Soliton Solutions to the Coupled Gerdjikov–Ivanov Equation with Rogue-Wave-Like Phenomena

Abstract

Bilinear forms of the coupled Gerdjikov–Ivanov equation are derived. The N-soliton solutions to the equation are obtained by Hirota’s method. It is interesting that the two-soliton solutions can generate the rogue-wave-like phenomena by selecting special parameters. The equation can be reduced to the Gerdjikov–Ivanov equation as well as its bilinear forms and its solutions.

Shifted nonlocal Kundu type equations: Soliton solutions

“…which is only valid for β = − 1 4 corresponding to CLL system (3) and ( 4), and for β = 0 corresponding to GI system ( 5) and (6). Soliton solutions of the GI system were also studied in [47] by Hirota method. But the Hirota bilinear form of the system and so the solutions given in [47] are not correct.…”

“…Soliton solutions of the GI system were also studied in [47] by Hirota method. But the Hirota bilinear form of the system and so the solutions given in [47] are not correct.…”

The relationship between the conservation laws and multi‐Hamiltonian structures of the Kundu equation

“…From Zhang et al, 26 we know that the operator Φ is hereditary and strong symmetrical and the isospectral hierarchy and nonisopectral hierarchy form an infinite-dimensional 𝜏-symmetry Lie algebra. Furthermore, the hierarchy (10) includes the Kaup-Newell equation, 27 the Chen-Lee-Liu equation, 28 the Gerdjikov-Ivanov equation, 29,30 the modified Korteweg-de Vries equation, the Sharma-Tasso-Olever equation, 31 and a new equation as special reductions. Now, let us work out the CLs from the Lax pairs.…”

Optical solitons to a perturbed Gerdjikov-Ivanov equation using two different techniques

“…A spectral problem and the associated perturbed GI hierarchy [32] of nonlinear evolution equations is presented and shown that the GI hierarchy is integrable in a Liouville sense and possesses bi-Hamiltonian structure. Numerous efficient and influential methods have been projected for obtaining solutions of GI equation, such as algebra-geometric solutions [33], soliton hierarchy [34], bifurcations and travelling wave [35], bright and dark soliton solutions [36], Darboux transformations [37] and many more being studied for more than a decade [38][39][40][41][42][43][44][45][46][47][48], and Kaur and Wazwaz [49] obtain the optical solitons for perturbed GI equation. In this paper we are going to apply the balanced modified extended tanh-function (METF) method as a new technique to get new exact solution for the perturbed GI equation.…”