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# Dynamics of Partial Differential Equations

## Volume 15 (2018)

### Number 3

### Invariant tori for a fifth order nonlinear partial differential equation with unbounded perturbation

Pages: 183 – 199

DOI: http://dx.doi.org/10.4310/DPDE.2018.v15.n3.a2

#### Authors

#### Abstract

In this paper, we are concerned with small perturbation of the nonlinear partial differential equation\[u_t = u_{5x} - \frac{5}{16} (8u^2_{xx} + 8u_x u_{xxx})\]under periodic boundary conditions. Using an abstract infinite dimensional KAM theorem, we obtain the existence of many two-dimensional invariant tori and thus many time quasi-periodic solutions for the above equation under sufficiently small Hamiltonian perturbation.

#### Keywords

quasi-periodic solution, KAM theory, normal form

#### 2010 Mathematics Subject Classification

35Q55, 37K55

Supported by NNSFC (11401041)(11601036), the University Science and Technology Foundation of Shandong Province Education Department (J14LI54) and Binzhou University (BZXYL1406).

Received 5 November 2017