# American Institute of Mathematical Sciences

December  2017, 37(12): 6471-6485. doi: 10.3934/dcds.2017280

## Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation

 1 Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China 2 Faculty of Information Technology, Macau University of Science and Technology, Macau, China

1Corresponding author

Received  June 2017 Published  August 2017

Fund Project: This work was partially supported by NNSFC (No.11671407), FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A030311004).

In this paper, we mainly study the Cauchy problem of an integrable dispersive Hunter-Saxton equation in periodic domain. Firstly, we establish local well-posedness of the Cauchy problem of the equation in $H^s (\mathbb{S}), s > \frac{3}{2},$ by applying the Kato method. Secondly, by using some conservative quantities, we give a precise blow-up criterion and a blow-up result of strong solutions to the equation. Finally, based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation. By using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equations, we get the travelling wave solutions of the original equation.

Citation: Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280
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