Communications in Mathematical Sciences

Volume 10 (2012)

Number 3

Wilton ripples in weakly nonlinear model equations

Pages: 1015 – 1024

(Fast Communication)



Benjamin F. Akers (Department of Mathematics and Statistics, Air Force Institute of Technology, Wright Patterson Air Force Base, Ohio, U.S.A.)

Wenxuan Gao (Department of Computer Science, University of Illinois at Chicago)


Traveling near-bichromatic solutions supported at resonant frequencies are computed in a family of nonlinear model equations. Wilton’s expansion is calculated for these solutions to all orders, and used as the basis for a perturbative numerical method for computing near-bichromatic traveling waves. The perturbative method relies on the analyticity of solutions with respect to wave amplitude. Results are compared to a non-perturbative continuation method, and a method for proving analyticity of solutions is proposed.


water waves, nonlinear waves, resonances

2010 Mathematics Subject Classification

35B34, 74J30, 76B15

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