The Generalized Coupled Kdv Equation Darboux Transformations And Exact Solutions

The soliton equation is one of the most prominent subject in the fields of nonliear science. In this paper, we consider the generalized coupled Korteweg-de Vries (KdV) soliton equationIt is well known that there are several systematic approaches to obtain solutions of soliton equations. Darboux transformation (DT) has been proved to be one of the most natural and beautiful method to get explicit solutions of some soliton equations from a trivial seed.In section 1, we introduce the initial Darboux transformation method and Darboux matrix method. Based on these methods, in the following paper, three kinds Darboux transformations are constructed.In section 2, consider Lax pairswhere u, v are two potentials, A is a constant spectral parameter, / and r are two constant parameters . Using the equation( U and U, V and Fhas the same form, except change u, v into u, v ) Proposition 2.1dx In a = --LiProposition 2.2I ■"V^ r>J ■"■& — LXJ^JV—L V-"- I i ' i ri'the first kind Darboux transformations with multi-parameters are derived/ A BT = T(X) = a[C DwhereN-\ N-\ N-l jV-1A = XN + Y, AkXk, B=Y, B^\ C = E CkX\ D="￡ Dk\\fc=0 Jt=O fc=0 k=0(a, Ak, -Bfc, Ck andDfc, (0 < k < N — 1) are functions of x and t).and give completly proof, from a trivial seed u and v are constants and v^O, we use the first kind Darboux transformation to generate new odd-soliton solutions of the generalized coupled Korteweg-de Vries (KdV) soliton equation, from a trivial seed u — 0, v = |, r = 0, / = — ~, we discuss the first two cases(N=l and N=2). When N = 2, suitably choosing parameters, we plot beautiful three-soliton collision picture, especially v[2] is the three-soliton solution, whose three-soliton collision picture has two peaks. It is a new type for soliton collision picture associated with 2x2 matrix spectral problem.In section 3 has the same Lax pairs with In section 2, we use the same equation and use Proposition 3.1dx\na = --ezProposition 3.2dt\na = DN_lx- -(lnAN^)xx - -[2ux + {ln-?-±)xx] + ^-[2ux - (lnv)the second kind Darboux transformations with multi-parameters are derived( A BT = T(A) = ayc dwhereJV-l JV-l N-l JV-1k-0 k=0 fc=0 fc=0(a, Ak, -8^, CjkandDfc (0 < A; < A^ — 1) arefunctions of x and ^).and give completly proof, from a trivial seed u and v are constants and v^O, we use the second kind Darboux transformation to generate new odd-soliton solutions of the generalized coupled KdV soliton equation, from a trivial seed u = 0, v = ~, r = 0, / = — |, we discuss the first two cases(N=l and N=2). When N = 2, suitably choosing parameters, we plot beautiful three-soliton collision picture, especially v[2] is the three-soliton solution, whose three-soliton collision picture has two peaks. It is a new type for soliton collision picture associated with 2x2 matrix spectral problem.In section 4 has the same Lax pairs with In section 2, we use the same equation and use Proposition 4.1dx ln(AN) = \dx2 \dx ln(l +2 vl I [v +Proposition 4.2vl I r + 4[k(1 + vl I (vVT(2T[ux - (lnv)xx) + AN_lx.2vlthe third kind Darboux transformations with multi-parameters are derivedT = T(A)A B C DwhereA = AN(X N f BkXk,fc=0 k=0 D = DkXk), k=0 k=0(An, Ak, Bk, Cka,ndDk (0 < k < N — 1) are functions of z and t). and give completly proof, from a trivial seed u and v are constants and v^0, we use the third kind Darboux transformation to generate new even-soliton solutions of the generalized coupled KdV soliton equation, from a trivial seed w = 0, v = ^, r = 0, I = — \, we discuss the first two cases(N=l and N—2). When N = 2, suitably choosing parameters, we plot beautiful four-soliton collision picture, especially v[2] is the four-soliton solution, whose four-soliton collision picture has three or four peaks. It is a new type for soliton collision picture associated with 2x2 matrix spectral problem.