Characteristics of the soliton solutions in a generalized
variable-coefficient nonlinear Schr\”{o}dinger equation
with single and double poles
Abstract
A generalized variable-coefficient nonlinear
Schr\”{o}dinger equation is investigated through the
Riemann-Hilbert approach based on inverse scattering transformation with
zero boundary conditions at infinity, and its various soliton solutions
are successfully derived. To derive the eigenfunction and scattering
matrix, and reveal their properties, the direct scattering problem is
studied. Then based on inverse scattering transformation, a
Riemann-Hilbert problem is constructed for the equation. For both cases
of single and double poles, the Riemann-Hilbert problem is solved, and
the formulae of soliton solutions are displayed. Finally, via evaluating
the impact of each parameters, the soliton solutions are analyzed
graphically involving $1$-, $2$- and $3$-soliton solutions.