  Within the (2 + 1)-dimensional Korteweg-de Vries equation framework, new
bilinear Backlund transformation and Lax pair are presented based on the binary
Bell polynomials and gauge transformation. By introducing an arbitrary
function, a family of deformed soliton and deformed breather solutions are
presented with the improved Hirotas bilinear method. Choosing the appropriate
parameters, their interesting dynamic behaviors are shown in three-dimensional
plots. Furthermore, novel rational solutions are generated by taking the limit
of obtained solitons. Additionally, two dimensional [2D] rogue waves (localized
in both space and time) on the soliton plane are presented, we refer to it as
deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as
a 2D analog of the Peregrine soliton on soliton plane, and its evolution
process is analyzed in detail. The deformed 2D rogue wave solutions are
constructed successfully, which are closely related to the arbitrary function.
This new idea is also applicable to other nonlinear systems.
