Multiple-scale analysis on the radiation within the nonlinearly coupled KdV equations Yezheng Li 1 , Joseph A. Biello 2 , Yerong Li 3 1 Applied Mathematics and Computational Science, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, PA 19104. 2 Department of Mathematics, University of California, Davis, CA 95616. 3 Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, CA 91125, USA E-mail: yezheng@sas.upenn.edu, biello@math.ucdavis.edu, yerong.li@caltech.edu Abstract. A multiple scale model of the nonlinearly coupled KdV equations is established to predict mechanism of interaction of equatorial Rossby waves and barotropic waves in certain case. Analytically, predicted precursor radiation is a centrosymmetric object and is shown in excellent quantitative agreement with numerical simulations; furthermore, the multiple scale model elucidates the salient mechanisms of the interaction of solitary waves and the mechanism for radiation. While the atmosphere-ocean science community is very interested in theoretical studies of tropical wave interactions and in developing reduced dynamical models that can explain some key features of equatorial phenomena, our analytic predictions quantitively explain formation of radiation during interaction in Biello’s model beyond qualitative level. PACS numbers: 91.10.Vr AMS classification scheme numbers: 35B40 Keywords : ”precursor” radiation, multiple scale analysis, fast-slow decomposition Submitted to: Nonlinearity arXiv:1609.09574v1 [nlin.PS] 30 Sep 2016