ON SMOOTH SOLUTIONS TO THE INITIAL-VALUE PROBLEM FOR THE MIXED NONLINEAR SCHRODINGER-EQUATIONS

Solutions to the initial value problem for the mixed nonlinear Schrodinger equation u(t) = i-alpha-u(xx) + beta-u2u(x)BAR + gamma\u\2 u(x) + ig(\u\2)u are considered. Conditions on the constants alpha, beta, gamma, function g(.) and initial data u(x, 0) are given so that, for this problem, the unique existence of smooth solutions is proved. In addition, the decay behaviours of the smooth solutions as \x\ --> + infinity are discussed.