Structure of optical solitons of resonant Schrödinger equation with quadratic cubic nonlinearity and modulation instability analysis

Bilinear Form, Soliton, Breather, Hybrid and Periodic-Wave Solutions for a (3+1)-Dimensional Korteweg-De Vries Equation in a Fluid

“…Hybrid solutions comprising the solitons and breathers for Eq. (1) have been constructed via N -Soliton Solutions (8) under Constraints (12). Via the Riemann theta function, Periodic-Wave Solutions (22) for Eq.…”

“…(1)Hybrid solutions consisting of Θ solitons and Λ breathers for Eq. (1) are derived with the following constraints in Solutions(8):N = 2Λ + Θ, a ν = a * Λ+ν = a ν1 + ia ν2 , b ν = b * Λ+ν = b ν1 + ib ν2 , c ν = c * Λ+ν = c ν1 + ic ν2 , ξ 0 ν = ξ 0 * Λ+ν = ξ 0 ν1 + iξ 0 ν2 , a s = a s1 , b s = b s1 , c s = c s1 , ξ 0 s = ξ 0 s1 , (ν = 1, 2, • • • , Λ, s = 2Λ + 1, 2Λ + 2, • • • , N )(12)where Θ and Λ are the positive integers, a s1 's, b s1 's, c s1 's and ξ 0 s1 's are the real constants. For example, the hybrid solutions comprising the first-order breather and one kink-type soliton are obtained from Solutions(8) with the parameters as N = 3, Λ = 1, Θ = 1, a 1 = a * 2 = a 11 + ia 12 , b 1 = b * 2 = b 11 + ib 12 , c 1 = c * 2 = c 11 + ic 12 , ξ 0 1 = ξ 0 * 2 = ξ 0 11 + iξ 0 12 , a 3 = a 31 , b 3 = b 31 , c 3 = c 31 , ξ 0 31 c 31…”

“…Interaction between the first-order breather and one kink-type soliton via Solutions(8) with α = 3, β = γ = δ = = 1, a 1 = a * 2 = 1 + 2i, b 1 = b * 2 = 1 + i, c 1 = c * 2 = 1 − 1 2 i,a 3 = c 3 = 2, b 3 = −1 and (a) z = 0; (b) y = 0; (c) x = 0. (a 1 ) t = − 1 t = 0 (c 3 ) t = 1 Interaction among the first-order breather and two kink-type solitons via Solutions (8) with α = c 4 = 3, β = γ = δ = = b 3 = a 4 = 1, a 1 = a * 2 = 1 + 2i, b 1 = b * 2 = 1 + i, c 1 = c * 2 = 1 + 1 2 i, a 3 = 2, c 3 = −1, b 4 = −2 and (a) z = 0; (b) y = 0; (c) x = 0.…”

Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg–de Vries equation in a fluid

“…[23], we construct the higher-order breather solutions for Eq. (1) with certain values of the parameters a ı 's, b ı 's, c ı 's, and ξ 0 ı 's in N -Soliton Solutions (8) with N being an even integer. We assume that…”

“…Fluids have been studied in such disciplines as atmospheric science, oceanography and astrophysics [1][2][3][4]. Analytic solutions for the nonlinear evolution equations (NLEEs) such as the soliton, breather and periodic-wave solutions have been applied in nonlinear optics, fluid mechanics and plasma physics [5][6][7][8][9][10]. Methods have been proposed to construct the analytic solutions, such as the bilinear method, Bäcklund transformation and Darboux transformation [11][12][13].…”