaystyle\psi^{\prime}(r)u(r)-u^{\prime}(r)\psi(r)$
$\displaystyle=\int_{0}^{r}(u^{3}+\rho(|u|^{2})u)\psi dx$
$\displaystyle-\int(\eta^{3}+\rho(|\eta|^{2})\eta-u^{3}-\rho(|u|^{2})u)udx$
$\displaystyle=\int(u^{2}+\rho(|u|^{2})-\eta^{2}-\rho(|\eta|^{2}))\eta udx.$

Observe that the left-hand side goes to $0$ as $r\rightarrow\infty$ whereas if
we assume that $\eta(r)>u(r)$ for all $r\geq 0$, the left-hand side converges
to a negative constant. So there exists $\gamma_{1}$ such that
$\eta(\gamma_{1})=u(\gamma_{1})$ (by the maximum principle, we can show that
$\rho(|u|^{2})-\rho(|\eta|^{2})<0$).

Next, we will show that it is in fact the only intersection point between $u$
and $\eta$. Indeed, suppose by contradiction that there exists
$\gamma_{2}>\gamma_{1}$ such that

$0<\eta(r)<u(r)\ \text{for}\ r\in(\gamma_{1},\gamma_{2}),\
u(\gamma_{2})=\eta(\gamma_{2}).$

This implies that

$\psi(r)<0\ \text{for}\ r\in(\gamma_{1},\gamma_{2}),\
\psi^{\prime}(\gamma_{1})<0,\ \psi^{\prime}(\gamma_{2})>0\ \text{and}\
\psi(\gamma_{1})=\psi(\gamma_{2}).$

Let $\xi$ be a solution to

(6.3)
$\begin{cases}&\xi^{\prime\prime}-(\lambda+r^{2})\xi+[p|u|^{p-1}+\partial_{u}(\rho(|u|^{2})u)]\xi=2ru,\
r>0\\\ &\xi(0)=0,\
\xi^{\prime}(0)=(\lambda-\rho(\alpha^{2}))\alpha-\alpha^{p}.\end{cases}$

In fact, we can think of $\xi$ as $u^{\prime}$ noticing that
$(\rho(|u|^{2})u)_{x}=u^{\prime}(\rho(|u|^{2})+u\partial_{u}(\rho(|u|^{2})))$.
Let

$\chi(r)=p|u|^{p-1}+\partial_{u}(\rho(|u|^{2})u)-\dfrac{|\eta|^{p-1}\eta-|u|^{p-1}u+\rho(|\eta|^{2})\eta-\rho(|u|^{2})u}{\eta-u}.$

Observe that the function $u\rightarrow\rho(|u|^{2})u$ is convex. Indeed
$\partial_{uu}(\rho(|u|^{2})u)=u\partial_{uu}\rho(|u|^{2})+2\partial_{u}\rho(|u|^{2})$
where $\partial_{u}\rho(|u|^{2}):=f$ is the solution to

$-f^{\prime\prime}-3\lambda((\rho(|u|^{2})^{\prime})^{2}f_{x})^{\prime}+bf=2u,$

and, $\partial_{uu}\rho(|u|^{2})=g$ is the solution to

$-g^{\prime\prime}-3\lambda((\rho(|u|^{2})^{\prime})^{2}g_{x})^{\prime}+bg=2+6\lambda((\partial_{u}\rho(|u|^{2}))^{2})^{\prime\prime}.$

By the maximum principle, we see that $f\geq 0$ and $g\geq 0$ (since by
comparison principle we can show that $\rho(tx_{1})\leq t\rho(x_{1})$ which
implies, using once more comparison principle that
$\rho(tx_{1}+(1-t)x_{2})\leq t\rho(x_{1})+(1-t)\rho(x_{2})$, for all
$x_{1},x_{2}\geq 0$ and $t\in[0,1]$). Using this and the convexity of $u^{p}$,
we see that $\chi(r)>0$ when $r\in(\gamma_{1},\gamma_{2})$. Taking the
difference of (6.2) multiplied by $\xi$ and (6.3) multiplied by $\psi$ and
integrating by parts on $[\gamma_{1},r]$, we find

(6.4)
$\xi(r)\psi^{\prime}(r)-\xi^{\prime}(r)\psi(r)=\xi(\gamma_{1})\psi^{\prime}(\gamma_{1})+\int_{\gamma_{1}}^{r}(\chi(s)\xi(s)\psi(s)-2su(s)\psi(s))ds.$

Taking $r=\gamma_{2}$ in the previous identity, we get

$\xi(r_{2})\psi^{\prime}(\gamma_{2})=\xi(\gamma_{1})\psi^{\prime}(\gamma_{1})+\int_{\gamma_{1}}^{\gamma_{2}}[\chi(s)\xi(s)\psi(s)-2su(s)\psi(s)]ds.$

This is a contradiction since the left-hand side is strictly negative while
the right-hand side is strictly positive. This establishes that $u$ and $\eta$
intersect exactly once.

Finally, we show that $\eta$ has to change sign. Suppose by contradiction that

$0<\eta(r)<u(r)\ \text{for}\ r\in(\gamma_{1},\infty).$

This implies that $\psi(r)<0$ for $r\in(\gamma_{1},\infty)$,
$\psi^{\prime}(\gamma_{1})<0$ and $\psi(\gamma_{1})=0$. Since $u,u^{\prime}$
and $u^{\prime\prime}$ go to $0$ as $r\rightarrow\infty$ (and the same for
$\eta$), we see that the right-hand side of (6.4) goes to $0$ taking
$r\rightarrow\infty$ whereas the righ-hand side converges to a positive
constant. Therefore, $\eta$ cannot be positive everywhere and consequently $u$
is the unique positive radial solution to our equation. ∎

We are finally in position to prove our bifurcation result, i.e. Proposition
1.

###### Proof of Proposition 1.

Since $\lambda_{0}$ is a simple eigenvalue, we can apply standard bifurcation
results (see for instance [11, Theorem 2.1]) to deduce that $(\lambda_{0},0)$
is indeed a bifurcation point and that the branch is unique provided that we
are sufficiently close to the bifurcation point. Next, the previous Lemma
guarantees that the minimizer of (1.15) $u_{c}$ actually belongs to this
branch at least for $c>0$ small enough. Finally, we use our uniqueness result
Proposition 2 to see that the set $\\{u_{c},c>0\\}$ is convex and therefore
included in the bifurcating branch. ∎

##  7\. Numerical simulations

In this section we perform some numerical simulations to illustrate our
results. We investigate the limit $\mu\to-\lambda_{0}$ mentioned in the
previous section, and analyse the behaviour of standing waves with the
variation of the intensity of the magnetic field $\mathbf{H}$.

###  7.1. Numerical method

Our first goal is to numerically approximate the standing waves (1.11),
according to the system (1.12). Following [12], we use a shooting method.
However, in the present case, the director field angle $\rho=\rho(|u|^{2})$
acts as an additional potential type term, depending on $u$ itself. Due to
this, we perform a Picard iteration and look for a fixed point $u$ of the
operator $\varphi\mapsto\Phi(\varphi)$, where $\Phi(\varphi)$ is the solution
of

(7.1) $\displaystyle u_{xx}-H^{2}x^{2}u+|u|^{2}u+\rho(\varphi)u=\mu u,$

with $\rho(\varphi)$ solving

(7.2) $\displaystyle-\rho_{xx}-\lambda(\rho^{3}_{x})_{x}+b\rho=\varphi$

and with boundary conditions $u(0)=u_{0}>0$, $\rho(0)=\rho_{0}>0$,
$u(\infty)=u^{\prime}(0)=\rho(\infty)=\rho^{\prime}(0)=0$.

According to the results in previous sections, we look for $u\in\mathbb{R}$
even, smooth, vanishing at infinity, strictly positive and decreasing with
$|x|$. For convenience, we shall denote the class of functions verifying these
conditions by $\mathcal{V}$. Although there is no result giving a similar
structure for $\rho(x)$, it is natural to assume that $\rho$ satisfies the
same hypotheses as $u$, at least for small $\lambda$, and so we look for
$u,\rho\in\mathcal{V}$.

We now describe our procedure in more detail. First, equations (7.1),(7.2) can
be recast as a first-order system:

(7.3) $\left\\{\begin{aligned} &u_{x}=w\\\
&w_{x}=H^{2}x^{2}u-|u|^{2}u-\rho(\varphi)u+\mu u\\\ &\rho_{x}=v\\\
&v_{x}=-\lambda(v^{3})_{x}+b\rho-\varphi^{2},\end{aligned}\right.$

with boundary conditions $u(0)=u_{0}>0$, $\rho(0)=\rho_{0}>0$, $v(0)=w(0)=0$,
and $\varphi\in\mathcal{V}$.

At each stage in the Picard iteration, we need, for a given
$\varphi\in\mathcal{V}$, to find $(u,w,\rho,v)$ solving (7.3). As mentioned,
we employ a shooting method, which we now describe. Suppose that we have
computed $\rho(\varphi),v(\varphi)$, and wish to compute $u,w$. The idea is to
adjust the initial value $u(0)=u_{0}$ so that $u(\infty)=0$. Following [12],
$u_{0}$ should verify $u_{0}=\sup\\{\beta>0:u(x;\beta)>0,x>0\\}$ where
$u(x;\beta)$ is the solution of (7.1) with $u(0)=\beta$, $u\in\mathcal{V}$. At
each step of the shooting method, we look for $u_{0}$ in an interval
$[a_{n},b_{n}]$. We set $u_{0,n}=(a_{n}+b_{n})/2$ and solve the first two
equations of (7.3) using an explicit Euler scheme (which is sufficient for our
purposes) with $w(0)=0$. Then, if $u$ attains negative values for some $x$, we
set $a_{n+1}=u_{0,n}$, $b_{n+1}=b_{n}$, thus decreasing $u_{0,n+1}$.
Conversely, if $u(x)$ is increasing at some point (so that it does not belong
in the class $\mathcal{V}$), we set $a_{n+1}=a_{n}$ and $b_{n+1}=u_{0,n}$,
which increases $u_{0,n+1}$.

The procedure to compute $\rho$ and $v$ is similar, except that the behaviour
of $\rho$ exhibits an inverse dependence on the initial value $\rho(0)$; thus
in each iteration of the shooting method the value of $\rho(0)$ is increased
when $\rho$ becomes negative, and decreased when $\rho$ becomes increasing.

Let us mention that on each iteration of the shooting method, the equation for
$v$ in (7.3) contains a nonlinear term when $\lambda\neq 0$. The discretized
equation reads

(7.4)
$\displaystyle\frac{v_{j+1}-v_{j}}{dx}=-\lambda\frac{(v_{j+1})^{3}-(v_{j})^{3}}{dx}+b\rho_{j}-(\varphi_{j})^{2},$

and so we use a Newton method at each step to approximately solve for
$v_{j+1}$.

As a starting point to the Picard iteration, we take
$u^{(0)}(x)\in\mathcal{V}$ as the solution with $\rho=0$, that is, $u^{(0)}$
solves $u_{xx}-H^{2}x^{2}u+|u|^{2}u=\mu u$ with $u^{(0)}\in\mathcal{V}$. With
an initial guess $u^{(0)}(x)\in\mathcal{V}$ for the Picard iteration in hand,
we compute $u^{(1)}(x),w^{(1)}(x)$, and so on, using the shooting method,
according to

(7.5) $\left\\{\begin{aligned} &u^{(n)}_{x}=w^{(n)}\\\
&w^{(n)}_{x}=H^{2}x^{2}u^{(n)}-(u^{(n)})^{3}-\rho(u^{(n-1)})u^{(n)}+\mu
u^{(n)},\end{aligned}\right.$

where $\rho(u^{(n-1)})$ solves

$\left\\{\begin{aligned} &\rho_{x}=v,\\\
&v_{x}=-\lambda(v^{3})_{x}+b\rho-(u^{(n-1)})^{2}\end{aligned}\right.$

(also using the shooting method), with boundary conditions $u(0)=u_{0}>0$,
$\rho(0)=\rho_{0}>0$, $v(0)=w(0)=0$.

###  7.2. Numerical results

Figure 1. Numerical approximation of the standing wave $u(x)$ and the director
field angle $\rho(x)$, solutions to (1.12), computed using a shooting method
and Picard iteration (Picard iterations in dashed lines). Parameters are
$H=1,\mu=-0.8,\lambda=0.1,b=1$.

In Fig. 1, we plot the standing wave $u(x)$ and the director field angle
$\rho(x)$ calculated according to the procedure described previously. The
dashed lines correspond to the iterations of the Picard method. For this
simulation, we have used a spatial step $dx=0.002$ (corresponding to $3000$
spatial points) and 15 Picard iterations.

Next, we illustrate the result of Proposition 3. First, note that it is easy
to see that $u^{*}(x)=e^{-\frac{H}{2}x^{2}}$ is the first eigenfunction of the
harmonic oscillator $-\partial_{xx}+H^{2}x^{2}$, with eigenvalue
$\lambda_{0}=H$. Note that in our notations, the parameter $-\mu$ plays the
role of $\lambda_{0}$. In parallel to [12], and in accordance with Proposition
3, we verify numerically that the $L^{2}$ norm of $u_{\mu}$ goes to zero as
$\mu\to-\lambda_{0}^{+}$. Taking $H=\lambda_{0}=2,$ we show in Fig. 2 the
numerical solutions of (1.12) for various values of $\mu\to-\lambda_{0}^{+}$.
We can see that the solutions appear to converge to zero, although the
convergence is very slow. In Fig. 3, we show how the $L^{2}$ norm of
$u=u_{\mu}$ varies as the Lagrange multiplier $\mu$ tends to the value
$-\lambda_{0}$. Our numerical tests indicate that, although slow, the
convergence to zero of the $L^{2}$ norm of $u_{\mu}$ is verified, in
accordance with Proposition 3.

Figure 2. Numerical approximation of the standing wave $u(x)$ and the director
field angle $\rho(x)$, solutions to (1.12), with $\mu\to-\lambda_{0}$.
Parameters are $H=\lambda_{0}=2,\lambda=0.1,b=1$. Figure 3. The norm
$\|u\|_{2}^{2}$ as a function on the Lagrange multiplier $\mu$ as
$\mu\to-\lambda_{0}=-H=-2$, in $\log$-$\log$ scale. The values of $\mu$ are
the same as in Fig. 2, but $\mu$ is ranging from $-1.9$ to $-1.9999153$,
taking 60 values (left). On the right is a zoom on the last 15 values of
$\mu$.

Next, we investigate numerically the behaviour of the standing wave when the
intensity of the magnetic field, $H$, is varied. It turns out that for each
set of parameters that we analyzed, there is a maximum (relatively small)
value of $H$ such that our numerical method diverges for larger values of $H$.
This may be related to the observation that the behaviour of $u$ (and $\rho$)
with respect to $u(0)$ is very sensitive to perturbations: any arbitrarily
small perturbation of the $u(0)$ found by the shooting method produces a
solution which (numerically at least) quickly blows up exponentially. The
desired solution appears to be unstable in this sense, and this effect appears
more markedly for larger values of $H$. Still, in Fig. (4) we show the
behaviour of the solution for $H$ between 0 and 2, which lets us nevertheless
see the general trend. In particular, it is clear that the director field
angle becomes more concentrated at the origin for larger values of $H$.

Figure 4. Numerical approximation of the standing wave $u(x)$ and the director
field angle $\rho(x)$, solutions to (1.12), with varying magnetic field
intensity $H$. Parameters are $\mu=0.2$, $\lambda=0.1,$ $b=2$.

### Acknowledgements

P. Amorim was partially supported by Conselho Nacional de Pesquisa (CNPq)
grant No. 308101/2019-7. J.-B. Casteras and J.P. Dias were supported by the
Fundação para a Ciência e a Tecnologia (FCT) through the grant
UIDB/04561/2020. J.P.Dias is indebted to R.Carles, H.Frid, E.Ducla Soares,
A.Farinha Martins and L.Sanchez for important suggestions and comments.

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