Scattering of solutions to the fourth-order nonlinear Schrödinger equation (Q2800058)

The authors consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation NEWLINE\[NEWLINE \begin{cases} i\partial_tu+\frac{1}{4}\partial^4_x u =\lambda| u| ^{\rho-1}u, \quad & t>0, \, x\in\mathbb R,\\ u(0,x) =u_0(x), \quad &x\in\mathbb R,\end{cases}NEWLINE\]NEWLINE where \(\lambda\in\mathbb C\), \(\rho >4\). They work in a weighted Sobolev space with norm NEWLINE\[NEWLINE \|\phi\| = \|\sqrt{1+x^2}\phi\|_{L^2} + \|(1-\partial_x^2)^{\delta/2}\phi\|_{L^2},\quad \delta>1/2.NEWLINE\]NEWLINE The first main theorem states that a unique global solution exists if \(\rho>5\) and \(\| u_0\|\) is small. Moreover, the large time asymptotic is described. The second theorem deals with odd initial data. Then \(\rho>4\) is sufficient.NEWLINENEWLINENEWLINEThe proof is based on the factorization techniques for the fourth-order Schrödinger evolution group and the commutation identities for the gauge invariant nonlinearities with the vector field \(x+it\delta_x^3\).