Rational Solutions for the Fokas System

Fokas system is the simplest (2+1)-dimensional extension of the nonlinear Schrodinger equation (Eq. (2), Inverse Problems 10 (1994) L19-L22). By using the bilinear transformation method, general rational solutions for the Fokas system are given explicitly in terms of two order-N determinants tau(n) (n = 0, 1) whose elements m(i,j)((n)) (n = 0, 1; 1 <= <= N) are involved with order-n, and order-n, derivatives. When N = 1, three kinds of rational solution, i.e., fundamental lump and fundamental rogue wave (RW) with n(1) >= 1, and higher-order rational solution with ni > 2, are illustrated by explicit formulas from tau(n) (n = 0, 1) and pictures. The fundamental RW is a line RW possessing a line profile on (x, y)-plane, which arises from a constant background with at t << 0 and then disappears into the constant background gradually at t >> 0. The fundamental lump is a traveling wave, which can preserve its profile during the propagation on (x, y)-plane. When N >= 2 and n(1) = n(2) = ... = n(N) = 1, given graphically.