Chin. Phys. B, 2017, Vol. 26(5): 050504    DOI: 10.1088/1674-1056/26/5/050504
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# Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation

Zhen Wang(王振)1, Yupeng Qin(秦玉鹏)1, Li Zou(邹丽)2,3
1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
2 School of Naval Architecture, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China;
3 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China
Abstract  We construct the Hirota bilinear form of the nonlocal Boussinesq (nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nlBq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nlBq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form. The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.
Keywords:  nonlocal Boussinesq equation      periodic wave solution      solitary waves      Riemann theta function
Received:  05 January 2017      Revised:  22 January 2017      Published:  05 May 2017
 PACS: 05.45.Yv (Solitons) 03.75.Lm (Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations) 02.30.Mv (Approximations and expansions)
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 51579040, 51379033, and 51522902), the National Basic Research Program of China (Grant No. 2013CB036101), and Liaoning Natural Science Foundation, China (Grant No. 201602172).
Corresponding Authors:  Zhen Wang     E-mail:  wangzhen@dlut.edu.cn