Rational Solutions for the Fokas System

Fokas system is the simplest (2+1)-dimensional extension of the nonlinear Schrödinger 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 τn (n = 0, 1) whose elements m_i,j⁽ⁿ⁾ (n = 0, 1; 1 ≤ i, j ≤ N) are involved with order-n_i and order-n_j derivatives. When N = 1, three kinds of rational solution, i.e., fundamental lump and fundamental rogue wave (RW) with n₁ = 1, and higher-order rational solution with n₁ ≥ 2, are illustrated by explicit formulas from τn (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 n₁ = n₂ = ··· = n_N = 1, several specific multi-rational solutions are given graphically.