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 sub(n) (n = 0, 1) whose elements m super((n)) sub(i,j) (n = 0, 1; 1 less than or equal to i, j less than or equal to N) are involved with order-n sub(i) and order-n sub(j) derivatives. When N = 1, three kinds of rational solution, i.e., fundamental lump and fundamental rogue wave (RW) with n sub(1) = 1, and higher-order rational solution with n sub(1) greater than or equal to 2, are illustrated by explicit formulas from tau sub(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 [Lt] 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 greater than or equal to 2 and n sub(1) = n sub(2) = . . . = n sub(N) = 1, several specific multi-rational solutions are given graphically.