Darboux transformation, soliton solutions of the variable coefficient nonlocal modified Korteweg–de Vries equation

Integrability, lump solutions, breather solutions and hybird solutions for the (2+1)-dimensional variable coefficient Korteweg-de Vries equation

“…Suppose that q(x, y,t) and q(x, y,t) are two different solutions of Eq. (17). The new transformations…”

“…Compared with the constant coefficient KdV equations, the coefficients in the equations studied are the function(s) of the space variable or time variable. Currently, the main methods for finding the exact solutions of variable coefficient KdV equations include Bäcklund transformation [12,13], Hirota's direct method [14,15], Darboux transformation [16,17], Painlevé analysis [13,18], nonlinear superposition principle [19] and auxiliary equation method [20,21].…”

Darboux transformation, soliton solutions of a generalized variable coefficients Hirota equation

“…In optical fibers, α, β, γ, δ represent the group dispersion velocity, third order dispersion, self-steepening and the amplification or absorption coefficient respectively [27]. Some related works associated with (1.1) have been researched, for example, when α, β, δ are all constants, multisolitons, breathers and rogue waves have been derived, see [28][29][30][31], α = δ = 0, β = β(t), γ = γ(t), the multisoliton solutions have been obtained, see [32][33][34][35], β = 0, α = α(t), δ = δ(t), γ = γ(t), multisoliton solutions, rogue wave solutions, semi-rational solutions, breathers are obtained, please see [36][37][38][39][40] for details.…”