Periodic and Quasi-Periodic Solutions for Reversible Unbounded Perturbations of Linear Schrödinger Equations

Abstract

In this paper, we consider a new class of derivative nonlinear Schrödinger equations with reversible nonlinearities of the form

$$\begin{aligned} \mathrm {i}u_t+u_{xx}+|u_x|^{4}u=0,\quad (t, x)\in {\mathbb {R}}\times {\mathbb {T}}. \end{aligned}$$

We obtain real analytic, linearly stable periodic solutions and quasi-periodic ones with two basic frequencies via infinite dimensional Kolmogorov–Arnold–Moser (KAM) theory for reversible systems. By investigating the gauge invariance and the compact form of vector fields, in our KAM iterative procedure, we remove the usual Diophantine restrictions on tangential frequencies and only use the Melnikov non-resonance conditions. In the proof, we also use Birkhoff normal form techniques due to the lack of external parameters in the equation above.

Appendix

1.1 Reversible Vector Field

We first give the properties of flow of vector field. Denote by $\phi _Y ^t$ the flow of analytic vector field $Y : {\mathscr {P}}^{a,p}_{{\mathbb {C}}}\supset D(s,r)\rightarrow {\mathscr {P}}^{a,p}_{{\mathbb {C}}}.$ Suppose there is a subdomain $D(s^ \prime , r^ \prime ) \subset D(s, r)$ such that for each $t \in [-1,1],$$\phi _Y ^t: D(s^ \prime , r^ \prime ) \rightarrow D(s, r)$ is well-defined. Define the transformed vector field of $X: D(s,r)\rightarrow {\mathscr {P}}^{a,q}_{{\mathbb {C}}}$ as follows

$$\begin{aligned} (\phi _Y^t)^*X=(D\phi _Y^t)^{-1}\cdot X\circ \phi _Y^t: D(s^ \prime , r^ \prime )\rightarrow {\mathscr {P}}^{a,q}_{{\mathbb {C}}}. \end{aligned}$$

Since $\frac{d}{dt}(\phi _Y^t)^*X=(\phi _Y^t)^*[X, Y],$ we obtain the following Taylor’s formula of $(\phi _Y ^1)^*X=(\phi _Y^t)^*X|_{t=1}$ with respect to t at 0 :

$$\begin{aligned} (\phi _Y ^1)^*X= & {} \sum \limits ^n_{l=0}\frac{1}{l !}ad^l_YX +\frac{1}{n !}\int \limits _0^1(1-t)^n(\phi _Y^t)^* ad^{n+1}_YX dt,\nonumber \\= & {} \sum \limits ^{\infty }_{l=0} \frac{1}{l!}ad^l_Y X , \end{aligned}$$

(6.1)

here $ad_YX:=[X, Y]$, and $ad^k_Y:=ad_Y ad^{k-1}_Y$, $ad^0_Y:=\text {Id}.$

Definition 6.1

(Reversible vector field) Suppose map S is an involution: $S^2=id.$ Vector field X is called reversible with respect to S (or S-reversible), if

$$\begin{aligned} DS\cdot X=- X\circ S, \end{aligned}$$

i.e.,

$$\begin{aligned} (DS(w))X(w)=-X(S(w)),\quad w\in D(s,r), \end{aligned}$$

where DS is the differential of S.

Lemma 6.1

[28] Suppose vector field X is S-reversible and vector field Y satisfies $Y\circ S = DS\cdot Y$. Then we have

• (1) $\phi _Y ^t \circ S=S \circ \phi _Y ^t.$

• (2) $(\phi _Y ^t)^*X$ and [X, Y] are S-reversible.

1.2 Technical Lemmas

We give several technical lemmas.

Lemma 6.2

[22] The convolution $w*z$ of two complex sequences w, z in $\ell ^{a,p}_{{\mathcal {I}}}$ is defined as $(w*z)_j=\sum \nolimits _{m} w_{j-m}z_m.$ If $a\ge 0,$$p> 1/2,$ then $\ell ^{a,p}_{{\mathcal {I}}}$ is a Hilbert algebra with respect to the convolution of sequences, and $\Vert w*z\Vert _{p}\le c\Vert w\Vert _{p}\Vert z\Vert _{p},$ the constant c depends only on p.

Lemma 6.3

Let $u_j, j=1, \ldots ,m,$ be m complex functions on ${\mathbb {T}}^n$ that are real analytic on $D(s) = \{\theta :|\text {Im} \theta | < s\}.$ Then

$$\begin{aligned} \sum \limits ^m_{j=1}\sup _{\theta \in D(s-\sigma )}|u_j(\theta )|\le \frac{4^n}{\sigma ^n}\sup _{\theta \in D(s)}\sum \limits ^m_{j=1}|u_j(\theta )| \end{aligned}$$

for $0<\sigma <s\le 1.$

Proof

Though the proof is essentially similar to that of Lemma M.2 in [16], we still give it here for convenience of the reader.

We first consider the case $n=1.$ For each $1\le j\le m$ there exists a point $\theta _j$ in the rectangle $Q=\{\theta :|\text {Re } \theta |\le \pi , |\text {Im} \theta |\le s-\sigma \}$ such that

$$\begin{aligned} \sup \limits _{\theta \in D(s-\sigma )}|u_j(\theta )|\le |u_j(\theta _j)|. \end{aligned}$$

By the Cauchy integral formula $u_j(\theta _j)=\frac{1}{2\pi \mathrm {i}}\int _\Gamma \frac{u_j(\zeta )}{\zeta -\theta _j}d\zeta$, here $\Gamma$ represents a rectangle with distance $0<\rho <\sigma$ around Q, independent of j. We then have

$$\begin{aligned} \begin{aligned} \sum \limits ^m_{j=1}\sup _{\theta \in D(s-\sigma )}|u_j(\theta )| \le&\frac{1}{2\pi }\int _\Gamma \sum \limits ^m_{j=1}|\frac{u_j(\zeta )}{\zeta -\theta _j}||d\zeta |\\ \le&\frac{4}{\rho }\sup _{\theta \in D(s)}\sum \limits ^m_{j=1}|u_j(\theta )|. \end{aligned} \end{aligned}$$

Let $\rho \rightarrow \sigma$ and we obtain the estimate for $n=1.$ The case $n>1$ follows similarly by the n-fold Cauchy integral. $\square$

With Lemma 6.3, we get the following lemma.

Lemma 6.4

Let $A=(A_{ij})_{ i\ge 1, 1\le j\le n}: {\mathbb {C}}^n\rightarrow l^2$ be a bounded operator which depends on $\theta \in {\mathbb {T}}^n$ such that all coefficients are analytic on D(s). Suppose $B=(B_{ij})_{ i\ge 1, 1\le j\le n}: {\mathbb {C}}^n\rightarrow l^2$ is another operator depending on $\theta$ whose coefficients satisfy

$$\begin{aligned} \sup _{\theta \in D(s)}|B_{ij}(\theta )|\le \frac{1}{i^p}\sup _{\theta \in D(s)}|A_{ij}(\theta )|,\quad p\ge 1. \end{aligned}$$

Then B is a bounded operator for each $\theta \in D(s),$ and

$$\begin{aligned} \sup _{\theta \in D(s-\sigma )}\Vert B(\theta )\Vert \le \frac{4^{n+1}}{\sigma ^n}\sup _{\theta \in D(s)}\Vert A(\theta )\Vert \end{aligned}$$

for $0<\sigma <s\le 1,$ where the operator norm $\Vert A\Vert =\sup \nolimits _{|v|=1} \Vert Av\Vert _{l^2}$, $|v|=\max \limits _{1\le j\le n}|v_j|.$

Proof

For $\theta \in D(s-\sigma )$ we have by Lemma 6.3

$$\begin{aligned} \begin{aligned} \sum \limits ^n_{j=1}|B_{ij}(\theta )|\le&\sum \limits ^n_{j=1} \frac{1}{i^p}\sup _{\theta \in D(s-\sigma )}|A_{ij}(\theta )|\\ \le&\frac{4^n}{i^p\sigma ^n}\sup _{\theta \in D(s)}\Vert A(\theta )\Vert . \end{aligned} \end{aligned}$$

Hence for $v\in {\mathbb {C}}^n,$$\Vert B(\theta )v\Vert ^2_{l^2}\le (\frac{4^{n+1}}{\sigma ^n}\sup _{\theta \in D(s)}\Vert A(\theta )\Vert )^2|v|^2.$ Thus we get the final estimate. $\square$

Lemma 6.5

[16] Let $A=(A_{ij})_{ i,j\in {\mathbb {Z}}_0}$ be a bounded operator on $\ell ^2$ which depends on $\theta \in {\mathbb {T}}^n$ such that all coefficients are analytic on D(s). Suppose $B=(B_{ij})_{ i,j\in {\mathbb {Z}}_0}$ is another operator on $\ell ^2$ depending on $\theta$ whose coefficients satisfy

$$\begin{aligned} \sup _{\theta \in D(s)}|B_{ij}(\theta )|\le \frac{1}{||i|-|j||}\sup _{\theta \in D(s)}|A_{ij}(\theta )|,\quad i\ne \pm j, \end{aligned}$$

and $B_{\pm jj}=0, j\in {\mathbb {Z}}_0.$ Then B is a bounded operator on $\ell ^2$ for each $\theta \in D(s),$ and

$$\begin{aligned} \sup _{\theta \in D(s-\sigma )}\Vert B(\theta )\Vert \le \frac{4^{n+1}}{\sigma ^n}\sup _{\theta \in D(s)}\Vert A(\theta )\Vert \end{aligned}$$

for $0<\sigma <s\le 1.$