General rogue waves in the three-wave resonant interaction systems

Abstract

General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction, respectively. It is shown that while the first family of solutions associated with a simple root exists for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, the ones generated by a $2\times 2$ block determinant in the double-root case, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained.

General rogue wave solutions to the Sasa-Satsuma equation

“…Remark 3. The function σ k (k = 0, 1) in Theorem 2.3 is a polynomial in x and t. It can be computed that for the N -th order rogue wave, the degree of σ k (k = 0, 1) is 2N (N + 1) for both variables x and t. Since the computations are very similar to those developed by Yang and Yang (see Appendix A in [48]), we omit the details.…”

“…then f, g and g satisfy the bilinear equations ( 22). Hence, with τ k = τ k0 , k = 0, 1, rational solutions to the Sasa-Satsuma equation ( 2) are obtained via the transformation (23).…”

“…the derivative NLS equation [22], the three-wave equation [23], the Manakov system [24,25] and the coupled Hirota system [26]. While rogue waves on the periodic background [27,28,29,30] have been extensively studied, rogue waves of infinite order have been revealed [31] with the help of Riemann-Hilbert approach [32].…”

“…We will adopt the Kadomtsev-Petviashvili (KP) hierarchy reduction method. This is a very powerful method to construct explicit solutions of integrable equations and has found extensive applications in the derivation of rogue wave or soliton solutions of various integrable equations [45,17,18,20,22,23]. However, several difficulties are encountered when applying this method to derive rogue wave solutions to the Sasa-Satsuma equation due to its complexity.…”

General higher—order breathers and rogue waves in the two‐component long‐wave–short‐wave resonance interaction model

“…It is known that the multi-component NLS equation admits more RW patterns in contrast to the scalar NLS equation [47][48][49][50][51][52], such as the mixed bounded RWs consisting of RWs of different types, and the degenerate bounded RWs. Very recently, Yang and Yang derived a family of new bounded RW solutions to the three-wave resonant interaction system [61], which consists of…”