In this paper, we prove the low-regularity global well-posedness of the adiabatic limit of the Quantum Zakharov system and consider its semi-classical limit, i.e., the convergence of the model equation as the quantum parameter tends to zero. We also show ill-posedness in negative Sobolev spaces and discuss the existence of ground-state soliton solutions in high spatial dimensions.

§1. Introduction. In this paper, we study well-posedness as well as ill-posedness of the adiabatic limit of the quantum Zakharov system and its nonlinear Schrödinger limit as the quantum parameter tends to zero. We are motivated by the classical Zakharov system <cit.>:

\begin{align}
\begin{cases}
i\partial_t E + \Delta E=nE,\: (x,t) \in \mathbb{R}^d \times \mathbb{R}\\
\frac{1}{\lambda^2}\partial_{tt}n-\Delta n=\Delta(|E|^2),\\
E(0)=E_0,\: n(0)=n_0,\: \partial_t n(0)=n_1,
\end{cases}
\end{align}

which describes the propagation of Langmuir waves in an ionised plasma. Here a complex-valued $E(x,t)$ describes a slowly-varying envelope of a rapidly oscillating electric field, and a real-valued $n(x,t)$ describes the deviation of the ion density from its mean. Classical solutions satisfy the conservation of mass and energy as follows:

\begin{align*}
\text{MASS}&=\lVert E(t)\rVert_{L^2}^2 = \text{constant}\\
\text{ENERGY}&=\lVert \nabla E(t)\rVert_{L^2}^2 + \frac{\lVert n(t)\rVert_{L^2}^2}{2}+\frac{\lVert \partial_t n\rVert_{\dot{H}^{-1}}}{2\lambda^2}+\int n(t)|E(t)|^2 = \text{constant}.
\end{align*}

The ion acoustic speed is proportional to $\lambda>0$, and the adiabatic limit $\lambda\rightarrow\infty$ was studied by Schochet-Weinstein <cit.>. In that regime, the second time-derivative term becomes negligible, at least formally, and under the assumption that $n+|E|^2$ vanishes at the infinity, the limiting function $E$ satisfies the cubic focusing nonlinear Schrödinger equation (NLS):

\begin{align}
i\partial_t E + \Delta E=-|E|^2E.
\end{align}

Physically the adiabatic limit corresponds to the assumption that the fluctuation in electric fields instantaneously affects that of plasma. Under some hypotheses, they proved <cit.> that $(1)$ is locally well-posed in some time interval $[0,T]$ (with $T$ independent of $\lambda$), and also identified rigorously the nonlinear Schrödinger limit, both in the strong and weak sense, as $\lambda\rightarrow\infty$. Later Ozawa-Tsutsumi found the optimal convergence rate for this nonlinear Schrödinger limit <cit.>.

On the other hand, the following more detailed model accounts for the quantum effects on the nonlinear interaction between Langmuir waves and ion-acoustic waves:

\begin{align}
\begin{cases}
i\partial_t E + \Delta E-\epsilon^2\Delta^2E=nE,\: (x,t)\in \mathbb{R}^d\times \mathbb{R}\\
\frac{1}{\lambda^2}\partial_{tt}n-\Delta n+\epsilon^2\Delta^2n=\Delta(|E|^2),\\
E(0)=E_0,\: n(0)=n_0,\: \partial_t n(0)=n_1,
\end{cases}
\end{align}

where $\epsilon = \frac{\hbar w_i}{k_B T_e}>0$; $\hbar$ is the Planck's constant divided by $2\pi$, $w_i$, ion plasma frequency, $k_B$, Boltzmann's constant, and $T_e$, the electrons' temperature. As before, the classical solutions of $(3)$ satisfy the mass and energy conservation as follows:

\begin{align*}
\text{MASS}&=\lVert E(t)\rVert_{L^2}^2 = \text{constant}\\
\text{ENERGY}&=\epsilon^2\lVert \Delta E(t)\rVert_{L^2}^2+\lVert \nabla E(t)\rVert_{L^2}^2 + \frac{\epsilon^2}{2}\lVert \nabla n(t)\rVert_{L^2}^2+\frac{\lVert n(t)\rVert_{L^2}^2}{2}+\frac{\lVert \partial_t n\rVert_{\dot{H}^{-1}}}{2\lambda^2}+\int n(t)|E(t)|^2 = \text{constant};
\end{align*}

for a quick variational treatment on conservation laws, see <cit.>. The consideration of $\epsilon$ would for example be experimentally relevant in the case of dense and cold plasmas, which occur in astrophysical scenarios; see <cit.> for more physical background.

To come to the focus of this note, we formally take $\lambda\rightarrow\infty$ in $(3)$, assuming $n+(I-\epsilon^2\Delta)^{-1}|E|^2\xrightarrow[|x|\rightarrow\infty]{}0$ and re-labelling the quantity of interest from $E$ to $u$, thereby obtaining the following integro-differential initial-value problem:

\begin{align}
\begin{cases}
i\partial_t u + \Delta u -\epsilon^2\Delta^2u=-(I-\epsilon^2\Delta)^{-1}(|u|^2)u,\: (x,t)\in \mathbb{R}^d\times \mathbb{R}\\
u(0)=u_0\in H^s(\mathbb{R}^d),
\end{cases}.
\end{align}

This can be thought of as a modified NLS (mNLS). More precisely, the extra bi-Laplacian term is expected to play a significant role for a long-time behaviour of solutions. On the other hand, the linear operator $J_\epsilon\coloneqq(I-\epsilon^2\Delta)^{-1}$ is a non-local Fourier multiplier that converges (in tempered distributions) to the delta function as $\epsilon\rightarrow 0$. By now it is folklore in the community of nonlinear equations that a PDE posed within a subcritical regime is solvable in the sense of well-posedness. Indeed the equation above is locally well-posed in $L^2(\mathbb{R})$ and moreover globally well-posed in $H^s(\mathbb{R})$ for $s\geq 2$, thanks to mass/energy conservation, Strichartz estimates and the Gagliardo-Nirenberg inequality (see proposition $2.5$ of <cit.> in particular). Nevertheless, we have not been able to find a global result regarding low regularity well-posedness in the literature- this is what we aim to achieve here. Before we more precisely state our main results, we briefly review the well-posedness theory of $(1)$ and $(3)$.

The study of well-posedness theory of $(1)$ by now is a mature subject - we give a list of (some) references in  chronological order for completeness: <cit.>. On the other hand, a lot of rigorous study of $(3)$, has taken place since $2010$. Guo-Zhang-Guo proved that $(3)$ is globally well-posed for data $(E_0,n_0,n_1)\in H^k(\mathbb{R}^d)\times H^{k-1}(\mathbb{R}^d)\times H^{k-3}(\mathbb{R}^d)$ where $k\geq 2$ and $d=1,2,3$, and that the classical limit holds as the quantum parameter tends to zero <cit.>. Jiang-Lin-Shao further obtained well-posedness results at lower regularities by carefully estimating the non-linear interactions of waves of different frequencies in the presence of the bi-Laplacian operator <cit.>. Fang-Lin-Segata proved that solutions of $(3)$ converge to those of $(1)$ at an optimal rate as the wave speed approaches infinity <cit.>. Chen-Fang-Wang proved global well-posedness of $(3)$ in $1D$ when the electric field component is only assumed to be square-integrable; this at least formally generalises the classical result of Colliander-Holmer-Tzirakis with the quantum parameter tending to zero <cit.>. Fang-Shih-Wang further contributed in obtaining low-regularity well-posedness results, which again formally recovered another classical result by Gibibre-Tsutsumi-Velo <cit.>. Finally Fang-Kuo-Shih-Wang generalised the result of <cit.> by obtaining a refined version of the semi-classical limit, with a convergence rate <cit.>.

A difficulty in obtaining global existence results for mNLS occurs in $H^s(\mathbb{R})$ when $s\in (0,\frac{1}{2}]$; for $s=0$, the global result is immediate due to mass conservation. For $s>\frac{1}{2}$, one can exploit the fact that $H^s(\mathbb{R})$ defines an algebra to obtain an exponential bound on the growth of Sobolev norm, which prevents finite time blow-up. To overcome this obstacle, we adopt the method of Fourier restricted norms, first initiated by Bourgain <cit.>. This method was also used to obtain well-posedness of KdV on $\mathbb{R}$ for $s>-\frac{3}{4}$ by Kenig-Ponce-Vega <cit.>, and a similar bilinear estimate was obtained in a negative Sobolev space for $1D$ NLS with quadratic nonlinearities <cit.>. The goal is to obtain a sufficient degree of smoothing, measured in Fourier restriction norm, for non-local nonlinearities. Though the argument follows closely those of the references above, a careful analysis on the quartic dispersion relation introduced by the bi-Laplacian is needed. To this end, we state our main result:

Theorem I. The mNLS is globally well-posed (in subcritical sense) in $H^s(\mathbb{R})$ for every $s\geq 0$. Furthermore, the data-to-solution map is Lipschitz in any closed ball in $H^s(\mathbb{R})$.

Then we consider the semi-classical limit of $(4)$. The smoothing estimate obtained above is to no avail since the implicit constant is not uniform with respect to $\epsilon>0$ as $\epsilon\rightarrow 0$; for the same reason, Strichartz estimates are not a useful tool in this context. However a direct estimate on the Duhamel formula yields the desired result if we are in the Sobolev algebra regime, i.e., $s>\frac{1}{2}$. We are able to obtain the desired convergence result locally in time, but we are only able to obtain a partial convergence result when the solution flow is not restricted to a finite-time interval. This long-time asymptotics case is much more difficult and interesting since it is known that the solutions to $(2)$ and $(4)$ do not scatter. For the topic of soliton existence for NLS, see the appendix of <cit.>, and for the existence and stability of solitions for mNLS in $d=1,2,3$, see <cit.>. By adopting the variational method used in <cit.>, we extend the result of Fang-Segata-Wu by showing that solitons exist in higher spatial dimensions as well. As for the first part of the next statement, it is of interest to ask whether the Sobolev algebra assumption can be removed.

Theorem II. Let $s>\frac{1}{2}$ and $\left\{u_0^{(\epsilon)}\right\}_{\epsilon>0}\subseteq B_{H^s(\mathbb{R})}(0,R)$ for some $R>0$ where $u_0^{(\epsilon)}\xrightarrow[\epsilon\to 0]{H^s} u_0$, and let $u^{(\epsilon)}$ and $u$ be the global solutions corresponding to $u_0^{(\epsilon)}$ and $u_0$, respectively. Then for every $T\in (0,\infty)$, $u^{(\epsilon)}\xrightarrow[\epsilon\to 0]{} u$ in $C([0,T],H^s(\mathbb{R}))$. Furthermore let $U(t) = e^{it\Delta}$ and $U_\epsilon(t) = e^{it(\Delta-\epsilon^2\Delta^2)}$ be the solution maps for the linearised equations of $(2)$ and $(4)$, respectively. Then for every $s\in \mathbb{R}$, there exists some $u_0\in H^s(\mathbb{R})$ such that $U_\epsilon(t)u_0\nrightarrow U(t)u_0$ in $C([0,\infty),H^s(\mathbb{R}))$ as $\epsilon\to 0$.

We also study the ill-posedness of $(4)$ in negative Sobolev spaces. Due to Galilean symmetry, the solution map for $(2)$ fails to be uniformly continuous for $s\in (-\infty,0)$ and even worse, the map exhibits norm inflation[We say that norm inflation occurs in $H^s(\mathbb{R}^d)$ if for every $\delta>0$, there exists an initial datum $f \in B_{H^s}(0,\delta)$ and $0<T<\delta$ such that the solution corresponding to $f$ exists on $[0,T]$ and $\lVert u[f](T)\rVert_{H^s}>\delta^{-1}$.] for $s\leq -\frac{1}{2}$; see <cit.>. However it is unknown whether there is Galilean invariance in the presence of bi-Laplacian operator, and therefore a direct proof of the failure of uniform well-posedness is unavailable. Instead we show that $u^{(\epsilon)}-u$ stays small for a short time. More precisely, we prove the following:

Theorem III. Let $s<0$ and $R,T>0$. Then there exists $\epsilon_0(s,R,T)>0$ such that  for all $\epsilon \in (0,\epsilon_0)$, if the solution $u^{(\epsilon)}$ has a well-defined data-to-solution map from $B_{H^s(\mathbb{R})}(0,R)$ to $C([0,T],H^s(\mathbb{R}))$, then such map fails to be uniformly continuous (in their usual Banach space topology).

We briefly outline how this paper is organised. In section 2, we introduce various norms and useful notations. In section 3, we prove theorem I by obtaining a smoothing estimate and a polynomial bound on the growth of Sobolev norm of the solution. In section 4, we consider the semi-classical limit. Using an estimate obtained in this section, we obtain an ill-posedness result in negative Sobolev spaces. In section 5, we apply the variational method to obtain soliton solutions in higher dimensions.

§2: Set-up and Notation. The quartic dispersion relation plays an important role in our analysis. We define $d_\epsilon(\xi)\coloneqq\xi^2+\epsilon^2\xi^4$ for $\epsilon\geq 0$. To run a fixed point argument, we work on the Fourier restriction space $X^{s,b}_{\pm(\epsilon)}$ (also known as dispersive Sobolev space) where $s,b\in\mathbb{R},\epsilon \geq 0$ and $u\in X^{s,b}_{\pm (\epsilon)}$ if

\begin{align*}
\lVert u \rVert_{X^{s,b}_{\pm (\epsilon)}}\coloneqq \lVert \langle \xi \rangle^s \langle \tau \pm d_\epsilon(\xi)\rangle^b \hat{u}(\xi,\tau)\rVert_{L^2_{\xi \tau}}<\infty,
\end{align*}

where we adopt the PDE convention of Fourier transform as follows:

\begin{align*}
\mathcal{F}[f](\xi)=\hat{f}(\xi) = \int f(x)e^{-i x\cdot \xi} dx;\: \mathcal{F}^{-1}[F](x)=\check{F}(x)=\frac{1}{(2\pi)^d}\int \hat{F}(\xi)e^{i\xi\cdot x}d\xi.
\end{align*}

The spaces $\mathscr{S}(\mathbb{R}^d)$ and $C^\infty_c(\mathbb{R}^d)$ denote the Schwartz class of rapidly decaying smooth functions and the set of smooth functions with compact support, respectively. We fix $\eta\in C^\infty_c(\mathbb{R})$ to be a function that is identical to one on $[-1,1]$ with a compact support on $[-2,2]$. The inhomogeneous and homogeneous differential operators are:

\begin{align*}
\langle \xi \rangle =
(1+|\xi|^2)^{1/2};\:\langle \nabla \rangle^s f = \mathcal{F}^{-1}(\langle \xi \rangle^s \Hat{f});\:| \nabla |^s f = \mathcal{F}^{-1}(| \xi |^s \Hat{f}).
\end{align*}

where $|\xi| = \Big(\sum\limits_{i=1}^d |\xi_i|^2\Big)^{1/2}$. To do a local-in-time argument, where $t \in [-\delta,\delta]$ for some $\delta \in (0,1]$, we need a restricted version of $X^{s,b}_{\pm (\epsilon)}$ as well. We denote such space by $X^{s,b}_{\pm (\epsilon),\delta}$ with its restricted norm being:

\begin{align*}
\lVert u \rVert_{X^{s,b}_{\pm (\epsilon),\delta}} = \inf \limits_{u=\Tilde{u}, t \in [-\delta,\delta]} \lVert \Tilde{u}\rVert_{X^{s,b}_{\pm (\epsilon)}}.
\end{align*}

The space of continuous spacetime functions $u:I\subseteq \mathbb{R} \rightarrow H^s(\mathbb{R}^d)$ is denoted by $C^0_{t,loc}H^s_x(I\times \mathbb{R}^d)$ where $u\in C^0_{t}H^s_x(I\times \mathbb{R}^d)$ if $\lVert u \rVert_{C^0_tH^s_x} \coloneqq \sup\limits_{t\in I} \lVert u(t)\rVert_{H^s(\mathbb{R}^d)}<\infty$. If $I=[0,T]$, we denote $\lVert u \rVert_{C^0_TH^s_x}\coloneqq \sup\limits_{t\in [0,T]}\lVert u(t)\rVert_{H^s}$ under which the space defines a complete metric space. We say $A \lesssim B$ if $A$ is bounded above by $B$ multiplied by a universal constant, i.e., if there exists $C>0$ such that $A \leq CB$. Similarly, say $A \sim B$ if $A \lesssim B$ and $B \lesssim A$. For $p\in[1,\infty]$, define $p^\prime = \frac{p}{p-1}$ to be the corresponding Hölder conjugate where $\frac{1}{\infty}=0$. We define $s+ = s+\epsilon$ for some universal $\epsilon <<1$; $s-$ is defined similarly.

§3: Low Regularity Well-posedness. The Duhamel integral formula corresponding to $(4)$ is as follows:

\begin{align}
u(t) = U_\epsilon(t)u_0 + i \int_0^t U_\epsilon(t-\tau)[J_\epsilon(|u|^2)u(\tau)]d\tau.
\end{align}

We say that $u$ is a $H^s_x$-strong solution of $(4)$ on $[-T,T]$ for $T>0$ if $u\in C^0_tH^s_x([-T,T]\times \mathbb{R})$ satisfies the Duhamel formula distributionally for all $t\in [-T,T]$. We say that $(4)$ is locally well-posed in $H^s(\mathbb{R})$ if for every $g\in H^s(\mathbb{R})$, there exists $\delta>0$, a subset $X\subseteq C^0_tH^s_x([-\delta,\delta]\times \mathbb{R})$ and an open ball $B\subseteq H^s(\mathbb{R})$ containing $g$ such that for every $u_0 \in B$, there exists a unique strong solution $u \in X$, and the data-to-solution map $u_0\mapsto u$ is continuous with respect to the topology of $H^s(\mathbb{R})$ and $C^0_tH^s_x([-\delta,\delta]\times \mathbb{R})$, respectively. If this $\delta>0$ can be chosen arbitrarily large, then we say that the well-posedness is global. If $\delta>0$ depends only on $\lVert g \rVert_{H^s}$ instead of $g$ itself, then the local well-posedness is in the subcritical sense. In practice, we show that the operator $u\mapsto U_\epsilon(t)u_0 + i \int_0^t U_\epsilon(t-\tau)[J_\epsilon(|u|^2)u(\tau)]d\tau$ defines a contraction on some Fourier restriction space, provided that we obtain appropriate nonlinear estimates on $N(u)\coloneqq J_\epsilon(|u|^2)u$. We state useful facts that we shall need in subsequent proofs.

Lemma 2.1. Let $\epsilon\geq 0, \delta \in (0,1]$ and $s,b\in \mathbb{R}$. Then,

* Linear estimate: $\lVert U_\epsilon(t)g\rVert_{X^{s,b}_{(\epsilon),\delta}}\lesssim_{s,b,\epsilon}\lVert g \rVert_{H^s}$.

* $X^{s,b-1}_{(\epsilon),\delta}$ is a companion space of $X^{s,b}_{(\epsilon),\delta}$ for $b\in (\frac{1}{2},1]$: $\left|\left|\int_0^t U_\epsilon(t-\tau)N(u)(\tau)d\tau\right|\right|_{X^{s,b}_{(\epsilon),\delta}}\lesssim_{s,b,\epsilon} \lVert N(u) \rVert_{X^{s,b-1}_{(\epsilon),\delta}}$.

* For $b>\frac{1}{2}$, we have the following continuous embedding $X^{s,b}_{(\epsilon),\delta}\hookrightarrow C^0_tH^s_x([-\delta,\delta]\times\mathbb{R})$.

* Let $-\frac{1}{2}<b^\prime\leq b<\frac{1}{2}$ and $s\in \mathbb{R}$. Then $\lVert u\rVert_{X^{s,b^\prime}_{(\epsilon),\delta}}\lesssim_{s,b^\prime,b,\epsilon}\delta^{b-b^\prime}\lVert u \rVert_{X^{s,b}_{(\epsilon),\delta}}$.

* Duality: $(X^{s,b}_{(\epsilon)})^* = X^{-s,-b}_{-(\epsilon)}$.

proof of Lemma 2.1. See section $2.6$ of <cit.>.

We also state elementary calculus facts that are used repeatedly:

Lemma 2.2.

* If $\beta \geq \gamma \geq 0$ and $\beta + \gamma>1$, then $\int \dfrac{d\tau}{\langle \tau-a_1 \rangle^\beta \langle \tau-a_2 \rangle^\gamma} \lesssim \langle a_1-a_2 \rangle^{-\gamma} \phi_\beta (a_1-a_2)$ where

$\phi_\beta (a) \sim \begin{cases}
1,& \beta > 1\\
\log (1+\langle a \rangle),              & \beta=1\\
\langle a \rangle^{1-\beta}, & \beta<1.
\end{cases}$

* Let $A>0$ and $0<a<1$. Then, $\int_A^\infty \frac{dz}{z(z-A)^a}\lesssim_{a} A^{-a}$.

proof of Lemma 2.2. For the first statement, see <cit.>. As for the second statement, let $t=z-A$. Then

\begin{align*}
\int_A^\infty \frac{dz}{z(z-A)^a}=\int_0^\infty \frac{dt}{(t+A)t^a} = \int_0^A \frac{dt}{(t+A)t^a}+\int_A^\infty \frac{dt}{(t+A)t^a}\leq (\frac{1}{1-a}+\frac{1}{a})A^{-a}.
\end{align*}

Lemma 2.3. Let $s\in \mathbb{R}, \epsilon>0$ and $X=H^s(\mathbb{R}^d)\:\text{or}\:L^p(\mathbb{R}^d)$ for $p\in [1,\infty)$. Then $\lVert J_\epsilon f \rVert_{X}\leq \lVert f \rVert_{X}$ for all $f\in X$. Moreover $J_\epsilon: H^s(\mathbb{R}^d)\rightarrow H^{s+2}(\mathbb{R}^d)$ is bounded with the best constant $\leq \epsilon^{-2}$. Lastly $J_\epsilon\xrightarrow[\epsilon\rightarrow 0]{}\delta$ in distribution where $\delta$ is the Dirac delta function.

proof of Lemma 2.3. See section $1.5$ of <cit.>. The convergence result follows from the dominated convergence theorem.

The majority of this section is devoted to obtaining the following estimate:

Proposition 2.4. Let $s \in [0,\infty), \gamma\in [\frac{1}{3},\frac{1}{2}),\delta \in (0,1]$ and $a\in [0,\frac{4}{3})$. Then for every $\epsilon>0$ and $b\in (\frac{1}{2},1-\gamma)$, we have $\lVert (J_\epsilon(u\overline{v})w \rVert_{X^{s+a,-\gamma}_{(\epsilon),\delta}}\lesssim_{s,a,\gamma,b,\epsilon} \lVert u \rVert_{X^{s,b}_{(\epsilon),\delta}}\lVert v \rVert_{X^{s,b}_{(\epsilon),\delta}}\lVert w \rVert_{X^{s,b}_{(\epsilon),\delta}}$.

By letting $u=v=w$, the proposition above immediately yields:

Corollary 2.5. Let $s,\gamma,\delta,a,\epsilon$ and $b$ be as above. Then $\lVert N(u)\rVert_{X^{s+a,-\gamma}_{(\epsilon),\delta}}\lesssim\lVert u\rVert_{X^{s,b}_{(\epsilon),\delta}}^3$.

A straightforward application of fixed point argument yields:

Corollary 2.6. The mNLS is locally well-posed (in subcritical sense) in $H^s(\mathbb{R})$ for every $s\geq 0$. Furthermore the data-to-solution map is Lipschitz in any closed ball in $H^s(\mathbb{R})$.

Remark. Local well-posedness can be established by an application of Strichartz estimates, which is by now well understood. Recall that a biharmonic admissible (B-admissible) pair is a pair $(q,r)$ such that

\begin{align*}
q,r \in [2,\infty],\: (q,r,d)\neq (2,\infty,4)\: \text{and}\: \frac{4}{q}+\frac{d}{r}=\frac{d}{2}.
\end{align*}

If $(q,r)$ and $(\Tilde{q},\Tilde{r})$ are B-admissible pairs and $u\in C([0,T],H^{-4}(\mathbb{R}^d))$ is a solution to

\begin{align*}
\begin{cases}
i\partial_t u + \Delta u -\epsilon^2\Delta^2u=F,\: (x,t)\in \mathbb{R}^d\times \mathbb{R}\\
u(0)=u_0\in L^2(\mathbb{R}^d),
\end{cases}
\end{align*}

then there exists a constant $C=C(q,r,\Tilde{q},\Tilde{r},d)>0$ such that

\begin{align*}
\lVert u \rVert_{L^q([0,T],L^r(\mathbb{R}^d))}\leq C (\epsilon^{-\frac{2}{q}}\lVert u_0\rVert_{L^2}+\epsilon^{-2(\frac{1}{q}+\frac{1}{\Tilde{q}})}\lVert F \rVert_{L^{\Tilde{q}^{\prime}}([0,T],L^{\Tilde{r}^{\prime}}(\mathbb{R}^d))}).
\end{align*}

Unfortunately the Strichartz estimate does not directly yield the global existence result for infinite energy data. Moreover the estimate blows up as $\epsilon\rightarrow 0$, and therefore this is an insufficient tool to study the semi-classical limit. For references on B-admissible Strichartz estimate, see <cit.>.

proof of Corollary 2.6. Let $s,\gamma,\delta,a,\epsilon$ and $b$ be as above. For $g\in H^s(\mathbb{R})$, define $X=\left\{u\in X^{s,b}_{(\epsilon),\delta}: \lVert u\rVert_{X^{s,b}_{(\epsilon),\delta}}\leq 2C \lVert g \rVert_{H^s}\right\}$ where $C$ is a fixed implicit constant that satisfies $\lVert U_\epsilon(t)\Tilde{g}\rVert_{X^{s,b}_{(\epsilon),\delta}}\lesssim \lVert \Tilde{g} \rVert_{H^s}$ for all $\Tilde{g}\in H^s(\mathbb{R})$. Define $\Gamma u=U_\epsilon(t)g+i\int_0^t U_\epsilon(t-\tau)N(u)(\tau)d\tau$. Then,

\begin{align*}
\lVert \Gamma u \rVert_{X^{s,b}_{(\epsilon),\delta}}&\lesssim \lVert g \rVert_{H^s}+\lVert N(u)\rVert_{X^{s,b-1}_{(\epsilon),\delta}}\lesssim \lVert g \rVert_{H^s}+\delta^{1-(b+\gamma)}\lVert N(u)\rVert_{X^{s,-\gamma}_{(\epsilon),\delta}}\lesssim \lVert g \rVert_{H^s}+\delta^{1-(b+\gamma)}\lVert u \rVert_{X^{s,b}_{(\epsilon),\delta}}^3\lesssim \lVert g \rVert_{H^s}+\delta^{1-(b+\gamma)}\lVert g \rVert_{H^s}^3\\
\Rightarrow \lVert \Gamma u \rVert_{X^{s,b}_{(\epsilon),\delta}}&\leq C\lVert g \rVert_{H^s}+C_1\delta^{1-(b+\gamma)}\lVert g\rVert_{H^s}^3.
\end{align*}

Let $\delta = \delta(\lVert g \rVert_{H^s})>0$ such that $\delta < \frac{C}{C_1}\lVert g \rVert_{H^s}^{-\frac{2}{1-(b+\gamma)}}$. Then, $\Gamma:X\rightarrow X$. Now we show that $\Gamma$ is a contraction on $X$, and hence there exists a unique fixed point, the desired strong solution. Note that uniqueness in $X$ implies that in $X^{s,b}_{(\epsilon),\delta}$ by shrinking $\delta$ if necessary.

First, note the following algebraic manipulation of nonlinearity:

\begin{align*}
\end{align*}

Let $u,v \in X$. Then,

\begin{align*}
\lVert \Gamma u-\Gamma v \rVert_{X^{s,b}_{(\epsilon),\delta}} &=\left|\left| \int_0^t U_\epsilon(t-\tau)(N(u)-N(v))(\tau)d\tau\right|\right|\lesssim \lVert N(u)-N(v)\rVert_{X^{s,b-1}_{(\epsilon),\delta}}\\&\lesssim \delta^{1-(b+\gamma)}\lVert N(u)-N(v) \rVert_{X^{s,-\gamma}_{(\epsilon),\delta}}\lesssim \delta^{1-(b+\gamma)}\lVert g \rVert_{H^s}^2 \lVert u-v \rVert_{X^{s,b}_{(\epsilon),\delta}},
\end{align*}

and therefore by shrinking $\delta$ if necessary, $\Gamma$ is a contraction on $X$.

To show continuous dependence on initial data, let $g,g_n \in H^s(\mathbb{R})$ such that $g_n\xrightarrow[n\rightarrow\infty]{}g$ and let $u,u_n$ the corresponding solutions, respectively. Let $u$ be defined on $[-\delta,\delta]$ and fix $0<T<\delta$. Then there exists a sufficiently large $N\geq 1$ such that $u_n$ is well-defined on $[-T,T]$ for all $n\geq N$. Arguing as above,

\begin{align*}
\lVert u-u_n \rVert_{X^{s,b}_{(\epsilon),T}}&\lesssim \lVert g-g_n \rVert_{H^s}+T^{1-(b+\gamma)}\lVert N(u)-N(u_n)\rVert_{X^{s,-\gamma}_{(\epsilon),T}}\\
&\lesssim \lVert g-g_n \rVert_{H^s}+T^{1-(b+\gamma)}\lVert g \rVert_{H^s}\cdot \lVert g_n \rVert_{H^s}\cdot\lVert g-g_n \rVert_{H^s},
\end{align*}

which proves the claim as $n\rightarrow \infty$. The Lipschitz regularity of data-to-solution map is proved similarly.

Similar to the Zakharov system, $(4)$ admits the conservation of $L^2$-norm and the energy defined as follows:

\begin{align*}
ENERGY[u(t)] = \frac{\epsilon^2}{2}\lVert \partial_{xx}u\rVert_{L^2}^2 + \frac{1}{2}\lVert \partial_x u \rVert_{L^2}^2 -\frac{1}{4}\lVert J_\epsilon(|u|^2)|u|^2\rVert_{L^1}.
\end{align*}

Hence the solutions for mNLS are globally well-posed in $L^2(\mathbb{R})$. On the other hand, by Gagliardo-Nirenberg inequality, one can show that smooth solutions in $H^2(\mathbb{R})$ are bounded for all times, from which global wellposedness in $H^2(\mathbb{R})$ is deduced. For $s\geq 2$, one can further show that the Sobolev norm of solutions grow exponentially in time by applying Gronwall's inequality to the Duhamel integral formula. In the following proposition, we show that the Sobolev norm of solutions grow polynomially in time for all $s\geq 0$ as an application of proposition $2.4$. We note, however, that our growth rate is not sharp.

Proposition 2.7. For all $s\geq 0$ and $\epsilon>0$, let $g\in H^s(\mathbb{R})$ and $u^{(\epsilon)}$ be the corresponding strong local solution as in Corollary $2.6$. Then there exists a non-decreasing function $C_s:[0,\infty)\rightarrow [0,\infty)$ such that $\lVert u^{(\epsilon)}(t)\rVert_{H^s}\leq C_s(\lVert g \rVert_{H^s})\langle t \rangle^{\frac{1}{2}\Big(3^{\lfloor \frac{3s}{4}\rfloor+1}-1\Big)}$ for all $t\in \mathbb{R}$.

proof of Theorem I. From corollary $2.6$, the mNLS is locally-wellposed in subcritical sense. Hence if $[0,T]$ is the maximum interval of existence for $u^{(\epsilon)}[g]$ for $T<\infty$, then $\varlimsup\limits_{t\rightarrow T-}\lVert u^{(\epsilon)}[g](t)\rVert_{H^s}=\infty$, which contradicts proposition $2.7$. Hence $T=\infty$.

To prove proposition $2.4$, we need a technical lemma, whose proof is contained in the appendix, that gives a lower and upper bound to the unique negative root $r(\xi_1)$ of a cubic polynomial in $\xi_2$:

\begin{align*}
P(\xi_2)\coloneqq 4\epsilon^2\xi_2^3 + |\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)\xi_2 + |(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau|.
\end{align*}

Lemma 2.8. Let $\xi>1, \tau\in (\frac{d_\epsilon(\xi)}{2},2d_\epsilon(\xi))$ and let $r(\xi_1)$ denote the unique negative root of $P$. Then,

* $r(\xi_1)=-\frac{|\xi_1|^{1/3}(\epsilon^2\xi_1^2+2)^{1/2}}{\sqrt{3}\epsilon}\sinh\bigg(\frac{1}{3}\sinh^{-1}\Big(3\sqrt{3}\epsilon \frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\Big)\bigg)$.

* $|r(\xi_1)|\gtrsim_\epsilon |\xi|^{\frac{4}{3}}$ for all $\xi_1 \in (-\infty,\infty)$ where the implicit constant does not depend on $\xi$.

* $|r(\xi_1)|\lesssim_\epsilon
\begin{cases}
d_\epsilon(\xi)^{1/3},& \text{if } |\xi_1|\in [0,\frac{\xi}{2}]\\
|\xi_1|^{1/3}\langle \epsilon \xi_1 \rangle,              & \textit{otherwise}.
\end{cases}$

Remark. The main idea of the next proof adopts that of [Lemma 2.4]kenig1996bilinear where the analysis of KdV (cubic dispersion) leads to an algebraic manipulation of a second-order polynomial; on the other hand, our model equation (quartic dispersion) demands an appropriate bound on a third-order polynomial.

proof of Proposition 2.4. We can neglect the $\delta$-dependence in our proof as the following shows. Suppose $\lVert N(u)\rVert_{X^{s+a,-\gamma}_{(\epsilon)}}\lesssim\lVert u\rVert_{X^{s,b}_{(\epsilon)}}^3$ for all $u \in X^{s,b}_{(\epsilon)}$. Fix $u\in X^{s,b}_{(\epsilon)}$ and let $\Tilde{u}\in X^{s,b}_{(\epsilon)}$ such that $u=\Tilde{u}$ on $t\in [0,\delta]$. Then,

\begin{align*}
\lVert N(u)\rVert_{X^{s+a,-\gamma}_{(\epsilon),\delta}}\leq \lVert \eta(t/\delta)N(\Tilde{u})\rVert_{X^{s+a,-\gamma}_{(\epsilon)}}\lesssim_{\eta}\lVert N(\Tilde{u})\rVert_{X^{s+a,-\gamma}_{(\epsilon)}}\lesssim\lVert \Tilde{u} \rVert_{X^{s,b}_{(\epsilon)}}^3.
\end{align*}

Noting that the implicit constant is independent of $\delta$, we take infimum over all possible $\Tilde{u}$, thereby obtaining the desired result. Henceforth we ignore the $\delta$-dependence and prove the estimate by duality. Let $u,v,w \in X^{s,b}_{(\epsilon)}$ and $\phi\in (X^{s+a,-\gamma}_{(\epsilon)})^*$. Define

\begin{align*}
f(\xi,\tau)&=|\hat{u}(\xi,\tau)|\langle \xi \rangle^s\langle \tau+d_\epsilon(\xi)\rangle^b,\:g(\xi,\tau)=|\hat{v}(\xi,\tau)|\langle \xi \rangle^s\langle \tau+d_\epsilon(\xi)\rangle^b;\\
h(\xi,\tau)&=|\hat{w}(\xi,\tau)|\langle \xi \rangle^s\langle \tau+d_\epsilon(\xi)\rangle^b,\:\psi(\xi,\tau)=|\hat{\phi}(\xi,\tau)|\langle \xi \rangle^{-(s+a)}\langle \tau-d_\epsilon(\xi)\rangle^\gamma.
\end{align*}

Then by Plancherel's theorem and   triangle inequality,

\begin{align*}
&\hspace{0.5cm}|\langle (I-\epsilon^2\partial_{xx})^{-1}(u\overline{v})w,\phi\rangle_{L^2_{xt}}|=|\langle \mathcal{F}[((I-\epsilon^2\partial_{xx})^{-1}(u\overline{v}))w],\mathcal{F}[\phi]\rangle_{L^2_{\xi \tau}}|\\
&\leq \int \frac{\langle\epsilon(\xi-\xi_1)\rangle^{-2}f(\xi-\xi_1-\xi_2,\tau-\tau_1-\tau_2)g(-\xi_2,-\tau_2)h(\xi_1,\tau_1)\psi(\xi,\tau)\langle \xi \rangle^{s+a}d\xi_1d\xi_2d\xi d\tau_1 d\tau_2 d\tau}{\langle \xi-\xi_1-\xi_2 \rangle^s \langle \tau-\tau_1-\tau_2+d_\epsilon(\xi-\xi_1-\xi_2)\rangle^b \langle \xi_2\rangle^s \langle \tau_2-d_\epsilon(\xi_2)\rangle^b \langle \xi_1 \rangle^s \langle \tau_1 +d_\epsilon(\xi_1)\rangle^b  \langle \tau -d_\epsilon(\xi)\rangle^{\gamma}}.
\end{align*}

We apply Cauchy-Schwarz inequality on the product of $f,g,h$ and the rest. The former, followed by Young's inequality, yields

\begin{align*}
\lVert f^2\ast \overline{g^2}\ast h^2 \rVert_{L^1_{\xi \tau}}^{1/2} \leq \lVert f \rVert_{L^2}\lVert g \rVert_{L^2}\lVert h \rVert_{L^2} = \lVert u \rVert_{X^{s,b}_{(\epsilon)}}\lVert v \rVert_{X^{s,b}_{(\epsilon)}}\lVert w \rVert_{X^{s,b}_{(\epsilon)}}.
\end{align*}

The remaining part of Cauchy-Schwarz inequality, followed by $L^1-L^\infty$ Hölder's inequality yields

\begin{align*}
\lVert \psi \rVert_{L^2}\cdot\Bigg(\sup_{\xi,\tau}\int \frac{\langle \epsilon(\xi-\xi_1)\rangle^{-4}\langle \xi \rangle^{2s+2a}\langle \tau -d_\epsilon(\xi)\rangle^{-2\gamma}d\xi_1 d\xi_2 d\tau_1d\tau_2}{\langle \xi-\xi_1-\xi_2 \rangle^{2s} \langle \tau-\tau_1-\tau_2+d_\epsilon(\xi-\xi_1-\xi_2)\rangle^{2b} \langle \xi_2\rangle^{2s} \langle \tau_2-d_\epsilon(\xi_2)\rangle^{2b} \langle \xi_1 \rangle^{2s} \langle \tau_1 +d_\epsilon(\xi_1)\rangle^{2b}}\Bigg)^{1/2}.
\end{align*}

After changing the variable $\xi_1\mapsto \xi_1+\xi$, it suffices to show

\begin{align*}
\sup_{\xi,\tau}\int \frac{\langle \epsilon\xi_1\rangle^{-4}\langle \xi \rangle^{2s+2a}\langle \tau -d_\epsilon(\xi)\rangle^{-2\gamma}d\xi_1 d\xi_2 d\tau_1d\tau_2}{\langle \xi_1+\xi_2 \rangle^{2s} \langle \tau-\tau_1-\tau_2+d_\epsilon(\xi_1+\xi_2)\rangle^{2b} \langle \xi_2\rangle^{2s} \langle \tau_2-d_\epsilon(\xi_2)\rangle^{2b} \langle \xi_1+\xi \rangle^{2s} \langle \tau_1 +d_\epsilon(\xi_1+\xi)\rangle^{2b}}<\infty.
\end{align*}

We further reduce this task by doing $\tau_1,\tau_2$ integration as follows (lemma 2.2.1):

\begin{align*}
\int \frac{d\tau_1}{\langle \tau_1 -(\tau-\tau_2+d_\epsilon(\xi_1+\xi_2))\rangle^{2b}\langle \tau_1 +d_\epsilon(\xi_1+\xi)\rangle^{2b}}&\lesssim \langle \tau_2-(\tau+d_\epsilon(\xi_1+\xi_2)+d_\epsilon(\xi_1+\xi))\rangle^{-2b}.\nonumber\\
\int \frac{d\tau_2}{\langle \tau_2-(\tau+d_\epsilon(\xi_1+\xi_2)+d_\epsilon(\xi_1+\xi))\rangle^{2b}\langle \tau_2 - d_\epsilon(\xi_2)\rangle^{2b}} &\lesssim \langle \tau+d_\epsilon(\xi_1+\xi_2)+d_\epsilon(\xi_1+\xi)-d_\epsilon(\xi_2) \rangle^{-2b}.
\end{align*}

Since $\langle \xi-A \rangle \langle \xi-B\rangle \gtrsim \langle A-B \rangle$, we have $\langle \xi_1+\xi_2\rangle^{2s}\langle \xi_2\rangle^{2s}\langle \xi_1+\xi \rangle^{2s}\gtrsim \langle \xi \rangle^{2s}$.
Thus, the problem reduces to showing

\begin{align*}
\sup\limits_{\xi,\tau}\frac{\langle\xi\rangle^{2a}}{\langle\tau-d_\epsilon(\xi)\rangle^{2\gamma}}\int \frac{\langle\epsilon \xi_1\rangle^{-4} d\xi_1 d\xi_2}{\langle\tau+d_\epsilon(\xi_1+\xi_2)+d_\epsilon(\xi_1+\xi)-d_\epsilon(\xi_2) \rangle^{2b}}<\infty.
\end{align*}

The expression $\tau+d_\epsilon(\xi_1+\xi_2)+d_\epsilon(\xi_1+\xi)-d_\epsilon(\xi_1) $ is a cubic polynomial in $\xi_2$ with an inflection point at $\xi_2 = -\frac{\xi_1}{2}$, and therefore after changing the variable $\xi_2\mapsto \xi_2-\frac{\xi_1}{2}$, the integral becomes

\begin{align*}
\int \frac{\langle\epsilon \xi_1\rangle^{-4} d\xi_1 d\xi_2}{\langle4\epsilon^2\xi_1\xi_2^3 + \xi_1(\epsilon^2\xi_1^2+2)\xi_2 + (1+\epsilon^2(\xi_1+\xi)^2)(\xi_1+\xi)^2+\tau\rangle^{2b}}<\infty.
\end{align*}

In doing $\xi_2$-integral, if $\xi_1<0$, then via another change of variable $\xi_2\mapsto -\xi_2$, the integral is invariant when

\begin{align*}
4\epsilon^2\xi_1\xi_2^3 + \xi_1(\epsilon^2\xi_1^2+2)\xi_2 + (1+\epsilon^2(\xi_1+\xi)^2)(\xi_1+\xi)^2+\tau
\end{align*}

is replaced with

\begin{align*}
4\epsilon^2|\xi_1|\xi_2^3 + |\xi_1|(\epsilon^2\xi_1^2+2)\xi_2 + (1+\epsilon^2(\xi_1+\xi)^2)(\xi_1+\xi)^2+\tau.
\end{align*}

Similarly the expression above can be replaced with

\begin{align*}
4\epsilon^2|\xi_1|\xi_2^3 + |\xi_1|(\epsilon^2\xi_1^2+2)\xi_2 + |(1+\epsilon^2(\xi_1+\xi)^2)(\xi_1+\xi)^2+\tau|,
\end{align*}

leaving the integral invariant.

Another change of variable $\xi_2\mapsto \frac{\xi_2}{|\xi_1|^{1/3}}$, followed by $\xi_1\mapsto -\xi_1$, eliminates the $\xi_1$-dependence in the leading coefficient of this cubic polynomial, and our task simplifies to showing the following estimate:

\begin{align}
\sup\limits_{\xi,\tau}\frac{\langle\xi\rangle^{2a}}{\langle\tau-d_\epsilon(\xi)\rangle^{2\gamma}}\int \frac{\langle\epsilon \xi_1\rangle^{-4}|\xi_1|^{-1/3} d\xi_1 d\xi_2}{\langle4\epsilon^2\xi_2^3 + |\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)\xi_2 + |(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau| \rangle^{2b}}<\infty.
\end{align}

It is easy to see that $\sup\limits_{\xi\in\mathbb{R}}$ can be reduced to $\sup\limits_{\xi>0}$, which we assume henceforth; if $\xi<0$, let $\xi^\prime = -\xi$ and do a change of variable $\xi_1\mapsto -\xi_1$ in the integral. However, we must consider $\tau>0,\tau<0$ separately.

Case I. $\tau<0$.

Since $\langle \xi \rangle^{2a}\langle \tau-d_\epsilon(\xi)\rangle^{-2\gamma}\leq \langle \xi \rangle^{2a}\langle d_\epsilon(\xi)\rangle^{-2\gamma}\lesssim_\epsilon 1$, it suffices to show

\begin{align}
\sup_{\xi,\tau}\iint \frac{|\xi_1|^{-1/3}d\xi_1 d\xi_2}{\langle 4\epsilon^2\xi_2^3 + |\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)\xi_2 + |(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau|\rangle^{2b}}<\infty.
\end{align}

We do the $\xi_2$-integral in three disjoint regions: $(-\infty,r)\cup(r,0)\cup (0,\infty)$.

$(i)$ Consider the Taylor expansion of $|P(\xi_2)|$ on $(-\infty,r)$ at $\xi_2=r$.

\begin{align*}
|P(\xi_2)| &=-(12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2))(\xi_2-r)-12\epsilon^2r(\xi_2-r)^2-4\epsilon^2(\xi_2-r)^3\nonumber\\
&\geq \max\Big(-|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)(\xi_2-r),-4\epsilon^2(\xi_2-r)^3\Big)\geq 0.
\end{align*}

Integrating these lower bounds, we obtain

\begin{align*}
|\xi_1|^{-1/3}\int_{-\infty}^r\frac{d\xi_2}{\langle p(\xi_2)\rangle^{2b}}&\leq|\xi_1|^{-1/3}\int_{-\infty}^r\frac{d\xi_2}{\langle |\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)(\xi_2-r) \rangle^{2b}}=|\xi_1|^{-1/3}\int_{0}^\infty\frac{d\xi_2}{\langle |\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)\xi_2 \rangle^{2b}}\\
&=\frac{|\xi_1|^{-1/3}}{|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)}\int_{0}^\infty\frac{d\xi_2}{\langle\xi_2 \rangle^{2b}}\simeq_b \frac{1}{|\xi_1|(\epsilon^2\xi_1^2+2)},
\end{align*}

\begin{align*}
|\xi_1|^{-1/3}\int_{-\infty}^r\frac{d\xi_2}{\langle p(\xi_2)\rangle^{2b}}&\leq|\xi_1|^{-1/3}\int_{-\infty}^r\frac{d\xi_2}{\langle 4\epsilon^2(\xi_2-r)^3\rangle^{2b}}\nonumber\\&=|\xi_1|^{-1/3}\int_{0}^\infty\frac{d\xi_2}{\langle 4\epsilon^2\xi_2^3\rangle^{2b}}\simeq_{b,\epsilon}|\xi_1|^{-1/3}.
\end{align*}

\begin{align}
|\xi_1|^{-1/3}\int_{-\infty}^r\frac{d\xi_2}{\langle p(\xi_2)\rangle^{2b}}\lesssim_{b,\epsilon}\min\Big(\frac{1}{|\xi_1|(\epsilon^2\xi_1^2+2)},|\xi_1|^{-1/3}\Big)
\end{align}

and the desired result follows by integrating with respect to $\xi_1$.

$(ii)$ Let $\xi_2 \in (r,0)$. Since $r$ is a root of $P(\xi_2)$,

\begin{align*}
&\geq (\xi_2-r)(|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2))\geq 0.
\end{align*}

Using this lower bound,

\begin{align*}
|\xi_1|^{-1/3}\int_r^0\frac{d\xi_2}{\langle p(\xi_2)\rangle^{2b}}&\leq |\xi_1|^{-1/3}\int_r^0 \frac{d\xi_2}{\langle (\xi_2-r)(|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2))\rangle^{2b}}=|\xi_1|^{-1/3}\int_0^{|r|}\frac{d\xi_2}{\langle \xi_2\cdot(|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2))\rangle^{2b}}\\
&\leq \frac{|\xi_1|^{-1/3}}{|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)}\int_0^\infty \frac{d\xi_2}{\langle \xi_2\rangle^{2b}}\simeq_{b}\frac{1}{|\xi_1|(\epsilon^2\xi_1^2+2)}.
\end{align*}

On the other hand,

\begin{align*}
&\geq 4\epsilon^2(\xi_2-r)^3\geq 0,
\end{align*}

where the inequality holds since $(12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2))(\xi_2-r)+12\epsilon^2r(\xi_2-r)^2\geq 0$ on $\xi_2 \in (r,0)$. Then,

\begin{align*}
|\xi_1|^{-1/3}\int_r^0\frac{d\xi_2}{\langle p(\xi_2)\rangle^{2b}}&\leq |\xi_1|^{-1/3}\int_r^0 \frac{d\xi_2}{\langle 4\epsilon^2(\xi_2-r)^3\rangle^{2b}}\nonumber\\
&=|\xi_1|^{-1/3}\int_0^{|r|}\frac{d\xi_2}{\langle 4\epsilon^2\xi_2^3\rangle^{2b}}\leq |\xi_1|^{-1/3}\int_0^{\infty}\frac{d\xi_2}{\langle 4\epsilon^2\xi_2^3\rangle^{2b}}\simeq_{b,\epsilon}|\xi_1|^{-1/3}.
\end{align*}

\begin{align}
|\xi_1|^{-1/3}\int_{r}^0\frac{d\xi_2}{\langle p(\xi_2)\rangle^{2b}}\lesssim_{b,\epsilon}\min\Big(\frac{1}{|\xi_1|(\epsilon^2\xi_1^2+2)},|\xi_1|^{-1/3}\Big)
\end{align}

and the desired result follows by integrating with respect to $\xi_1$.

$(iii)$ Similarly on $\xi_2\in (0,\infty)$,

\begin{align}
|\xi_1|^{-1/3}\int_{0}^\infty\frac{d\xi_2}{\langle p(\xi_2)\rangle^{2b}}\lesssim_{b,\epsilon}\min\Big(\frac{1}{|\xi_1|(\epsilon^2\xi_1^2+2)},|\xi_1|^{-1/3}\Big),
\end{align}

where we use the two lower bounds of $|P(\xi_2)|=P(\xi_2)\geq \max\Big(4\epsilon^2\xi_2^3,|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)\xi_2\Big)$ to argue as before. This concludes the proof for case I.

Case II. $\tau \in [0,\frac{d_\epsilon(\xi)}{2}]\cup [2d_\epsilon(\xi),\infty)$.

If $\tau \in [0,\frac{d_\epsilon(\xi)}{2}]$, then $d_\epsilon(\xi)-\tau \geq \frac{d_\epsilon(\xi)}{2}\geq 0$, and therefore $\langle \tau-d_\epsilon(\xi)\rangle^{-2\gamma}\leq \langle \frac{d_\epsilon(\xi)}{2}\rangle^{-2\gamma}$. If $\tau \in [2d_\epsilon(\xi),\infty)$, then $\tau-d_\epsilon(\xi)\geq d_\epsilon(\xi)$, and therefore $\langle \tau-d_\epsilon(\xi)\rangle^{-2\gamma}\leq \langle d_\epsilon(\xi)\rangle^{-2\gamma}$. In either cases, the analysis reduces to showing

\begin{align}
\sup_{\xi,\tau}\iint \frac{|\xi_1|^{-1/3}d\xi_1 d\xi_2}{\langle 4\epsilon^2\xi_2^3 + |\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)\xi_2 + (1+\epsilon^2(\xi_1-\xi))(\xi_1-\xi)^2+\tau\rangle^{2b}}<\infty,
\end{align}

which can be done as in case I.

Case III. $\tau \in (\frac{d_\epsilon(\xi)}{2},2d_\epsilon(\xi))$.

On this region, $\langle \tau - d_\epsilon(\xi)\rangle^{-2\gamma}\leq 1$, and therefore, we need to extract an algebraic decay in $\xi$ from the double integral. As before, we derive good lower bounds on $|p(\xi_2)|$ on three disjoint regions, $(-\infty,r)\cup (r,0)\cup (0,\infty)$ and bound the integral separately. Moreover we can assume $\xi>1$ without loss of generality since $\left\{(\xi,\tau):\xi \in [0,1], \tau \in [\frac{d_\epsilon(\xi)}{2},2d_\epsilon(\xi)]\right\}$ is a compact subset of $\mathbb{R}^2$, and therefore, extreme value theorem applies on our double integral.

$(i)$ On $\xi_2\in (-\infty,r)$,

\begin{align*}
|P(\xi_2)| &=-(12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2))(\xi_2-r)-12\epsilon^2r(\xi_2-r)^2-4\epsilon^2(\xi_2-r)^3\\
&\geq -(12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2))(\xi_2-r)\geq 0\\
\Rightarrow |\xi_1|^{-1/3}\int_{-\infty}^r\frac{d\xi_2}{\langle P(\xi_2)\rangle^{2b}}&\leq |\xi_1|^{-1/3}\int_0^\infty \frac{d\xi_2}{\langle( 12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2))\xi_2\rangle^{2b}}\simeq_{b}\frac{|\xi_1|^{-1/3}}{12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)}.
\end{align*}

Consider a change of variable $z=\xi_1^{\frac{8}{3}}+c_\epsilon \xi^{\frac{8}{3}}$ where $c_\epsilon>0$ is to be determined and $p\in [1,3)$. Then,

\begin{align*}
\int_0^\infty \frac{\langle \epsilon\xi_1\rangle^{-4}|\xi_1|^{-1/3}d\xi_1}{12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)}&=
\int_0^\xi \frac{\langle \epsilon\xi_1\rangle^{-4}|\xi_1|^{-1/3}d\xi_1}{12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)}+\int_\xi^\infty \frac{\langle \epsilon\xi_1\rangle^{-4}|\xi_1|^{-1/3}d\xi_1}{12\epsilon^2r^2+|\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)}\\
&\lesssim_\epsilon |\xi|^{-\frac{8}{3}}\lVert \langle\epsilon \xi_1\rangle^{-4}\rVert_{L^{p^\prime}_{(0,\xi)}}\lVert \xi_1^{-1/3}\rVert_{L^p_{(0,\xi)}}+\langle \epsilon\xi\rangle^{-4}\xi^{-1/3}\int_\xi^\infty \frac{d\xi_1}{\xi_1^{8/3}+c_\epsilon \xi^{8/3}}\\
&\lesssim_p \langle\xi\rangle^{-(3-\frac{1}{p})}+\langle \epsilon\xi\rangle^{-4}\xi^{-1/3}\int_{c_\epsilon\xi^{8/3}}^\infty \frac{dz}{z(z-c_\epsilon\xi^{8/3})^{\frac{5}{8}}}\lesssim \langle \xi \rangle^{-(3-\frac{1}{p})},
\end{align*}

where we use lemma 2.8.2 as a lower bound on $|r|$ and lemma 2.2.2 in the last inequality.

$(ii)$ On $\xi_2\in (r,0)$, we have $|P(\xi_2)|=P(\xi_2)\geq \frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}{|r|}\xi_2+(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau\geq 0$, and therefore,

\begin{align*}
|\xi_1|^{-1/3}\int_r^0 \frac{d\xi_2}{\langle \frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}{|r|}\xi_2+(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau\rangle^{2b}}
&\lesssim_b \frac{|\xi_1|^{-1/3}|r(\xi_1)|}{\langle (1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2 + \frac{d_\epsilon(\xi)}{2}\rangle}.
\end{align*}

We change variable $z=\xi_1^4 + \frac{\xi^4}{2}$ and integrate with respect to $\xi_1$ as in the previous case $(i)$, and use the upper bound in lemma 2.8.3.

\begin{align*}
\int_{-\infty}^\infty\int_r^0\frac{\langle \epsilon\xi_1\rangle^{-4}|\xi_1|^{-1/3}d\xi_2 d\xi_1}{\langle p(\xi_2)\rangle^{2b}}&
\lesssim \int_0^{\xi}\frac{\langle \epsilon\xi_1\rangle^{-4}|\xi_1|^{-1/3}d_\epsilon(\xi)^{1/3}d\xi_1}{\langle\frac{d_\epsilon(\xi)}{2}\rangle}+\int_{\xi}^\infty \frac{\langle \epsilon\xi_1\rangle^{-3} d\xi_1}{\langle (1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\frac{d_\epsilon(\xi)}{2}\rangle}\\
&\lesssim_b \langle \xi \rangle^{-(3-\frac{1}{p})}+\frac{\langle\epsilon\xi\rangle^{-3}}{\epsilon^2}\int_0^\infty \frac{ d\xi_1}{\xi_1^4+\frac{\xi^4}{2}}\lesssim\langle \xi \rangle^{-(3-\frac{1}{p})}.
\end{align*}

$(iii)$ On $\xi_2\in (0,\infty)$, we note

\begin{align*}
|P(\xi_2)|=P(\xi_2)\geq 4\epsilon^2\xi_2^3+\frac{d_\epsilon(\xi)}{2}\geq 0.
\end{align*}

Consider the following change of variable: $z=4\epsilon^2\xi_2^3+\frac{d_\epsilon(\xi)}{2}$. Then,

\begin{align*}
|\xi_1|^{-1/3}\int_0^\infty\frac{d\xi_2}{\langle P(\xi_2)\rangle^{2b}}&\leq |\xi_1|^{-1/3}\int_0^\infty\frac{d\xi_2}{\langle 4\epsilon^2\xi_2^3+\frac{d_\epsilon(\xi)}{2}\rangle^{2b}}\\
&\simeq_{\epsilon} |\xi_1|^{-1/3}\int_{\frac{d_\epsilon(\xi)}{2}}^\infty \frac{dz}{\langle z\rangle^{2b}(z-\frac{d_\epsilon(\xi)}{2})^{2/3}}\lesssim_b |\xi_1|^{-\frac{1}{3}}|d_\epsilon(\xi)|^{-\frac{2}{3}},
\end{align*}

where in the last inequality, we combine lemma $2.2.2$ with $\langle z\rangle^{-2b}\lesssim |z|^{-1}$ on the region of integration. On the other hand, we use $4\epsilon^2\xi_2^3+(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\frac{d_\epsilon(\xi)}{2}\geq 0$ as another lower bound to derive a similar estimate for $|\xi_1|>2\xi$ on which $|\xi_1-\xi|\geq \frac{|\xi_1|}{2}$.

\begin{align*}
|\xi_1|^{-1/3}\int_0^\infty\frac{d\xi_2}{\langle P(\xi_2)\rangle^{2b}}&\lesssim_{b,\epsilon}\frac{|\xi_1|^{-1/3}}{\langle (1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\frac{d_\epsilon(\xi)}{2}\rangle^{2/3}}\leq \frac{|\xi_1|^{-1/3}}{\langle\frac{\epsilon^2\xi_1^4}{16}+\frac{d_\epsilon(\xi)}{2}\rangle^{2/3}}\lesssim \frac{|\xi|^{-1/3}}{\Big(\frac{\epsilon^2\xi_1^4}{16}+\frac{d_\epsilon(\xi)}{2}\Big)^{2/3}},
\end{align*}

and the remaining $\xi_1$ integral proceeds as before.

proof of Proposition 2.7. We prove by induction on $k \geq 0$. Let $I_k = [\frac{4}{3}k,\frac{4}{3}(k+1))$ and consider the following statement: for all $s\in I_k$, there exists $C_s$, a non-decreasing function, such that $\lVert u^{(\epsilon)}(t)\rVert_{H^s}\leq C_s(\lVert g \rVert_{H^s})\langle t \rangle^{\frac{1}{2}(3^{k+1}-1)}$ for all $t\in \mathbb{R}$; it suffices to assume $t\geq 0$ by time-reversal symmetry of solutions. Let $\alpha_k = \frac{1}{2}(3^{k+1}-1)$ for $k\geq 0$; note that $\alpha_k = 3\alpha_{k-1}+1$.

Let $k=0$ and fix $b\in (\frac{1}{2},1-\gamma), \gamma \in [\frac{1}{3},\frac{1}{2})$ once and for all. By the local theory in $L^2(\mathbb{R})$ with $\delta\simeq \lVert g\rVert_{L^2}^{-\rho}$ for some $\rho = \rho(b,\gamma)>0$,

\begin{align*}
\lVert u^{(\epsilon)}(t)\rVert_{H^s}&\leq \lVert g \rVert_{H^s} + C\lVert u^{(\epsilon)} \rVert_{X^{0,b}_{(\epsilon),\delta}}^3\leq \lVert g \rVert_{H^s}+C^\prime \lVert g \rVert_{L^2}^3, t \in (0,\delta].
\end{align*}

Time evolving $u^{(\epsilon)}$ iteratively for $j=1,2,...$ and $t \in ((j-1)\delta,j\delta]$,

\begin{align*}
\lVert u^{(\epsilon)}(t)\rVert_{H^s}&\leq \lVert g \rVert_{H^s}+C^\prime j \lVert g \rVert_{L^2}^3< \lVert g \rVert_{H^s}+C^\prime (1+\frac{t}{\delta})\lVert g \rVert_{L^2}^3\leq
\lVert g \rVert_{H^s}+C^\prime (1+\frac{t}{\delta})\lVert g \rVert_{H^s}^3\\
&\leq\lVert g \rVert_{H^s}+C^\prime\lVert g \rVert_{H^s}^3 +C^{\prime\prime}\lVert g \rVert_{H^s}^{3+\rho}t.
\end{align*}

Noting that $\langle t \rangle \simeq 1+|t|$ and that $C^\prime, C^{\prime\prime}$ are universal constants that only depend on the given parameters, there exists a non-decreasing function $C_s$ such that

\begin{align*}
\lVert g \rVert_{H^s}+C^\prime\lVert g \rVert_{H^s}^3 +C^{\prime\prime}\lVert g \rVert_{H^s}^{3+\rho}t &\leq
C_s(\lVert g \rVert_{H^s})\langle t \rangle.
\end{align*}

For example, one can explicitly check that $C_s(\zeta) =C^{\prime\prime}\zeta^{3+\rho}+ C^\prime\zeta^3+\zeta$ would do.

Now, suppose $k\geq 1$ and that the inductive hypothesis holds for $j=0,1,...,k-1$, and let $s \in I_k$. Fix $a\in (0,\frac{4}{3})$ such that $s-ja \in I_{k-j}$ for $j=1,2,...,k$ and $s-(k+1)a\leq 0$. We again use the $L^2$ local theory on $[0,\delta]$ where $\delta = \Tilde{c}\lVert g \rVert_{L^2}^{-\rho}$ for some $\Tilde{c}>0$. Iteratively applying smoothing estimate for $N(u^{(\epsilon)})$ and using triangle inequaltiy $(a+b)^3\lesssim a^3+b^3$ for $a,b\geq 0$, we obtain

\begin{align*}
t \in (0,\delta]: \lVert u^{(\epsilon)}(t) \rVert_{H^s}&\leq \lVert g \rVert_{H^s}+C(\lVert g \rVert_{H^{s-a}}^3+\dots+\lVert g \rVert_{H^{s-ka}}^{3^k})+C\lVert g \rVert_{L^2}^{3^{k+1}}\\
t\in(\delta,2\delta]:\lVert u^{(\epsilon)}(t)\rVert_{H^s}&\leq \lVert g \rVert_{H^s}
+C(\lVert g \rVert_{H^{s-a}}^3+\dots+\lVert g \rVert_{H^{s-ka}}^{3^k})+2C\lVert g \rVert_{L^2}^{3^{k+1}}
\\&\hspace{3cm}+C\Big(C_{s-a}(\lVert g \rVert_{H^{s-a}})^3\langle\delta\rangle^{3\alpha_{k-1}}+\dots+C_{s-ka}(\lVert g \rVert_{H^{s-ka}})^{3^k}\langle\delta\rangle^{3^k\alpha_{0}}\Big)\\
&\leq \lVert g \rVert_{H^s}
+C(\lVert g \rVert_{H^{s-a}}^3+\dots+\lVert g \rVert_{H^{s-ka}}^{3^k})+2C\lVert g \rVert_{L^2}^{3^{k+1}}
\\&\hspace{3cm}+C\Big(C_{s-a}(\lVert g \rVert_{H^{s}})^3+\dots+C_{s-ka}(\lVert g \rVert_{H^{s}})^{3^k}\Big)\langle\delta\rangle^{3\alpha_{k-1}}\\
t\in((j-1)\delta,j\delta]:\lVert u^{(\epsilon)}(t)\rVert_{H^s}&\leq \lVert g \rVert_{H^s}
+C(\lVert g \rVert_{H^{s-a}}^3+\dots+\lVert g \rVert_{H^{s-ka}}^{3^k})+jC\lVert g \rVert_{L^2}^{3^{k+1}}
\\
&\hspace{3cm}+(j-1)C\Big(C_{s-a}(\lVert g \rVert_{H^{s}})^3+\dots+C_{s-ka}(\lVert g \rVert_{H^{s}})^{3^k}\Big)\langle (j-1)\delta\rangle^{3\alpha_{k-1}}.\\
\end{align*}

Hence for all $t\geq 0$,

\begin{align*}
\lVert u^{(\epsilon)}(t)\rVert_{H^s}&<\lVert g \rVert_{H^s}
+C(\lVert g \rVert_{H^{s}}^3+\dots+\lVert g \rVert_{H^{s}}^{3^k})+C(1+\frac{\lVert g \rVert_{H^s}^\sigma}{\Tilde{c}}t)\lVert g \rVert_{H^s}^{3^{k+1}}
\\
&\hspace{3cm}+C\frac{\lVert g \rVert_{H^s}^\sigma}{\Tilde{c}}\Big(C_{s-a}(\lVert g \rVert_{H^{s}})^3+\dots+C_{s-ka}(\lVert g \rVert_{H^{s}})^{3^k}\Big)\langle t\rangle^{3\alpha_{k-1}+1},
\end{align*}

and therefore, there exists a non-decreasing function $C_s$ such that

\begin{align*}
\lVert u^{(\epsilon)}(t)\rVert_{H^s}\leq C_s(\lVert g \rVert_{H^s})\langle t \rangle^{\alpha_k}.
\end{align*}

§4: Nonlinear Schrödinger Limit.

In this section, we study the $\epsilon\rightarrow 0$ problem. Heuristically the mNLS is a perturbation of NLS, and it is our goal to make this statement rigorous. We first study the convergence of linear evolution.

Proposition 4.1. Let $\epsilon> 0, s\in\mathbb{R}, t \in \mathbb{R}\setminus\left\{0\right\}$. Consider $U_\epsilon(t)$ and $U(t)$ as unitary operators on $H^s(\mathbb{R}^d)$. Then,

* $U_\epsilon(t)\nrightarrow U(t)$ in norm operator topology as $\epsilon\rightarrow 0$.

* $U_\epsilon(t)\xrightarrow[\epsilon\rightarrow 0]{} U(t)$ in strong operator topology. More generally if $u_0^{(\epsilon)}, u_0\in H^s(\mathbb{R}^d)$ and $u_0^{(\epsilon)} \xrightarrow[\epsilon\rightarrow 0]{} u_0$, then $U_\epsilon(t)u_0^{(\epsilon)}\xrightarrow[\epsilon\to 0]{} U(t)u_0$ in $C([0,T],H^s(\mathbb{R}^d))$ for every $0<T<\infty$.

* For all $0<T<\infty$ and $u\in C([0,T],H^s(\mathbb{R}^d))$, we have $U_\epsilon(t)u(t)\xrightarrow[\epsilon\rightarrow 0]{H^s} U(t)u(t)$ uniformly in $t\in [0,T]$. In particular, for every initial data $u_0\in H^s(\mathbb{R}^d)$, we have $U_\epsilon(t)u_0 \xrightarrow[\epsilon\rightarrow 0]{} U(t)u_0$ in $C([0,T],H^s(\mathbb{R}^d))$.

* There exists some $u_0\in H^s(\mathbb{R}^d)$ such that $U_\epsilon(t)u_0\nrightarrow U(t)u_0$ in $C([0,\infty),H^s(\mathbb{R}^d))$ as $\epsilon\to 0$.

proof of Proposition 4.1. We use the characterisation of Fourier multiplier on $L^2(\mathbb{R}^d)$ where the operator norm of a Fourier multiplier equals the $L^\infty$ norm of the corresponding symbol. Since $H^s(\mathbb{R}^d)$ is unitarily isomorphic to $L^2(\mathbb{R}^d)$ via $f \mapsto \langle \xi \rangle^s \hat{f}$, the unitary action $U_\epsilon(t)$ on $H^s(\mathbb{R}^d)$ is unitarily isomorphic to a multiplication operator on $L^2(\mathbb{R}^d)$ via $F(\xi) \mapsto e^{-it(|\xi|^2+\epsilon^2 |\xi|^4)}F(\xi)$. Hence to show $U_\epsilon(t) \nrightarrow U_0(t)$ in norm topology, it suffices to show $\lVert e^{-it|\xi|^2}(e^{-i\epsilon^2 t |\xi|^4}-1)\rVert_{L^\infty_\xi} \nrightarrow 0$ as $\epsilon \rightarrow 0$. Indeed $\lVert e^{-it|\xi|^2}(e^{-i\epsilon^2 t |\xi|^4}-1)\rVert_{L^\infty_\xi} =2$ for all $\epsilon>0$ and $t\in \mathbb{R}\setminus \left\{0\right\}$. This proves the first claim.

Noting that

\begin{align*}
\lVert U_\epsilon(t)u_0^{(\epsilon)}-U(t)u_0\rVert_{H^s}\leq \lVert u_0^{(\epsilon)}-u_0\rVert_{H^s}+\lVert(U_\epsilon(t)-U(t))u_0\rVert_{H^s},
\end{align*}

by triangle inequality and unitarity, it suffices to show that the latter tends to zero uniformly in $t\in [0,T]$. Since $|e^{-i\epsilon^2t |\xi|^4}-1|^2=2(1-\cos(\epsilon^2t|\xi|^4))$, we will show $\lim\limits_{\epsilon \rightarrow 0+}I=0$ where $I=\int (1-\cos(\epsilon^2t|\xi|^4)) \langle \xi \rangle^{2s}|\widehat{u_0}|^2d\xi$. We first observe that $1-\cos(\epsilon^2t|\xi|^4)$ converges to zero as $\epsilon \rightarrow 0$ for a fixed $t$ and $\xi$. For $n \in \mathbb{N}$, we observe that $\xi_n=\xi_n(\epsilon,t) = (\cos^{-1}(\frac{n-1}{n}))^{1/4}\epsilon^{-1/2}t^{-1/4}$ solves $1-\cos(\epsilon^2t\xi_n^4)=\frac{1}{n}$ and $0\leq 1-\cos(\epsilon^2t|\xi|^4) \leq \frac{1}{n}$ whenever $|\xi|\leq \xi_n(\epsilon,t)$. Estimating $I$ in two different regions, we obtain

\begin{align*}
I &= \int_{|\xi|\leq \xi_n(\epsilon,t)} + \int_{|\xi|>\xi_n(\epsilon,t)} \leq \frac{\lVert u_0 \rVert_{H^s}^2}{n}+2\int_{|\xi|>\xi_n(\epsilon,T)} \langle\xi\rangle^{2s}|\hat{u_0}|^2d\xi.\\
\Rightarrow \varlimsup\limits_{\epsilon\rightarrow 0} I &\leq \frac{\lVert u_0 \rVert_{H^s}^2}{n}, \forall n \in \mathbb{N}.
\end{align*}

which yields the desired result as $n\rightarrow \infty$. This proves the second claim.

Let $I=\int (1-\cos(\epsilon^2t|\xi|^4)) \langle \xi \rangle^{2s}|\widehat{u(t)}|^2d\xi$. On $t\in [0,T]$, we have $0\leq 1-\cos(\epsilon^2t|\xi|^4) \leq \frac{1}{n}$ whenever $|\xi|\leq \xi_n(\epsilon,T)$. Hence,

\begin{align*}
I &= \int_{|\xi|\leq \xi_n(\epsilon,T)} + \int_{|\xi|>\xi_n(\epsilon,T)} \leq \frac{\lVert u \rVert_{C^0_TH^s_x}^2}{n}+2\int_{|\xi|>\xi_n(\epsilon,T)} \langle\xi\rangle^{2s}|\widehat{u(t)}|^2d\xi.
\end{align*}

We define $F_\epsilon(t) = \int_{|\xi|>\xi_n(\epsilon,T)} \langle\xi\rangle^{2s}|\widehat{u(t)}|^2d\xi$ and show $\lim\limits_{\epsilon\to 0}\sup\limits_{t\in [0,T]}F_\epsilon(t)=0$ by Arzelà-Ascoli argument. Since $u\in C([0,T],H^s(\mathbb{R}))$, it is straightforward to see $F_\epsilon\in C([0,T],\mathbb{R})$ with a pointwise bound; in fact, $F_\epsilon(t)\rightarrow 0$ as $\epsilon\rightarrow 0$ pointwise. For $t_1,t_2 \in [0,T]$,

\begin{align*}
|F_\epsilon(t_1)-F_\epsilon(t_2)|&= \int_{|\xi|>\xi_n(\epsilon,T)}\langle\xi\rangle^{2s}\Big(|\widehat{u(t_1)}|^2-|\widehat{u(t_2)}|^2\Big)d\xi\\
&\leq \Bigg(\bigg(\int_{|\xi|>\xi_n(\epsilon,T)}\langle\xi\rangle^{2s}|\widehat{u(t_1)}|^2d\xi\bigg)^{\frac{1}{2}}+\bigg(\int_{|\xi|>\xi_n(\epsilon,T)}\langle\xi\rangle^{2s}|\widehat{u(t_2)}|^2d\xi\bigg)^{\frac{1}{2}}\Bigg)\cdot\lVert u(t_1)-u(t_2)\rVert_{H^s}\leq 2\lVert u\rVert_{C^0_TH^s_x}\lVert u(t_1)-u(t_2)\rVert_{H^s},
\end{align*}

and hence uniform equicontinuity on $[0,T]$. This proves the third claim.

Lastly, define $u_0 = \langle \nabla\rangle^{-s}|\nabla|^{\frac{1-d}{2}}e^{-\frac{|x|^2}{2}}$. Then we claim

\begin{align*}
\sup\limits_{t\in\mathbb{R}}\int_{-\infty}^\infty (1-\cos(\epsilon^2t\xi^4))\langle \xi \rangle^{2s}|\widehat{u_0}|^2d\xi=\sup\limits_{t\in[0,\infty)}\int_{-\infty}^\infty (1-\cos(\epsilon^2t\xi^4))\langle \xi \rangle^{2s}|\widehat{u_0}|^2d\xi=2\pi, \forall \epsilon>0.
\end{align*}
First, observe that

\begin{align*}
\langle \xi \rangle^s\widehat{u_0}(\xi)&=\sqrt{2\pi}|\xi|^{\frac{1-d}{2}}e^{-\frac{|\xi|^2}{2}}\in L^2(\mathbb{R}^d).
\end{align*}

By a direct computation,

\begin{align*}
\int_{-\infty}^\infty (1-\cos(\epsilon^2t\xi^4))\langle \xi \rangle^{2s}|\widehat{u_0}|^2d\xi&=2\pi|S^{d-1}| \int_0^\infty (1-\cos(\epsilon^2 t r^4)) e^{-r^2}dr\\
&=\frac{\pi^{3/2}}{4}|S^{d-1}| \left(\frac{\sqrt{2 \pi } \left(\sin \left(\frac{1}{8} \left(\pi -\frac{1}{t \epsilon ^2}\right)\right) J_{\frac{1}{4}}\left(\frac{1}{8 t \epsilon ^2}\right)-\cos \left(\frac{1}{8} \left(\frac{1}{t \epsilon ^2}+\pi \right)\right) J_{-\frac{1}{4}}\left(\frac{1}{8 t \epsilon ^2}\right)\right)}{\sqrt{t} \epsilon }+4\right),
\end{align*}

where $J_{\pm \frac{1}{4}}(x)$, the Bessel functions of the first kind, are the two linearly independent solutions to

\begin{align*}
x^2y^{\prime\prime} + x y^\prime + (x^2-\frac{1}{16})y=0;
\end{align*}

here the $\prime$ is the derivative with respect to $x$. The right-hand side of our direct computation is known to be an increasing function in $t\in [0,\infty)$ whose limit as $t\rightarrow\infty$ is $\pi^{\frac{3}{2}}|S^{d-1}|$ for all $\epsilon>0$.

To study the convergence of nonlinear evolution, we need to control the nonlinear term in the Duhamel formula, and it suffices, but most likely not necessary, to assume that our solutions are continuous in space.

Lemma 4.2. Let $s>\frac{1}{2}$ and $u^{(\epsilon)}$ be the global solution corresponding to the initial data $u_0^{(\epsilon)}$ where $\left\{u_0^{(\epsilon)}\right\}\subseteq B_{H^s(\mathbb{R})}(0,R)$ for some $R>0$. Then there exists $C=C(R,s)>0$ such that $\sup\limits_{\epsilon>0}\lVert u^\epsilon(t) \rVert_{H^s}\leq Re^{Ct}$ for all $t\geq 0$.

proof of Lemma 4.2. When $s>\frac{1}{2}$, we can establish (unconditional) local wellposedness of mNLS in $C([0,T],H^s(\mathbb{R}))$ such that $\lVert u^{(\epsilon)} (t)\rVert_{H^s}\leq 2R$ for $t\in [0,T]$, by using that $H^s(\mathbb{R})$ is a Sobolev algebra, where $T\simeq R^{-2}$ where the implicit constant is independent of $\epsilon>0$. Then by writing the solution in an integral form, and by applying Gronwall's inequality, one can deduce

\begin{align*}
\sup\limits_{\epsilon>0}\lVert u^\epsilon(t) \rVert_{H^s}\leq Re^{Ct}, \forall t \in [0,T],
\end{align*}

for some $C=C(R,s)$. In fact, we show that the inequality above holds for all $t\geq 0$. For a contradiction, suppose the desired inequality is false for some time element and define

\begin{align*}
T_1&\coloneqq\sup\left\{t\geq 0: \sup\limits_{\epsilon>0}\lVert u^{(\epsilon)}(\tau) \rVert_{H^s}\leq Re^{C\tau}, \forall\tau\in[0,t]\right\}\\
T_2&\coloneqq\inf\left\{t\geq 0:\sup\limits_{\epsilon>0}\lVert u^{(\epsilon)}(t) \rVert_{H^s}> Re^{Ct}\right\}.
\end{align*}

Then $T_0\coloneqq T_1=T_2>0$ where the strict inequality is by the previous local wellposedness argument. If $T_0$ were to satisfy the desired inequality, then we can run another local wellposedness argument, which would contradict the maximality of $T_1$. Hence there exists $\epsilon_0>0$ such that $\lVert u^{(\epsilon_0)}(T_0)\rVert_{H^s}>Re^{CT_0}$. However by continuity, $\lVert u^{(\epsilon_0)}(T_0)\rVert_{H^s} = \lim\limits_{t\rightarrow T_0-}\lVert u^{(\epsilon_0)}(t)\rVert_{H^s}\leq Re^{CT_0}$, a contradiction.

proof of Theorem II. The second part of the statement is proposition 4.1.4. As for the first, the strategy is to write the solutions in the integral form and show that the difference goes to zero as $\epsilon\rightarrow 0$ by applying Gronwall's inequality.

Let $\delta>0$. By proposition 4.1.2, there exists $\epsilon_0>0$ such that if $\epsilon\in (0,\epsilon_0)$, then

\begin{align*}
\sup\limits_{t\in [0,T]}\lVert U_\epsilon(t)u_0^{(\epsilon)}-U(t)u_0\rVert_{H^s}<\delta.
\end{align*}

To proceed with the nonlinear estimates, note that

\begin{align*}
\int_0^t U_\epsilon(t-\tau)\Big(J_\epsilon(|u^{(\epsilon)}|^2)u^{(\epsilon)}\Big)(\tau)-U(t-\tau)\Big(|u|^2u\Big)(\tau)d\tau &= I_1+I_2+I_3+I_4,\:\text{where}\\
I_1 = \int_0^t U_\epsilon(t-\tau)\bigg( J_\epsilon\Big(|u^{(\epsilon)}|^2\Big)(u^{(\epsilon)}-u)\bigg)d\tau;\:I_2&=\int_0^t U_\epsilon(t-\tau)\bigg(J_\epsilon\Big(|u^{(\epsilon)}|^2-|u|^2\Big)u\bigg)d\tau.\\
I_3 = \int_0^t U_\epsilon(t-\tau)\Bigg(\bigg(J_\epsilon\Big(|u|^2\Big)-|u|^2\bigg)u\Bigg)d\tau;\: I_4&= \int_0^t \Big(U_\epsilon(t-\tau)-U(t-\tau)\Big)(|u|^2u)d\tau.
\end{align*}

We estimate one by one.

\begin{align*}
\lVert I_1\rVert_{H^s}\leq \int_0^t \lVert J_\epsilon\Big(|u^{(\epsilon)}|^2\Big)(u^{(\epsilon)}-u)\rVert_{H^s}d\tau\lesssim_s \Big(\sup\limits_{\epsilon>0}\lVert u^{(\epsilon)}\rVert_{C^0_TH^s_x}\Big)^2\int_0^t \lVert u^{(\epsilon)}-u\rVert_{H^s}d\tau,
\end{align*}

and similarly

\begin{align*}
\lVert I_2\rVert_{H^s}\lesssim_s (\sup\limits_{\epsilon>0}\lVert u^{(\epsilon)}\rVert_{C^0_TH^s_x}+\lVert u \rVert_{C^0_T H^s_x})\lVert u \rVert_{C^0_TH^s_x}\int_0^t \lVert u^{(\epsilon)}-u\rVert_{H^s}d\tau.
\end{align*}

As for $I_3$, we have

\begin{align*}
\lVert I_3\rVert_{H^s}\lesssim_s \lVert u \rVert_{C^0_TH^s_x}\int_0^t \lVert (J_\epsilon-I)\Big(|u|^2\Big)\rVert_{H^s}d\tau.
\end{align*}

By Arzelà-Ascoli argument, we show that the right-hand side converges to $0$ uniformly on $t\in [0,T]$. Let $F_\epsilon(t)=\int_0^t \lVert (J_\epsilon-I)\Big(|u|^2\Big)\rVert_{H^s}d\tau$. By dominated convergence, $F_\epsilon(t)\xrightarrow[\epsilon\rightarrow 0]{} 0$ pointwise. Since $|F_\epsilon(t)-F_\epsilon(t^\prime)|\lesssim\lVert u \rVert_{C^0_TH^s_x}^2 |t-t^\prime|$, the family $\left\{F_\epsilon\right\}\subseteq C([0,T],\mathbb{R})$ is uniformly equicontinuous, which proves the claim. Then there exists $\epsilon_1>0$ such that if $\epsilon \in (0,\epsilon_1)$, then

\begin{align*}
\lVert u \rVert_{C^0_TH^s_x}\int_0^t \lVert (J_\epsilon-I)\Big(|u|^2\Big)\rVert_{H^s}d\tau < \delta,
\end{align*}

uniformly on $t\in [0,T]$.

Lastly, by a change of variable, we have

\begin{align*}
\lVert I_4 \rVert_{H^s}\leq \int_0^t \left|\left|\Big(U_\epsilon(t-\tau)-U(t-\tau)\Big)(|u|^2u)\right|\right|_{H^s} d\tau = \int_0^t \left|\left|\Big(U_\epsilon(\tau)-U(\tau)\Big)(|u(t-\tau)|^2u(t-\tau))\right|\right|_{H^s} d\tau.
\end{align*}

We claim $ \left|\left|\Big(U_\epsilon(\tau)-U(\tau)\Big)(|u(t-\tau)|^2u(t-\tau))\right|\right|_{H^s}\xrightarrow[\epsilon\to 0]{}0$ uniformly for $(t,\tau)\in \Omega_T = \left\{(t^\prime,\tau^\prime)\in\mathbb{R}^2: 0\leq t^\prime\leq T, 0\leq \tau^\prime \leq t^\prime\right\}$. Define $w(\tau)\coloneqq |u(t-\tau)|^2u(t-\tau)$ for $(t,\tau)\in \Omega_T$. With $\xi_n(\epsilon,\tau)$ defined as before,

\begin{align*}
\left|\left| \Big(U_\epsilon(\tau)-U(\tau)\Big)w(\tau)\right|\right|_{H^s}^2&\leq \frac{2}{n}\int_{|\xi|\leq \xi_n(\epsilon,\tau)}\langle\xi\rangle^{2s}|\widehat{w(\tau)}|^2d\xi+4\int_{|\xi|>\xi_n(\epsilon,\tau)}\langle\xi\rangle^{2s}|\widehat{w(\tau)}|^2d\xi\\
&\leq \frac{C_s \lVert u \rVert_{C^0_TH^s_x}^6}{n}+4\int_{|\xi|>\xi_n(\epsilon,T)}\langle\xi\rangle^{2s}|\widehat{w(\tau)}|^2d\xi.
\end{align*}

Choose $N$ sufficiently big such that $\frac{C_s \lVert u \rVert_{C^0_TH^s_x}^6}{n}<\frac{\delta}{2}$ for all $n\geq N$. It suffices to show  $F_\epsilon(t,\tau)\coloneqq \int_{|\xi|>\xi_N(\epsilon,T)}\langle\xi\rangle^{2s}|\widehat{w(\tau)}|^2d\xi\xrightarrow[\epsilon\to 0]{}0$ uniformly on $\Omega_T$. Let $(t,\tau),(t^\prime,\tau^\prime)\in\Omega_T$. Then

\begin{align*}
|F_\epsilon(t,\tau)-F_\epsilon(t^\prime,\tau^\prime)|&\leq \bigg(\Big(\int_{|\xi|>\xi_N(\epsilon,T)}\langle\xi\rangle^{2s}|\widehat{w(\tau)}|^2d\xi\Big)^{\frac{1}{2}}+\Big(\int_{|\xi|>\xi_N(\epsilon,T)}\langle\xi\rangle^{2s}|\widehat{w(\tau^\prime)}|^2d\xi\Big)^{\frac{1}{2}}\bigg)\Big(\int_{|\xi|>\xi_N(\epsilon,T)}\langle\xi\rangle^{2s}|\widehat{w(\tau)}-\widehat{w(\tau^\prime)}|^2d\xi\Big)^{\frac{1}{2}}\\
&\lesssim_s \lVert u \rVert_{C^0_TH^s_x}^3\Big(\int_{|\xi|>\xi_N(\epsilon,T)}\langle\xi\rangle^{2s}|\widehat{w(\tau)}-\widehat{w(\tau^\prime)}|^2d\xi\Big)^{\frac{1}{2}}\leq \lVert u \rVert_{C^0_TH^s_x}^3\lVert w(\tau)-w(\tau^\prime)\rVert_{H^s}\\
&\leq \lVert u \rVert_{C^0_TH^s_x}^3\bigg(\left|\left| \Big(|u(t-\tau)|^2+|u(t^\prime-\tau^\prime)|^2\Big)\Big(u(t-\tau)-u(t^\prime-\tau^\prime)\Big)\right|\right|_{H^s}+\left|\left| u(t-\tau)u(t^\prime-\tau^\prime)\cdot\overline{u(t-\tau)-u(t^\prime-\tau^\prime)}\right|\right|_{H^s}\bigg)\\
&\lesssim_s \lVert u\rVert_{C^0_TH^s_x}^5\lVert u(t-\tau)-u(t^\prime-\tau^\prime)\rVert_{H^s}.
\end{align*}

This estimate combined with the fact that $F_\epsilon(t,\tau)\xrightarrow[\epsilon\to 0]{}0$ pointwise on $\Omega_T$ by Dominated Convergence implies the desired result by Arzelà-Ascoli.

Hence there exists $\epsilon_2>0$ such that if $\epsilon \in (0,\epsilon_2)$, then

\begin{align*}
\left|\left|\Big(U_\epsilon(\tau)-U(\tau)\Big)(|u(t-\tau)|^2u(t-\tau))\right|\right|<\delta, \forall (t,\tau)\in \Omega_T,
\end{align*}

and therefore

\begin{align*}
\int_0^t \left|\left|\Big(U_\epsilon(\tau)-U(\tau)\Big)(|u(t-\tau)|^2u(t-\tau))\right|\right|_{H^s} d\tau<\delta t.
\end{align*}

By restricting $\epsilon \in (0,\min(\epsilon_0,\epsilon_1,\epsilon_2))$, we obtain

\begin{align*}
\lVert u^{(\epsilon)}(t)-u(t)\rVert_{H^s}\lesssim_{R,s,T} \delta\langle t\rangle +\int_0^t \lVert u^{(\epsilon)}(\tau)-u(\tau)\rVert_{H^s}d\tau, \forall t \in [0,T],
\end{align*}

from which Gronwall's inequality yields the following:

\begin{align*}
\lVert u^{(\epsilon)}(t)-u(t)\rVert_{H^s}\leq C\delta \langle t \rangle e^{Ct}, \forall t \in [0,T],
\end{align*}

where $C = C(R,s,T)$. Since $\delta>0$ is arbitrary, the proof is complete.

From the following lemma follows the failure of uniform continuity of the data-to-solution map corresponding to $(2)$:

Lemma 4.3 [Exercise 3.5]tao2006nonlinear. Let $s\in (-\infty,0)$. For every $0<\delta\ll \epsilon<1$, there exists $f_1,f_2 \in \mathscr{S}(\mathbb{R})$ with their $H^s$-norms of $O(\epsilon)$ and the separation of $O(\delta)$ such that there exists $T_\epsilon = O(\epsilon)$ for which $\lVert u[f_1](T_\epsilon)-u[f_2](T_\epsilon)\rVert_{H^s}\simeq \epsilon$.

proof of Theorem III.. We study the time-evolution for the difference $v\coloneqq u^{(\epsilon)}-u$ where $u^{(\epsilon)}(0)=u(0)$ via various Sobolev embeddings and Gronwall's inequality to obtain an upper bound on $\lVert v(t)\rVert_{H^s}$. Then $v$ satisfies

\begin{align*}
\begin{cases}
i\partial_t v= (\epsilon^2\partial_{xx}-\partial_{xx})v+N(v)+F(x,t),\: (x,t) \in \mathbb{R} \times \mathbb{R}\\
v(0) = 0.
\end{cases}
\end{align*}

where $F(x,t) = \epsilon^2\partial_{xx}^2 u +(I-J_\epsilon)(|u^{(\epsilon)}|^2)u^{(\epsilon)}$ and $N(v)=-(|u^{(\epsilon)}|^2+|u|^2)v+u^{(\epsilon)} u \overline{v}$. Observe that the linear contribution of the solution is always zero since $v(0)=0$. Hence any contribution to the solution directly comes from $F$ and $N$ at least for a short time.

Writing the solution in the Duhamel form and using the unitarity of $U_\epsilon(t)$ in Sobolev spaces, we obtain

\begin{align*}
\lVert v(t)\rVert_{H^s}=\left|\left| \int_0^t U_\epsilon(t-\tau)\Big(N(v)+F\Big)(\tau)d\tau\right|\right|_{H^s}\leq \int_0^t \lVert N(v)\rVert_{H^s}d\tau \int_0^t \lVert F(\tau)\rVert_{H^s}d\tau.
\end{align*}

where $t\in [0,T]$ for some $T$ to be determined later.

We first estimate $\lVert F(\tau)\rVert_{H^s}$. The complete integrability of cubic NLS on $\mathbb{R}$ gives $\lVert \partial_{xx}^2 u\rVert_{H^s}\leq C(\lVert u_0\rVert_{H^4})$. On the other hand, note that $(I-J_\epsilon)(|u^{(\epsilon)}|^2) = \epsilon^2\partial_{xx}J_\epsilon(|u^{(\epsilon)}|^2)$. By Theorem II, we obtain $\lVert u^{(\epsilon)}(\tau) \rVert_{H^j} \leq C(\lVert u_0\rVert_{H^j})$ for $j=1,2$ on $\tau\in [0,T]$, if $\epsilon$ is sufficiently small depending on $T$ and $u_0$. Hence

\begin{align*}
\lVert(I-J_\epsilon)(|u^{(\epsilon)}|^2)u^{(\epsilon)} \rVert_{H^s}&\leq \lVert (I-J_\epsilon)(|u^{(\epsilon)}|^2)u^{(\epsilon)}\rVert_{L^2}\leq \lVert (I-J_\epsilon)(|u^{(\epsilon)}|^2)\rVert_{L^2}\lVert u^{(\epsilon)} \rVert_{L^\infty}\leq \epsilon^2 \lVert \partial_{xx}(|u^\epsilon|^2)\rVert_{L^2}\lVert u^{(\epsilon)}\rVert_{L^\infty}\\
&\lesssim \epsilon^2 \lVert u^{(\epsilon)} \rVert_{H^1}\lVert u^{(\epsilon)} \rVert_{H^2}^2\leq \epsilon^2 C(\lVert u_0\rVert_{H^2}).
\end{align*}

Hence the Duhamel contribution by $F$ is at most $C(\lVert u_0\rVert_{H^4})\epsilon^2 t$ for $t\in [0,T]$ and $0<\epsilon<\epsilon_0$, for some $\epsilon_0=\epsilon_0(T,u_0)>0$.

To estimate $\lVert N(v)(\tau)\rVert_{H^s}$, let $s^\prime = \lceil -s \rceil$. Then

\begin{align*}
\lVert |u|^2 v \rVert_{H^s}\lesssim \lVert |u|^2 \rVert_{H^{s^\prime}}\lVert v \rVert_{H^s}\lesssim \lVert u \rVert_{H^{s^\prime}}^2\lVert v \rVert_{H^s} \leq C(\lVert u_0\rVert_{H^{s^\prime}})\lVert v \rVert_{H^s},
\end{align*}

where the first inequality is by [Theorem 8.1]behzadan2015multiplication. Similarly by Theorem II and the complete integrability of cubic NLS, we obtain

\begin{align*}
\lVert N(v)(\tau)\rVert_{H^s}\leq C(\lVert u_0\rVert_{H^{s^\prime}})\lVert v(\tau) \rVert_{H^s},
\end{align*}

for $t\in [0,T]$ and for sufficiently small $\epsilon$. This yields

\begin{align*}
\lVert v(t)\rVert_{H^s}\leq \epsilon^2 C(\lVert u_0\rVert_{H^2}) t+\int_0^t C(\lVert u_0\rVert_{H^{s^\prime}})\lVert v(\tau)\rVert_{H^s}d\tau,
\end{align*}

which by Gronwall's inequality implies

\begin{align}
\lVert v(t)\rVert_{H^s}\leq \epsilon^2\frac{C(\lVert u_0\rVert_{H^2})}{C(\lVert u_0\rVert_{H^{s^\prime}})} \Big(e^{C(\lVert u_0\rVert_{H^{s^\prime}})t}-1\Big), \forall t \in [0,T].
\end{align}

Pick $\epsilon^\prime>0$ small enough so that $\epsilon^\prime\lesssim R$ and $T_{\epsilon^\prime}\lesssim T$ where $T_{\epsilon^{\prime}}$ is as in lemma $4.3$. Then for every $0<\delta\ll\epsilon^\prime$, there exists a pair of classical solutions (to NLS flow) $u_1,u_2$ such that $\lVert f_1-f_2\rVert_{H^s}\lesssim \delta$, where $f_i = u_i(0), i=1,2$, and $\lVert u_1(T_{\epsilon^\prime})-u_2(T_{\epsilon^\prime})\rVert_{H^s}\simeq \epsilon^\prime$. Fix any such $\delta>0$. Let $T=T_{\epsilon^{\prime}}$ in $(37)$. From Gronwall's inequality, there exists $\epsilon_0>0$ such that if $\epsilon \in (0,\epsilon_0)$, we obtain

\begin{align*}
\lVert u^{(\epsilon)}[f_1](T_{\epsilon^\prime})-u^{(\epsilon)}[f_2](T_{\epsilon^\prime})\rVert_{H^s}\gtrsim \frac{\epsilon^\prime}{2},
\end{align*}

which proves the desired claim.

§5. Soliton Solutions in High Dimensions.

We consider solutions of the form $u^{(\epsilon)}(x,t) = e^{i\tau t} Q_{\epsilon,\tau}(x)$ where $u$ satisfies $(4)$, which after substitution yields

\begin{align}
\epsilon^2\Delta^2 Q_{\epsilon,\tau} -\Delta Q_{\epsilon,\tau} - J_\epsilon(|Q_{\epsilon,\tau}|^2)Q_{\epsilon,\tau} + \tau Q_{\epsilon,\tau}=0,
\end{align}

where we consider $\tau>0$ and assume $Q_{\epsilon,\tau}$ is real-valued; we drop the subscript notation whenever $\epsilon,\tau$ are fixed. Whereas Fang-Segata-Wu use the method of constrained minimisation to obtain groud state solutions in $d=1,2,3$ (see <cit.> and references therein), we use the mountain-pass theorem to obtain ground state solutions in $1\leq d \leq 9$. Our method directly comes from that in <cit.> where they studied the soliton solutions to a Schrödinger-type equation generated by the second harmonic generation, a nonlinear optical process that has applications in laser physics. Due to a sufficient amount of overlap with the work of Zhao-Zhao-Shi, we give a sketch of our proof, highlighting the key difference in technical details that rises from a different choice in the function space due to the singular perturbation from the bi-Laplacian operator.

We approach the analysis of $(13)$ from variational point of view. For real-valued $u\in H^2(\mathbb{R}^d)$, $v\in H^1(\mathbb{R}^d)$, define the action functional:

\begin{align*}
I(u,v) = \int_{\mathbb{R}^d}\frac{\epsilon^2 (\Delta u)^2}{2}+\frac{|\nabla u|^2}{2}+\frac{\tau u^2}{2}+ \frac{\epsilon^2 |\nabla v|^2}{4}+\frac{v^2}{4}-\frac{u^2v}{2} dx
\end{align*}

where for convenience, we denote the integrand as $\mathscr{L}$ for Lagrangian. Then formally, we obtain the following Euler-Lagrange equation:

\begin{align}
\begin{cases}
\epsilon^2\Delta^2 u - \Delta u + \tau u -uv=0\\
-\epsilon^2\Delta v + v - u^2=0,
\end{cases}
\end{align}

where we require $(u,v)$ to vanish at the infinity. We are interested in the existence of strictly positive solution that obtains the minimum value of the action.

Proposition 5.1. For $1\leq d \leq 9$. there exists a pair $(\overline{u},\overline{v})\in H^2(\mathbb{R}^d)\oplus H^1(\mathbb{R}^d)$ of smooth, strictly positive solution to $(14)$ that minimises the action. There exists no non-trivial smooth solution when $d\geq 12$.

From the proposition above, $\overline{u}$ is our desired solution for $(13)$. Since the action functional $I$ is sufficiently smooth, in fact $I\in C^2(\mathscr{H},\mathbb{R})$ where $\mathscr{H}=H^2(\mathbb{R}^d)\oplus H^1(\mathbb{R}^d)$, every weak solution to $(14)$ equipped with the boundary condition $\lim\limits_{|x|\rightarrow\infty}u(x)=\lim\limits_{|x|\rightarrow\infty}v(x)=0$ rises from the critical points of the action, or the collection of $x\in \mathscr{H}$ such that $I^\prime(x)=0$.[By $\oplus$, we mean the Hilbert space direct sum where $\langle (u,v),(\phi,\psi)\rangle_{\mathscr{H}}=\langle u,\phi\rangle_{H^2}+\langle v,\psi\rangle_{H^1}$. We shall drop the subscript notation whenever the context is clear. Having equipped $\mathscr{H}$ with an inner product, we can discuss how regular the action is. Recall that $I\in C^1(\mathscr{H},\mathbb{R})$ if for every $x\in\mathscr{H}$, there exists unique $I^\prime(x)\in \mathscr{H}^\ast$ such that $I(x+h)=I(x)+\langle I^\prime(x),h\rangle+o(h)$ as $h\xrightarrow[]{\mathscr{H}} 0$ where $\langle \cdot,\cdot\rangle$ is the Hilbert space dual pairing, and the map $x\mapsto I^\prime(x)$ is continuous under the Hilbert space topology. Further recall that $I\in C^2(\mathscr{H},\mathbb{R})$ if $I\in C^1(\mathscr{H},\mathbb{R})$ and for every $x\in \mathscr{H}$, there exists unique $I^{\prime\prime}(x)\in \mathscr{B}(\mathscr{H},\mathscr{H}^\ast)$ such that $I^{\prime}(x+h) = I^\prime(x)+I^{\prime\prime}(x)h+o(h)$ as $h\xrightarrow[\mathscr{H}]{}0$, and the map $x\mapsto I^{\prime\prime}(x)$ continuous where $\mathscr{B}(\mathscr{H},\mathscr{H}^\ast)$ is endowed with the norm operator topology.] Moreover since the linear part of the Lagrangian has a moderate growth rate, in fact quadratic in $u,v$ and their higher derivatives, whereas the nonlinear part is smooth in $(u,v)$, every weak solution is a classical solution. Hence our goal is to find a ground state $(\overline{u},\overline{v})\in \mathscr{H}\setminus \left\{0\right\}$ such that $I^\prime(\overline{u},\overline{v})=0$ and $I(\overline{u},\overline{v}) = \inf\left\{I(u,v): I^\prime(u,v)=0, (u,v)\in\mathscr{H}\setminus\left\{0\right\}\right\}$. We first check that $I$ is well-defined.

Lemma 5.2. Let $1\leq d \leq 10$. For all $u,v \in H^2(\mathbb{R}^d),\:w\in H^1(\mathbb{R}^d)$, we have $\lVert uvw\rVert_{L^1}\lesssim_d \lVert u \rVert_{H^2}\lVert v \rVert_{H^2}\lVert w \rVert_{H^1}$. This estimate fails for $d\geq 11$.

Since the estimates in the proof below are used throughout, we include them explicitly.

proof of Lemma 5.2. For $d=1,2$, a straightforward application of Sobolev embedding yields

\begin{align*}
\lVert uvw\rVert_{L^1}\leq \lVert u \rVert_{L^3}\lVert v \rVert_{L^3}\lVert w \rVert_{L^3}\lesssim \lVert u \rVert_{H^2}\lVert v \rVert_{H^2}\lVert w \rVert_{H^1}.
\end{align*}

Likewise for $d=3,\dots,10$,

\begin{align*}
\lVert uvw\rVert_{L^1}\leq \lVert u \rVert_{L^{\frac{4d}{d+2}}}\lVert v \rVert_{L^{\frac{4d}{d+2}}}\lVert w \rVert_{L^{\frac{2d}{d-2}}}\lesssim \lVert u \rVert_{H^2}\lVert v \rVert_{H^2}\lVert w \rVert_{H^1}.
\end{align*}

However the nonlinear estimate $\lVert uvw\rVert_{L^1}\lesssim \lVert u \rVert_{H^2}\lVert v \rVert_{H^2}\lVert w \rVert_{H^1}$ fails for $d\geq 11$. To see this, let $\hat{\phi}\in C^\infty_c(\mathbb{R}^d)$ be a smooth radial bump with the Fourier support $|\xi|\in [0,1]$. Let $\hat{u}(\xi) = \hat{\phi}(\xi)-\hat{\phi}(2\xi)$. For $k\in\mathbb{N}$, define $u_k(x) = \mathcal{F}^{-1}[\hat{u}(\frac{\cdot}{2^k})](x) = 2^{dk}u(2^kx)$. An explicit computation yields

\begin{align*}
\lVert u_k\rVert_{L^3}^3 \simeq 2^{2dk};\: \lVert u_k\rVert_{H^s}\simeq 2^{(s+\frac{d}{2})k},
\end{align*}

where $s\in \mathbb{R}$ and the implicit constant only depends on $\phi$. If the nonlinear estimate were to hold, then

\begin{align*}
2^{2dk}\lesssim 2^{2(2+\frac{d}{2})k}2^{(1+\frac{d}{2})k}=2^{(5+\frac{3d}{2})k}\Rightarrow 1\lesssim 2^{(5-\frac{d}{2})k} \Rightarrow d \leq 10.
\end{align*}

The version of mountain-pass theorem that we use [Theorem 1.15]willem1997minimax assumes that the action is $C^2$, which can be checked by the following direct computation:

Lemma 5.3. Let $(u,v),(\phi,\psi), h=(h_1,h_2)\in \mathscr{H}$. Then the first two derivatives are as follows:

\begin{align*}
\langle I^\prime (u,v),(\phi,\psi)\rangle&=\int \epsilon^2\Delta u \Delta \phi + \nabla u \cdot \nabla \phi+\tau u \phi+ \frac{\epsilon^2}{2}\nabla v\cdot \nabla \psi + \frac{v\psi}{2} - \frac{u^2\psi}{2} - uv\phi.\\
\langle I^{\prime\prime}(u,v)h,(\phi,\psi)\rangle&=\langle I^\prime (h_1,h_2),(\phi,\psi)\rangle - \int (uh_2+h_1v)\phi+uh_1\psi.
\end{align*}

The non-existence of non-trivial solutions in $d\geq 12$ immediately follows from the following Pohozaev's identity, which can be proved by multiplying the first and second equation of $(14)$ by $x\cdot\nabla u$ and $x\cdot\nabla v$, respectively, and integrating by parts.

\begin{align}
-2\epsilon^2(d-4)\int (\Delta u)^2 - 2(d-2)\int |\nabla u|^2 - 2\tau d \int u^2 - \epsilon^2 (d-2)\int |\nabla v|^2 - d \int v^2+ 2d \int u^2 v=0.
\end{align}

If $(u,v)\in\mathscr{H}$ is a critical point, then $\langle I^\prime(u,v),(u,v)\rangle=0$, and writing this explicitly,

\begin{align}
\int \epsilon^2 (\Delta u)^2 + |\nabla u|^2 +\tau u^2 +\frac{\epsilon^2}{2}|\nabla v|^2+ \frac{v^2}{2}-\frac{3}{2}u^2v =0.
\end{align}

Combining $(15)$ and $(16)$, we obtain

\begin{align}
(8-\frac{2d}{3})\epsilon^2\int (\Delta u)^2 + (4-\frac{2d}{3})\int |\nabla u|^2 + (2-\frac{d}{3})\epsilon^2 \int |\nabla v|^2=\frac{2\tau d}{3}\int u^2+\frac{d}{3}\int v^2,
\end{align}

from which we deduce that for $d\geq 12$, only trivial solution $(0,0)$ exists, and this proves the second part of proposition $5.1$. Note that if $\epsilon=0$, then non-trivial solutions do not exist in $d\geq 6$.

For $d=11$, the action is unbounded and for $d=10$, we happen to be at the borderline case of the concentration-compactness lemma, a crucial tool in our argument, and therefore we focus on $1\leq d \leq 9$.

Lemma 5.4 [Lemma 1.1]lions1984concentration. Let $p\in (1,\infty]$ with $p\neq \frac{2d}{d+2}$ if $p<d$. Let $p^* = \frac{dp}{d-p}$ if $p<d$ and $p^*=\infty$ if $p\geq d$. Suppose $\left\{u_n\right\}\subseteq L^2(\mathbb{R}^d)$ is bounded and $\left\{\nabla u_n\right\}\subseteq L^p(\mathbb{R}^d)$ is bounded. If $\lim\limits_{n\rightarrow\infty}\sup_{y\in\mathbb{R}^d}\int_{B(y,1)}|u_n|^2=0$, then $u_n\xrightarrow[n\rightarrow\infty]{}0$ in $L^\alpha(\mathbb{R}^d)$ for all $\alpha \in (2,p^*)$.

Before we apply the mountain-pass argument on $I$, we recall some useful notions. For $c\in\mathbb{R}$, a sequence $\left\{x_n\right\}\subseteq\mathscr{H}$ is called a Palais-Smale sequence at $c$, or $(PS)_c$ sequence for short, if $I(x_n)\xrightarrow[n\rightarrow\infty]{}c$ and $I^\prime(x_n)\xrightarrow[n\rightarrow\infty]{\mathscr{H}^\ast}0$. In practice, we generate a sequence of approximate solutions that is of $(PS)_c$ where $c$ is defined such that we can minimise the action using the properties of Nehari manifold:

\begin{align*}
N\coloneqq\left\{x\in\mathscr{H}\setminus \left\{0\right\}:\langle I^\prime(x),x\rangle=0\right\}.
\end{align*}

Lemma 5.5. If $x\in N$, then $I(x) = \max\limits_{t\geq 0}I(tx)$.
Moreover let $c\coloneqq \inf\limits_{\gamma\in \Gamma} \max\limits_{t\in [0,1]}I(\gamma(t))$ where $\Gamma =\left\{\gamma\in C([0,1],\mathscr{H}): \gamma(0)=0,I(\gamma(1))<0\right\}$. Then we have the following minimax identity:

\begin{align*}
c=\inf\left\{I(u,v):(u,v)\in N\right\} = \inf\left\{I(u,v): I^\prime(u,v)=0, (u,v)\in\mathscr{H}\setminus\left\{0\right\}\right\}.
\end{align*}

Remark. The analogue of the result above is proved in <cit.>. However it is straightforward to check that essentially the same proof works for our Lagrangian that has an additional Laplacian term.

A non-trivial critical point is necessarily an element in the Nehari manifold though the converse need not be true. The following lemma is likely to be well-known, but we provide a proof for the sake of completion.

Lemma 5.6. If $x \in N$ and $I(x)=c$, then $I^\prime (x)=0$.

proof of Lemma 5.6. Assume $I^\prime(x)\neq 0$. By continuity of $I^\prime$, there exists a small neighbourhood $S\subseteq \mathscr{H}$ around $x$ such that $\left\{I^\prime(y):y\in S\right\}$ is uniformly bounded away from $0\in \mathscr{H}^*$. By the following deformation lemma ([Lemma 2.3]willem1997minimax), there exists $\eta \in C([0,1]\times \mathscr{H},\mathscr{H})$ that lowers the sublevel set of $I$ at the value $c$; more precisely, we will use properties (i), (v) and (vi) of [Lemma 2.3]willem1997minimax.

On the other hand, by Lemma $5.2$ we obtain $I(u,v)\gtrsim \lVert (u,v)\rVert_{\mathscr{H}}^2$, and hence $I$ has a strict local minimum at the origin, which implies $c>0$. As in [Lemma 2.4]zhao2015higher, one can show that $f(t) \coloneqq I(tx)$ is a $C^2$-function that obtains the global maximum at $t=1$, $\lim\limits_{t\rightarrow\infty}f(t) = -\infty$ and $f^{\prime\prime}(t)<0$ for all $t\in (0,\infty)$. We consider $g(t)\coloneqq I(\eta(1,tx))$ as a function of $t\in [0,\infty)$. By (v), $g(t)\leq f(t)$. Near the origin, by (i) we have $g(t)=f(t)$ and therefore is strictly increasing. Moreover for all $t>T$ for $T>0$ sufficiently large, $g(t)=f(t)$ and therefore monotonically decreases to $-\infty$. Around $t=1$, $g(t)<c$ by (vi). Note that $\gamma(t)\coloneqq\eta(1,tx)$, or more precisely an appropriate re-parameterisation of $\gamma$, contributes to the mountain-pass value, and therefore

\begin{align*}
c\leq \max_{t \geq 0} g(t)<c.
\end{align*}

proof sketch of Proposition 5.1. We first claim that $I$ has at least one non-trivial critical point $(\overline{u},\overline{v})\in \mathscr{H}\setminus\left\{0\right\}$ such that $I(\overline{u},\overline{v})\leq c$. Invoking the definition of $c$ defined in Lemma $5.5$, we apply the mountain-pass theorem to obtain a $(PS)_c$ sequence $(u_n,v_n)$, which is bounded in $\mathscr{H}$. By compactness, there exists $\delta\geq 0$ and a subsequence of $(u_n,v_n)$ (without re-labelling the index) such that

\begin{align*}
\delta = \lim_{n\rightarrow\infty}\sup_{y \in \mathbb{R}^d} \int_{B_1(y)} u_n^2+v_n^2.
\end{align*}

If $\delta>0$, then our argument proceeds verbatim as in [Theorem 2.2]zhao2015higher, thereby proving our claim; the sequence $(u_n,v_n)$ weakly converges to some non-trivial critical point $(\overline{u},\overline{v})$ up to translation and by Fatou's lemma, one can show $I(\overline{u},\overline{v})\leq c$. Then by Lemma $5.5$, we have $I(\overline{u},\overline{v})=\inf\left\{I(u,v): I^\prime(u,v)=0, (u,v)\in\mathscr{H}\setminus\left\{0\right\}\right\}$, a ground state solution. Going back to the equation $(14)$, $\overline{v}>0$ by strong maximum principle. If $|\overline{u}|\in H^2(\mathbb{R}^d)$, then it is straightforward to check $I(\overline{u},\overline{v}) = I(|\overline{u}|,\overline{v})$ and $(|\overline{u}|,\overline{v})\in N$, which by Lemma $5.6$ implies $I^\prime(|\overline{u}|,\overline{v})=0$. We can choose $\overline{u}\geq 0$ without loss of generality, which then implies $\overline{u}>0$ by another application of strong maximum principle. Hence it suffices to show $|\overline{u}|\in H^2(\mathbb{R}^d)$ and $\delta>0$[If $\epsilon=0$, then the appropriate energy class is $H^1(\mathbb{R}^d)$, and it is generally true that $f\in H^1(\mathbb{R}^d)\Rightarrow |f|\in H^1(\mathbb{R}^d)$. This is not true when $\epsilon>0$ and thus the appropriate energy class is $H^2(\mathbb{R}^d)$.].

Recall that $\overline{u}$ is a classical solution that satisfies $A\overline{u}\coloneqq(\epsilon^2 \Delta^2 - \Delta +\tau) \overline{u} = J_\epsilon(\overline{u}^2)\overline{u}$ pointwise. Hence $|\overline{u}|\in H^1(\mathbb{R}^d)$ satisfies $A|\overline{u}| = J_\epsilon(\overline{u}^2)|\overline{u}|$ a.e., or more precisely, on an open subset $\left\{x\in\mathbb{R}^d:\overline{u}\neq 0\right\}$. Since $A$ is a linear elliptic operator of order $4$, if $J_\epsilon(\overline{u}^2)|\overline{u}|\in H^\sigma(\mathbb{R}^d)$ for some $\sigma \leq 1$, then $|\overline{u}|\in H^{\sigma+4}(\mathbb{R}^d)$. In fact, this process can be iterated infinitely many times to show $|\overline{u}|\in H^\infty(\mathbb{R}^d)\coloneqq \bigcap\limits_{s\in\mathbb{R}}H^s(\mathbb{R}^d)$[Since $\lVert \overline{u}^2\rVert_{H^2}\lesssim \lVert \overline{u}\rVert_{H^2}\lVert \overline{u}\rVert_{L^\infty}<\infty$, we have $J_\epsilon(\overline{u}^2)\in H^4(\mathbb{R}^d)$ by Lemma $2.3$. By keeping track of multiplication properties of Sobolev spaces, we can show $J_\epsilon(\overline{u}^2)|\overline{u}|\in H^{-1}(\mathbb{R}^d)$, which implies $|\overline{u}|\in H^3(\mathbb{R}^d)$; see [Theorem 8.1]behzadan2015multiplication. Another iteration of elliptic regularity argument is sufficient to conclude $|\overline{u}|\in H^{\frac{9}{2}+}(\mathbb{R}^d)$, a Sobolev algebra for $1\leq d \leq 9$. We iteratively use this bootstrapping argument to conclude $|\overline{u}|\in H^\infty(\mathbb{R}^d)$.].

If $\delta=0$, then from

\begin{align*}
c&=\lim_{n\rightarrow 0}I(u_n,v_n) = \lim_{n\rightarrow 0} \Big(I(u_n,v_n)-\frac{1}{2}\langle I^\prime(u_n,v_n),(u_n,v_n)\rangle\Big) = \frac{1}{4}\lim_{n\rightarrow 0}\int u_n^2v_n,
\end{align*}

and from Lemma $5.4$, we conclude $c=0$, a contradiction by the strict local minimality of $I$ at the origin. To elaborate, we first consider the case $1\leq d \leq 5$ where

\begin{align*}
\lVert u_n^2 v_n\rVert_{L^1}\leq \lVert u_n\rVert_{L^3}^2\lVert v_n\rVert_{L3}\xrightarrow[n\rightarrow \infty]{}0,
\end{align*}

where the last limit follows from Lemma $5.4$ where we use $(p,\alpha) = (2,3)$. However for $6\leq d \leq 9$, we need to use the fact that $\left\{u_n\right\}$ is bounded not only in $H^1(\mathbb{R}^d)$, but also in $H^2(\mathbb{R}^d)$ to rule out $\delta=0$. In particular, consider the following:

\begin{align*}
\lVert u_n^2v_n \rVert_{L^1}\leq \lVert u_n\rVert_{L^{\frac{4d}{d+2}}}^2\lVert v_n\rVert_{L^{2^*}}\lesssim \lVert u_n\rVert_{L^{\frac{4d}{d+2}}}^2\lVert v_n\rVert_{H^1}\xrightarrow[n\rightarrow\infty]{}0,
\end{align*}

where the last limit follows from Lemma $5.4$, where we use $(p,\alpha)=(\frac{2d}{d-2},\frac{4d}{d+2})$, since $2<\frac{4d}{d+2}<\frac{2d}{d-4}$, and from an apriori uniform bound of $\left\{v_n\right\}$ in $H^1(\mathbb{R}^d)$. Unfortunately $d=10$ is the borderline case where the concentration compactness lemma does not apply.

§6. Appendix.

proof of Lemma 2.8. The first statement is a hyperbolic trigonometric representation of a cubic root for a unique real root, which can be verified by a direct substitution.

For the second statement, since $|r(-\xi_1)|\geq |r(\xi_1)|$ for all $\xi_1 \geq 0$, it suffices to show $|r(\xi_1)|\gtrsim_\epsilon |\xi|^{4/3}$ for $\xi_1\geq 0$. Observe that $|r|$ is a decreasing function on $\xi_1\in (0,\xi)$ since $P(0)=\Big(1+\epsilon^2(\xi_1-\xi)^2\Big)(\xi_1-\xi)^2+\tau$ is decreasing and $\partial_{\xi_2}P(0) = |\xi_1|^{2/3}(\epsilon^2\xi_1^2+2)$ is increasing on $\xi_1\in(0,\xi)$; sketch a graph to see this. Hence for $\xi_1 \in (0,\xi]$,

\begin{align*}
|r(\xi_1)|&\gtrsim_{\epsilon} |\xi|^{1/3}\langle \epsilon \xi\rangle \sinh\bigg(\frac{1}{3}\sinh^{-1}\Big(3\sqrt{3}\epsilon\frac{\frac{d_\epsilon(\xi)}{2}}{|\xi|(\epsilon^2\xi^2+2)^{3/2}}\Big)\bigg)\gtrsim_\epsilon |\xi|^{4/3}.
\end{align*}

For $\xi_1 \in (\xi,2\xi]$,

\begin{align*}
|r(\xi_1)|&\simeq_{\epsilon} |\xi_1|^{1/3}\langle \epsilon \xi_1\rangle \sinh\bigg(\frac{1}{3}\sinh^{-1}\Big(3\sqrt{3}\epsilon\frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\Big)\bigg)\nonumber\\
&\geq |\xi|^{1/3}\langle \epsilon \xi\rangle \sinh\bigg(\frac{1}{3}\sinh^{-1}\Big(3\sqrt{3}\epsilon\frac{\frac{d_\epsilon(\xi)}{2}}{2|\xi|(4\epsilon^2\xi^2+2)^{3/2}}\Big)\bigg)
\gtrsim_\epsilon |\xi|^{4/3},
\end{align*}

where the second inequality holds since for $\xi>1$,

\begin{align*}
\frac{\frac{d_\epsilon(\xi)}{2}}{2|\xi|(4\epsilon^2\xi^2+2)^{3/2}}\gtrsim \frac{d_\epsilon(\xi)}{\xi^4}\gtrsim_\epsilon 1.
\end{align*}

For $\xi_1 \in (2\xi,\infty)$, we make use of $\frac{\xi_1-\xi}{\xi_1}\geq \frac{1}{2}$ to show that the argument inside $\sinh^{-1}$ is bounded below by a positive constant as follows:

\begin{align*}
\frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}} \geq \frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\gtrsim_\epsilon \frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2}{\xi_1^4}\gtrsim_\epsilon 1,
\end{align*}

and this proves our claim as in the previous case for $\xi_1\in (\xi,2\xi)$.

To show the third statement, we recall that $|r|$ is decreasing for $\xi_1\in [0,\xi)$, and therefore for such $\xi_1$

\begin{align*}
|r(\xi_1)|\leq |r(0)|\simeq_{\epsilon}(d_\epsilon(\xi)+\tau)^{1/3}\lesssim d_\epsilon(\xi)^{1/3}
\end{align*}

Furthermore for $\xi_1 \in [0,\frac{\xi}{2})$, we claim $|r(-\xi_1)|\lesssim_\epsilon |r(\xi_1)|$. Consequently, $|r(\xi_1)|\lesssim_\epsilon d_\epsilon(\xi)^{1/3}$ for $|\xi_1|\in [0,\frac{\xi}{2})$.

It suffices to show that there exists a constant $C>0$ such that the following holds:

\begin{align*}
\frac{\sinh\bigg(\frac{1}{3}\sinh^{-1}\Big(3\sqrt{3}\epsilon\frac{(1+\epsilon^2(\xi_1+\xi)^2)(\xi_1+\xi)^2+\tau}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\Big)\bigg)}{\sinh\bigg(\frac{1}{3}\sinh^{-1}\Big(3\sqrt{3}\epsilon\frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\Big)\bigg)}\leq C
\end{align*}

Let $X,Y$ be the arguments inside the numerator and denominator of the hyperbolic sine, respectively. Then,

\begin{align*}
\frac{\sinh(X)}{\sinh(Y)} &= \frac{e^X-e^{-X}}{e^Y-e^{-Y}}\leq \frac{e^X}{e^Y-e^{-Y}}\lesssim_\epsilon \frac{e^X}{e^Y}
\end{align*}

where the last inequality follows since $Y$ is bounded below by a positive constant since

\begin{align*}
\frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\geq \frac{\frac{d_\epsilon(\xi)}{2}}{\frac{|\xi|}{2}(\frac{\epsilon^2\xi^2}{4}+2)^{3/2}}\gtrsim_\epsilon 1.
\end{align*}

Then using the following identity

\begin{align*}
\sinh^{-1}(t) = \ln (t+\sqrt{1+t^2}), \forall t \in \mathbb{R},
\end{align*}

and letting $\alpha_1,\alpha_2$ be the arguments in the numerator and denominator of $\sinh^{-1}$, respectively,

\begin{align*}
\bigg(\frac{e^X}{e^Y}\bigg)^3&=\frac{\alpha_1+\sqrt{1+\alpha_1^2}}{\alpha_2+\sqrt{1+\alpha_2^2}}\leq \frac{\alpha_1+\sqrt{1+\alpha_1^2}}{\alpha_2}\lesssim_\epsilon \frac{\alpha_1}{\alpha_2},
\end{align*}

since $\alpha_1$ is bounded below by a positive constant (similar to $Y\gtrsim_{\epsilon} 1$). Note that our hypothesis on $\xi_1$ implies

\begin{align*}
-3\leq\frac{\xi_1+\xi}{\xi_1-\xi}\leq -1.
\end{align*}

\begin{align*}
\frac{\alpha_1}{\alpha_2}=
\frac{(1+\epsilon^2(\xi_1+\xi)^2)(\xi_1+\xi)^2+\tau}{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}&= \frac{(1+\epsilon^2(\xi_1+\xi)^2)(\xi_1+\xi)^2}{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}+\frac{\tau}{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}\\
&\leq \frac{(1+\epsilon^2(\xi_1+\xi)^2)(\xi_1+\xi)^2}{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2}+1\leq 9\Big(1+\Big(\frac{\xi_1+\xi}{\xi_1-\xi}\Big)^2\Big)+1\leq 91,
\end{align*}

as desired.

For $\xi_1>\frac{\xi}{2}$, we claim $\frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+\tau}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\lesssim\frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+d_\epsilon(\xi)}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\lesssim_\epsilon 1$, which yields the claim.
Note that our hypothesis on $\xi_1$ implies

\begin{align*}
-1\leq \frac{\xi_1-\xi}{\xi_1}\leq 1,
\end{align*}

from which

\begin{align*}
\frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2+d_\epsilon(\xi)}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}&=\frac{(1+\epsilon^2(\xi_1-\xi)^2)(\xi_1-\xi)^2}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}+\frac{d_\epsilon(\xi)}{|\xi_1|(\epsilon^2\xi_1^2+2)^{3/2}}\\
&\lesssim \frac{\Big(1+\epsilon^2(\xi_1-\xi)^2\Big)(\xi_1-\xi)^2}{\epsilon^3\xi_1^4}+c_\epsilon\lesssim \frac{1}{\epsilon^3\xi^2}+\frac{1}{\epsilon}+c_\epsilon\lesssim_{\epsilon}1,
\end{align*}

as desired. Arguing as above, one can show $|r(-\xi_1)|\lesssim_\epsilon |\xi_1|^{1/3}\langle\epsilon\xi_1\rangle$ for $\xi_1>\frac{\xi}{2}$ since for such $\xi_1$, $1\leq \frac{\xi_1+\xi}{\xi_1}\leq 3$.

Missing 'biblatex' package
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journaltitlearXiv preprint arXiv:1512.07379
titleMultiplication in Sobolev spaces, revisited

journaltitleJournal of Functional Analysis
titleConvolutions of singular measures and applications to the Zakharov system

IOP Publishing

titleOn the 2D Zakharov system with L2 Schrödinger data

journaltitleComptes Rendus de l'Académie des Sciences-Series I-Mathematics
titleDispersion estimates for fourth order Schrödinger equations

journaltitleGeometric and Functional Analysis
titleFourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations

Hindawi Publishing Corporation

journaltitleInternational Mathematics Research Notices
titleOn wellposedness of the Zakharov system

American Mathematical Soc.

titleSemilinear schrödinger equations

journaltitleTaiwanese Journal of Mathematics
titleLow regularity global well-posedness for the quantum Zakharov system in 1D

Johns Hopkins University Press

journaltitleAmerican journal of mathematics
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journaltitleTransactions of the American Mathematical Society
titleLow regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

Mathematical Sciences Publishers

journaltitleAnalysis & PDE
titleSmoothing and global attractors for the Zakharov system on the torus

journaltitleZeitschrift für angewandte Mathematik und Physik
titleThe fourth-order nonlinear Schrödinger limit for quantum Zakharov system

journaltitleJournal of Mathematical Analysis and Applications
titleOn the standing waves of quantum Zakharov system

The Mathematical Society of the Republic of China

journaltitleTaiwanese Journal of Mathematics
titleSemi-classical Limit for the Quantum Zakharov System

journaltitlePhysics of Plasmas
titleModified Zakharov equations for plasmas with a quantum correction

journaltitleJournal of Functional Analysis
titleOn the Cauchy problem for the Zakharov system

journaltitleZeitschrift für angewandte Mathematik und Physik
titleGlobal well-posedness and the classical limit of the solution for the quantum Zakharov system

Springer Science & Business Media

titleQuantum plasmas: An hydrodynamic approach

journaltitlePhysical Review E
titleQuantum and classical dynamics of Langmuir wave packets

journaltitlearXiv preprint arXiv:1412.2882
titleOn one dimensional quantum Zakharov system

journaltitleJournal of the American Mathematical Society
titleA bilinear estimate with applications to the KdV equation

journaltitleJournal of Functional Analysis
titleOn the Zakharov and Zakharov-Schulman systems

Duke University Press

journaltitleDuke Mathematical Journal
titleOn the ill-posedness of some canonical dispersive equations

journaltitleTransactions of the American Mathematical Society
titleQuadratic forms for the 1-D semilinear Schrödinger equation

journaltitlearXiv preprint arXiv:1806.10066
titleA remark on norm inflation for nonlinear Schr$\backslash$" odinger equations

booktitleAnnales de l'Institut Henri Poincare (C) Non Linear Analysis
titleThe concentration-compactness principle in the calculus of variations. The locally compact case, part 2

Research Institute forMathematical Sciences

journaltitlePublications of the Research Institute for Mathematical Sciences
titleExistence and smoothing effect of solutions for the Zakharov equations

Khayyam Publishing, Inc.

journaltitleDifferential and Integral Equations
titleThe nonlinear Schrödinger limit and the initial layer of the Zakharov equations

journaltitleCommunications in Mathematical Physics
titleThe nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence

American Mathematical Soc.

titleNonlinear dispersive equations: local and global analysis

Springer Science & Business Media

titleMinimax theorems

journaltitleSov. Phys. JETP
titleCollapse of Langmuir waves

journaltitleCalculus of Variations and Partial Differential Equations
titleHigher dimensional solitary waves generated by second-harmonic generation in quadratic media