Quasideterminant solutions of 2-component non–commutative complex coupled integrable dispersionless system

We present a non–commutative extension of two-component complex coupled integrable disspersionless (NC-CCID) system. We studied matrix Darboux transformation and generate multiple soliton solutions by using the notation of quasideterminants. Further, we obtain single and double soliton solutions in terms of quasideterminants for NC-CCID system. In the commutative limit, we obtain soliton solutions for CCID system as ratio of ordinary determinants.


Introduction
The nonlinear differential equations (integrable or non-integrable) describe various phenomenon in physics and mathematics, such as, plasma physics, fiber optics, condensed matter physics, fluid mechanics, string theory etc [1][2][3]. During last three decades, there has been an increasing interest in the study of dispersionless integrable systems, due to their abundant applications in the fields of mathematics and physics such as string theory, theories of fields, conformal maps over complex plane, theory of solitons, etc [4][5][6][7][8]. Ordinary integrable systems with a dispersion term reduce into dispersionless integrable systems under semi-classical limits. The coupled integrable dispersionless (CID) system is an important equation of two-dimensional field theory. The coupled integrable equations have attracted a great deal of attraction during recent past [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In this work, we present a non-commutative extension 2 of two-component complex coupled integrable dispersionless (NC-CCID) system: If we take * = r r (a real scalar field), we obtain a non-commutative real coupled integrable dispersionless (NC-CID) system recently reported in [21]: which represents usual commutative two-component CCID system [22]. The NC-CCID system (1.1) can be represented as consistency condition of following Wadati Konno Ichikawa (WKI) type Lax pair 4 and the 4×4 coefficients U V , are given by , , along with Note: O 2 and I 2 are 2×2 null and identity matrices respectively. The integrability condition of (1.5)-(1.6) becomes a zero-curvature condition i.e.
If we substitute U and V into (1.8) we obtain coupled system (1.1). Darboux transformation (DT) is a powerful solution generating technique [23][24][25][26][27][28]. In this paper, we apply DT and construct multiple soliton solutions of 2-component NC-CCID system (1.1) in terms of quasideterminants. In section 2, we apply DT on j (a matrix-valued solution) to the linear system (1.5)-(1.6) and generate multi-soliton solutions in terms of quasideterminants [29][30][31]. In commutative limit, we obtain multi-soliton solutions as ratios of ordinary determinants for CCID system (1.4). In section 3, we compute explicit expressions of single and double soliton solutions for NC-CCID system (1.1). In order to illustrate our results we also draw different profiles of soliton solutions for CCID system (1.4). Section 4, deals with concluding remarks and future work.

Darboux transformation and multiple soliton solutions
DT enables us to compute a family of new solutions to linear eigenvalue problem from a known one. If j is a known solution of linear system (1.5)-(1.6), then under the action of DT the new solution j [ ] 1 is given by and Γ be a 4×4 matrix-valued function to be determined. The new solution also satisfies system (1.5)-(1.6), that is, Lets choose Γ-matrix as l l l l The matrix of particular solutions Ω satisfies following linear system It is straightforward to verify that Γ given by (2.10) satisfies both conditions (2.5) and (2.6). Therefore, G = WL W is the required one-fold DT for NC-CCID system (1.1).
Using properties of quasideterminants (for more details see Appendix), one can re-express equation ( Note: I 4 and O 4 are 4×4 identity and null matrices respectively. The M-times successive application of DT on j leads into the following quasideterminant formula Similarly, M-times iteration of DT on matrix-valued potential Z becomes Further expressions (2.20)-(2.21) give us following transformations on scalar potentials q and r i : From above transformations, we obtain following reduction conditions The quasideterminant formulae (2.22) permit us to compute multiple soliton solutions of NC-CCID as well as for commutative CCID. In next section, we'll compute explicit expressions of single and double solitons by using quasideterminant formulae (2.22). The integration of above system yields

Explicit soliton solutions
, here α be a constant of integration.

Single-soliton solution
To construct an explicit expression of single-soliton solution take M=1 into (2.22), we obtain For one-soliton solution, we have The matrix element G ( ) 11 1 is given by Now substitute G ( ) Similarly, one can easily compute following expressions In commutative limit, equations (3.6)-(3.8) reduce into ratios of ordinary determinants, i.e, In what follows next, we would like to compute an explicit expression of single-soliton solutions in terms of elementary function. Let us take a a a a ñ = -   The reduction conditions (2.23) allow us to take * l l = -  Similarly, one can easily verify other conditions given by expression (2.23). Using G G  represents single-soliton solution for commutative CCID system (1.4). The different profiles of dark and bright soliton solutions (3.11)-(3.13) have been plotted in figure 1.
Our results are consistent with the results obtained in [18].

Double-soliton solution
For M=2 equation      The above expressions represent interaction of two-soliton for NC-CCID system (1.1). In commutative limit, we obtain 3.15 phenomena of dispersionless slow light propagations in periodic medium. It would be fascinating to compute soliton solutions of other non-commutative and supersymmetric integrable systems by using matrix Darboux transformation.