Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential

Abstract

Abstract In this paper, the nonlinear wave equation with singular Legendre potential utt−uxx+VL(x)u+mu+perturbation=0 $$ u_{tt}-u_{xx}+V_{L}(x)u + mu + \mathit{perturbation}=0 $$ subject to certain boundary conditions is considered, where m is a positive real number and VL(x)=−12−14tan2x $V_{L}(x)=-\frac{1}{2}-\frac{1}{4}\tan ^{2} {x}$, x∈(−π2,π2) $x\in (-\frac{\pi }{2},\frac{\pi }{2})$. By means of the partial Birkhoff normal form technique and infinite-dimensional Kolmogorov–Arnold–Moser theory, it is proved that, for every m∈R+∖{14} $m\in \mathbb{R}_{+}\setminus \{\frac{1}{4}\}$, the above equation admits plenty of quasi-periodic solutions with three frequencies.

Keywords

• Kolmogorov–Arnold–Moser theory
• Quasi-periodic solutions
• Singular differential operator
• Birkhoff normal form