Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 4

Orbital stability of solitary wave solutions of Zakharov–Rubenchik equation

Pages: 693 – 710

DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n4.a4


Xiao Hua Liu (School of Data Science and Information Engineering, Guizhou Minzu University, Guizhou, Guiyang, China)


We study orbital stability of bell-shaped solitary wave solutions of the Zakharov–Rubenchik equation for the interaction of high-frequency and low-frequency waves in an arbitrary medium. Our approach is based on the theories of orbital stability presented by Grillakis, Shatah and Strauss, and relies on a reformulation of the coupled equations in Hamiltonian form. We investigate stability of solitary wave solutions by ascertaining the number of negative eigenvalues of the linear operator and the number of positive eigenvalues of its Hessian of the scalar function.


solitary wave solution, undeterminded coefficient method, orbital stability, the Zakharov–Rubenchik equation

2010 Mathematics Subject Classification

35B35, 35Q20, 76Bxx

This project is supported by Fund of Reform of Teaching Content and Curriculum System in Colleges and Universities of Guizhou Education Department (20161111040), Key Subjects of Graduate Education and Teaching Reform of Guizhou Education Department (JG [2016] 07).

Received 26 November 2015

Published 21 December 2018