good illustration of the combination of the three theories [@bib:hall2; @bib:hall6; @bib:hall9; @bib:hall10]. The existence and form of these classical solutions are fairly independent of the details of the theory, making them useful to probe the string behavior. In fact these solitons are the (lower-dimensional) D-branes of string theory manifested in a field theory limit while still capturing many string features.

Starting from the celebrated paper of Gopakumar, Minwalla and Strominger [13]{}, there are many works to study soliton solutions of noncommutative field theory and integrable systems in the background of noncommutative spaces \[23-30\]. Although Derrick’s theorem forbids solitons in ordinary 2+1 dimensional scalar field theory [@bib:33], solitons in noncommutative scalar field theory on the plane were constructed in terms of projection operators in [@13]. It was soon realized that noncommutative solitons represent D-branes in string field theory with a background $B$ field, and many of Sen’s conjectures [@18; @19] regarding tachyon condensation in string field theory have been beautifully confirmed using properties of noncommutative solitons. Gopakumar, Minwalla and Strominger makde an important finding that in a noncommutative space, a projector may correspond to a soliton in the field theory [@13], which proves the significance of the study of projection 