WEAK SOLVABILITY AND WELL-POSEDNESS OF A COUPLED SCHRODINGER-KORTEWEG-DE-VRIES EQUATION FOR CAPILLARY-GRAVITY WAVE INTERACTIONS

An interaction equation of the capillary-gravity wave is considered. W e show that the Cauchy problem of the coupled Schrodinger-KdV equation , [GRAPHICS] is locally well-posed for weak initial data u(0) x v(0) i s an element of L-2(R) x H-1/2(R). We apply the analogous method for e stimating the nonlinear coupling terms developed by Bourgain and refin ed by Kenig, Ponce, and Vega.