Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions

Abstract

Higher order and multicomponent generalizations of the nonlinear Schrödinger equation are important in various applications, e.g., in optics. One of these equations, the integrable Sasa-Satsuma equation, has particularly interesting soliton solutions. Unfortunately the construction of multisoliton solutions to this equation presents difficulties due to its complicated bilinearization. We discuss briefly some previous attempts and then give the correct bilinearization based on the interpretation of the Sasa-Satsuma equation as a reduction of the three-component Kadomtsev-Petviashvili hierarchy. In the process we also get bilinearizations and multi-soliton formulae for a two component generalization of the Sasa-Satsuma equation (the Yajima-Oikawa-Tasgal-Potasek model), and for a (2 + 1)-dimensional generalization.

Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation

“…(1) reduces to the standard NLS equation which has only the terms describing lowest order dispersion and self-phase modulation. The soliton solutions have been presented on the zero background in [18][19][20]. Here, we study rational solutions on a CW background,…”

Novel energy sharing collisions of multicomponent solitons

“…A standard bilinearization procedure will result in a greater number of bilinear equations than the number of bilinearising variables, which results in soliton solutions with less number of arbitrary parameters. In order to get more general soliton solutions we introduce an auxiliary function during the bilinearization of the m-CCNLS system which gives equal number of bilinear equations and variables [23][24][25][26][27]. To be more clear with the presentation, we give below the bilinear equations for the m-CCNLS system (5)…”

Oscillatory solitons of U(1)-invariant modified Korteweg-de Vries equations. II. Asymptotic behavior and constants of motion

“…First, the Hirota equation (2.1) and the Sasa-Satsuma equation (2.2) are known to be gauge-equivalent to third-order NLS equations [13] qt ± i v/3(3qxx + α|q| 2 q) + α|q| 2 qx + β(|q| 2 )xq + qxxx = 0 (6.1) through the Galilean-phase transformatioñ t = t,x = x + vt, u(t, x) = q(t,x) exp ± i v/3(x − (2v/3)t) (6.2) where v > 0 is a speed parameter, with α = 24, β = 0 in the Hirota case and α = 12, β = 6 in the Sasa-Satsuma case. Under this transformation, the oscillatory 1-solitons (2. q 1 (ξ 1 ,ξ 2 ) = exp(∓i v/3ξ 1 )f 1 (ξ 1 ,ξ 2 ),q 2 (ξ 1 ,ξ 2 ) = exp(∓i v/3ξ 2 )f 2 (ξ 1 ,ξ 2 ), (6.9) where the functionsf 1 (ξ 1 ,ξ 2 ) andf 2 (ξ 1 ,ξ 2 ) are given in Proposition 2 for the oscillatory 2-solitons (2.14)-(2.21) of the Hirota and Sasa-Satsuma equations.…”

“…[10,11,12,13]. This work amounts to deriving the 1-soliton and 2-soliton formulas in a mathematically equivalent but less physically useful envelope form, without any analysis of the asymptotic behaviour and the constants of motion for these solutions.…”