Communications in Mathematical Sciences

Volume 17 (2019)

Number 5

Dedicated to the memory of Professor David Shen Ou Cai

Asymmetric behavior of surface waves induced by an underlying interfacial wave

Pages: 1333 – 1351



Shixiao W. Jiang (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Gregor Kovačič (Mathematical Sciences Department, Rensselaer Polytechnic Institute, Troy, New York, U.S.A.)

Douglas Zhou (School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)


We develop a weakly nonlinear model to study the spatiotemporal manifestation and the dynamical behavior of surface waves in the presence of an underlying interfacial solitary wave in a two-layer fluid system. We show that interfacial solitary-wave solutions of this model can capture the ubiquitous broadening of large-amplitude internal waves in the ocean. In addition, the model is capable of capturing three asymmetric behaviors of surface waves: (i) Surface waves become short in wavelength at the leading edge and long at the trailing edge of an underlying interfacial solitary wave. (ii) Surface waves propagate towards the trailing edge with a relatively small group velocity, and towards the leading edge with a relatively large group velocity. (iii) Surface waves become high in amplitude at the leading edge and low at the trailing edge. These asymmetric behaviors can be well quantified in the theoretical framework of ray-based theories. Our model is relatively easily tractable both theoretically and numerically, thus facilitating the understanding of the surface signature of the observed internal waves.


interfacial waves, surface waves, ray-based theory

2010 Mathematics Subject Classification

35L05, 65M22, 76B07, 76B55

Received 5 April 2019

Accepted 20 August 2019

Published 6 December 2019