Darboux Transformation Of The Dnls Hierarchy And Explicit Solutions Of Four (2+1)-dimensional Soliton Equations

The soliton theory is an important part of the nonlinear science. There are many nonlinear partial differential equations that have soliton properties in the pure and applied science. Therefore, solving soliton equations (especially, (2+1)-dimensional equations) is important in the field of theory and application.With the help of Darboux transformation, the explicit solutions of the (1+1)-dimensional DNLS equations can be obtained. Through the relation between (2+1)-dimensional DNLS, MKP-type, CMKP-type equations and the (1+1)-dimensional DNLS equations, we get the explicit solutions of the DNLS, MKP-type, CMKP-type equations. The equations areandThere are four sections in the paper. The first section is an introduction to the history of the soliton theory and the Darboux transformation. In the second section, we shall decompose (2+1)-dimensional DNLS, MKP-type, CMKP-type equations into the first two nontrivial (1+1)-dimensional soliton equations in DNLS hierarchy. In the third section, we study the Darboux transformation of the (1+1)-dimensional DNLS equations and obtain their Darboux Matrix and the explicit solutions of the (2+1)-dimensional DNLS, MKP-type, CMKP-type equations. The Darboux Matrix iswithand A_t,B_m,C_n,D_t(0≤k≤N;0≤m≤N-1;0≤n≤N-2)are the functions of x, y, t with A_ND_N=1,A^*_N-1=D_N-1,B^*_N-1=C_N-2. The four section, taking Q = 0, we get the explicit solutions of the (1+1)-dimensional DNLS equations as N = 1, 2. In final section, we obtain the explicit solutions of the (2+1)-dimensional DNLS, MKP-type, CMKP-type equations as N=1, 2.