Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities

Abstract

We study the Cauchy problem to the semilinear fourth-order Schrödinger equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t u+\partial _x^4u=G\left( \left\{ \partial _x^{k}u\right\} _{k\le \gamma },\left\{ \partial _x^{k}{\bar{u}}\right\} _{k\le \gamma }\right) , &{} t>0,\ x\in {\mathbb {R}},\\ \ \ \ u|_{t=0}=u_0\in H^s({\mathbb {R}}), \end{array}\right. }\quad \quad (4\mathrm{NLS}) \end{aligned}$$

where $\gamma \in \{1,2,3\}$ and the unknown function $u=u(t,x)$ is complex valued. In this paper, we consider the nonlinearity G of the polynomial

$$\begin{aligned} G(z)=G(z_1,\ldots ,z_{2(\gamma +1)}) :=\sum _{m\le |\alpha |\le l}C_{\alpha }z^{\alpha }, \end{aligned}$$

for $z\in {\mathbb {C}}^{2(\gamma +1)}$, where $m,l\in {\mathbb {N}}$ with $3\le m\le l$ and $C_{\alpha }\in {\mathbb {C}}$ with $\alpha \in ({\mathbb {N}}\cup \{0\})^{2(\gamma +1)}$ is a constant. The purpose of the present paper is to prove well-posedness of the problem (4NLS) in the lower order Sobolev space $H^s({\mathbb {R}})$ or with more general nonlinearities than previous results. Our proof of the main results is based on the contraction mapping principle on a suitable function space employed by Pornnopparath (J Differ Equ, 265:3792–3840, 2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by Bejenaru et al. (Ann Math 173:1443–1506, 2011). Moreover we discuss scattering of global solutions and the optimality for the regularity of our well-posedness results, namely we prove that the flow map is not smooth in several cases.

Derivation of an important 4NLS model with third order derivative nonlinearities

In this appendix, we derive the important 4NLS model with third order derivative nonlinearities ($\gamma =3$), that is, (1.1)–(1.5), from the second ($n=2$) of the derivative nonlinear Schrödinger (DNLS) hierarchy (1.6). To describe the DNLS hierarchy (1.6) more precisely, we give the definitions of several notations. For a complex-valued function $u=u(x)$ on ${\mathbb {R}}$, we introduce a ${\mathbb {C}}^2$-valued function $U=U(x)$ on ${\mathbb {R}}$ defined by $U:=(u,{\overline{u}}){}^{\text {T}}$. Let $\sigma _3$ be the third Pauli matrix given by

$$\begin{aligned} \sigma _3:=\begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad -1 \\ \end{pmatrix}. \end{aligned}$$

For a ${\mathbb {C}}^2$-valued smooth function $(v,w){}^{\text {T}}$ on ${\mathbb {R}}$, we introduce a first order differential operator ${\mathfrak {D}}_1$ defined by

$$\begin{aligned} {\mathfrak {D}}_1\begin{pmatrix} v \\ w \\ \end{pmatrix}(x) :=\sigma _3\frac{d}{dx}\begin{pmatrix} v \\ w \\ \end{pmatrix}(x) =\begin{pmatrix} v_x(x) \\ -w_x(x)\\ \end{pmatrix}. \end{aligned}$$

Moreover, for a ${\mathbb {C}}^2$-valued smooth function $(v,w){}^{\text {T}}$ decaying 0 as $|x|\rightarrow \infty$, we introduce a linear operator ${\mathfrak {D}}_2$ defined by

$$\begin{aligned} {\mathfrak {D}}_2\begin{pmatrix} v \\ w \\ \end{pmatrix}(x)&:=-U(x)\int _x^{\infty }U(y)^*\frac{d}{dy}\begin{pmatrix} v(y) \\ w(y) \\ \end{pmatrix}dy\\&=-\begin{pmatrix} u(x) \\ {\overline{u}}(x)\\ \end{pmatrix} \int _x^{\infty }\left\{ \overline{u(y)}v_y(y)+u(y)w_y(y)\right\} dy. \end{aligned}$$

We note that for a smooth function $u=u(x)$ decaying 0 as $|x|\rightarrow \infty$, the identity

$$\begin{aligned} {\mathfrak {D}}_2U(x) =\begin{pmatrix} |u(x)|^2u(x) \\ |u(x)|^2\overline{u(x)}\\ \end{pmatrix} \end{aligned}$$

holds for any $x\in {\mathbb {R}}$. Indeed, this identity follows from the following identities

$$\begin{aligned} -\int _x^{\infty }\left\{ \overline{u(y)}u_y(y)+u(y)\overline{u_y(y)}\right\} =-\int _x^{\infty }\frac{d}{dy}|u(y)|^2dy=|u(x)|^2. \end{aligned}$$

For a smooth ${\mathbb {C}}^2$-valued function $(v,w){}^{\text {T}}$ decaying 0 as $|x|\rightarrow \infty$, we define the recursion operator $\Lambda$ given by

$$\begin{aligned} \Lambda \begin{pmatrix} v \\ w\\ \end{pmatrix} :=\frac{i}{2}\left( {\mathfrak {D}}_1+i{\mathfrak {D}}_2\right) \begin{pmatrix} v \\ w\\ \end{pmatrix}. \end{aligned}$$

(A.1)

By using this operator, we can write the n-th of the derivative nonlinear Schrödinger hierarchy as

$$\begin{aligned} i\partial _t U(t,x)+\partial _x\left\{ (-2i\Lambda )^{2n-1}U\right\} (t,x)=0,\ \ \ (t,x)\in {\mathbb {R}}\times {\mathbb {R}}, \end{aligned}$$

(A.2)

where $U=U(t,x)=\left( u(t,x),\overline{u(t,x)}\right) {}^{\text {T}}$ is a smooth solution decaying 0 as $|x|\rightarrow \infty$ and $n\in {\mathbb {N}}$.

In the following, we only consider the case of $n=2$. By a simple calculation, the identity

$$\begin{aligned} (-2i\Lambda )^3&=({\mathfrak {D}}_1+i{\mathfrak {D}}_2)^3\\&={\mathfrak {D}}_1^3 -{\mathfrak {D}}_1{\mathfrak {D}}_2^2 -{\mathfrak {D}}_2({\mathfrak {D}}_1{\mathfrak {D}}_2+{\mathfrak {D}}_2{\mathfrak {D}}_1)\\&\quad +i\left\{ {\mathfrak {D}}_1({\mathfrak {D}}_1{\mathfrak {D}}_2+{\mathfrak {D}}_2{\mathfrak {D}}_1)+{\mathfrak {D}}_2{\mathfrak {D}}_1^2-{\mathfrak {D}}_2^3\right\} \end{aligned}$$

holds. By a simple calculation and taking the first component of the Eq. (A.2), we can derive the Eqs. (1.1) with (1.5).