Exact Solutions for a Coupled Korteweg–de Vries System

2.1 Rational Solutions

In order to obtain the rational solutions for (1), we assume that

where l_m′s (m=0, 1, 2, 3) and ξ are all the real numbers. Substituting Expression (10) into Expressions (7 and 8), we derive out the rational solutions of (1),

2.2 Soliton Solutions

In order to attain the first-order soliton-like solutions of (1), we assume that

where η₁=k₁x+w₁t+δ₁ and γ=kx−wt, while Q₁, P₁, R, k₁, w₁, k, w, and δ₁ are all the real constants. Substituting Expression (13) into Expressions (7 and 8), we derive out the relationship between k₁, w₁, k, and w as

which is called the nonlinear dispersion relation [32].

Similarly, in order to derive the second-order soliton-like solutions, we set

where η_j=k_jx+w_jt+δ_j (j=1, 2), while Q₂, P₂, k₂, w₂, and δ₂ are all the real constants. Substituting Expression (15) into Expressions (7 and 8), we have

Thus, we can obtain the second-order soliton-like solutions for (1) as

where u21=3(3−σ)[−kRsin[k(2kt+x)]+k1(−e−xk1 − 2tk13 − δ1P1 + exk1 + 2tk13 + δ1Q1)+k2(−e−xk2 − 2tk23 − δ2P2+exk2 + 2tk23 + δ2Q2)]2+3(σ−2) [1+Rcos[k(2kt+x)]+e−xk1 − 2tk13 − δ1P1+e−xk2 − 2tk23 − δ2P2+exk1 + 2tk13 + δ1 Q1+exk2 + 2tk23 + δ2Q2] [−k2Rcos[k(2kt+x)]+k12(e−xk1 − 2tk13 − δ1P1 + exk1 + 2tk13 + δ1Q1)+ k22(e−xk2 − 2tk23 − δ2P2+exk2 + 2tk23 + δ2Q2)],

u22={1+Rcos[k(2kt+x)]+e−xk1 − 2tk13 − δ1P1+e−xk2 − 2tk23 − δ2P2 + exk1 + 2tk13 + δ1Q1+exk2 + 2tk23 + δ2Q2}2,

v21=3{−[−kRsin[k(2kt+x)]+k1(−e−xk1 − 2tk13 − δ1P1+exk1 + 2tk13 + δ1Q1) + k2(−e−xk2 − 2tk23 − δ2P2+exk2 + 2tk23 + δ2Q2)]2+[1+Rcos[k(2kt+x)] + e−xk1 − 2tk13 − δ1P1+e−xk2 − 2tk23 − δ2P2+exk1 + 2tk13 + δ1Q1+exk2 + 2tk23 + δ2Q2] [−k2Rcos[k(2kt+x)]+k12(e−xk1 − 2tk13 − δ1P1+exk1 + 2tk13 + δ1Q1) + k22(e−xk2 − 2tk23 − δ2P2+exk2 + 2tk23 + δ2Q2)]}.

This process can be continued for us to derive the third-order soliton-like solutions, which can be written as

where u31=3(3−σ){−kRsin[k(2kt+x)]−e−xk3 − 2tk33 − δ3k3P3 + k1(−e−xk1 − 2tk13 − δ1P1+exk1 + 2tk13 + δ1Q1)+k2(−e−xk2 − 2tk23 − δ2P2 + exk2 + 2tk23 + δ2Q2)+exk3 + 2tk33 + δ3k3Q3}2+3(−2+σ) {1+Rcos[k(2kt+x)]+e−xk1 − 2tk13 − δ1P1+e−xk2 − 2tk23 − δ2P2 + e−xk3 − 2tk33 − δ3P3+exk1 + 2tk13 + δ1Q1+exk2 + 2tk23 + δ2Q2 + exk3 + 2tk33 + δ3Q3}{−k2Rcos[k(2kt+x)]+e−xk3 − 2tk33 − δ3k32P3 + k12(e−xk1 − 2tk13 − δ1P1+exk1 + 2tk13 + δ1Q1)+k22(e−xk2 − 2tk23 − δ2P2 + exk2 + 2tk23 + δ2Q2)+exk3 + 2tk33 + δ3k32Q3},

u32=1+Rcos[k(2kt+x)]+e−xk1 − 2tk13 − δ1P1+e−xk2 − 2tk23 − δ2P2 +e−xk3 − 2tk33 − δ3P3+exk1 + 2tk13 + δ1Q1+exk2 + 2tk23 + δ2Q2 +exk3 + 2tk33 + δ3Q3,

v31=3{−(−kRsin[k(2kt+x)]−e−xk3 − 2tk33 − δ3k3P3+k1(− e−xk1 − 2tk13 − δ1P1+exk1 + 2tk13 + δ1Q1)+k2(−e−xk2 − 2tk23 − δ2P2+ exk2 + 2tk23 + δ2Q2)+exk3 + 2tk33 + δ3k3Q3)2+(1+Rcos[k(2kt+x)]+ e−xk1 − 2tk13 − δ1P1+e−xk2 − 2tk23 − δ2P2+e−xk3 − 2tk33 − δ3P3+exk1 + 2tk13 + δ1Q1+ exk2 + 2tk23 + δ2Q2+ exk3 + 2tk33 + δ3Q3)(−k2Rcos[k(2kt+x)]+ e−xk3 − 2tk33 − δ3k32P3 + k12(e−xk1 − 2tk13 − δ1P1+exk1 + 2tk13 + δ1Q1)+ k22(e−xk2 − 2tk23 − δ2P2+ exk2 + 2tk23 + δ2Q2)+ exk3 + 2tk33 + δ3k32Q3)}.

If we continue such process, the Nth-order soliton-like (N≥1) solutions of (1) are

where f=1+∑j = 1N(Qjekjx + 2kj3t + δj+Pje−kjx − 2kj3t − δj)+Rcosγ, while Q_j, P_j, k_j and δ_j (j=1, 2, …, N) are all the real constants.

We note that the values of Q_j, P_j, and R can affect the shape, propagation, and interaction of the solitons. Thus, three type shapes (soliton-like, kink and bell, anti-bell, and bell shapes) of the soliton solutions for (1) will be obtained in the following.

Special case I

If Q_jP_jR≠0 in Expressions (21 and 22), the N-order soliton-like solutions of (1) will be obtained. For example, when Q₁=P₁=R=1, the first-order soliton-like solutions are

where γ=kx+2k³t and η1=k1x+2k13t+δ1.

Special case II

If we take R=P_j=f_j=0 and ϵ=Q_j=1 in Expressions (21 and 22), the Nth-order kink and bell-shape soliton-like solutions for (1) can be attained. Such as, the one-order solutions are

Special case III

If we take R=f_j=0 and ϵ=Q_j=P_j=1 in Expressions (21 and 22), the Nth-order anti-bell and bell-shape soliton solutions for (1) can be obtained. For example, the one solitons are

3.1 Propagation and Interaction of the Special Case I

Figure 1 displays that the propagation of the soliton-like solutions. Potential u and v are both have the periodic properties, which are caused by the term R cos γ in Expression (13).

Figure 1:

First-order soliton-like solutions for u and v via Expressions (23 and 24) with the parameters k₁=0.9, k=1.2, δ₁=σ=0, Q₁=P₁=1, R=0.6, and (a) of u; (b) of v.

Figure 2 shows the interaction of the second-order soliton-like solutions. Figures 1 and 2 display that the potential u is a dark soliton-like while the potential v is a bright soliton-like.

Figure 2:

Second-order soliton-like solutions for u and v via Expressions (21 and 22) with the parameters N=2, k₁=0.5, k₂=1.1, k=1.6, σ=δ₁=δ₂=0, Q₁=Q₂=P₁=P₂=1, R=0.6, and (a) of u; (b) of v.

3.2 Propagation and Interaction of the Special Case II

Figure 3 shows that the shape of u is affected by the parameter σ. Figure 3(b) exhibits that the potential u is a shock wave when σ=2, while the potential u is composed of two shock waves when σ=0 as displayed in Figure 3(a) and (c). Besides, the direction of the one soliton-like propagation is decided by the sign of k₁, i.e. Figure 3(a) and (c) have the counter directions of the soliton-like propagation, when k₁ has the different sign.

Figure 3:

One soliton-like solutions for u via Expression (25) with the parameters δ₁=1.3 and (a) of k₁=1.4, σ=0; (b) of k₁=1.4, σ=2; and (c) of k₁=−1.4, σ=0.

Figure 4 shows that the parameters k₁ and σ have no effect on the shape and direction of the soliton-like propagation for the potential v. Besides, potential v is a bell shape soliton-like.

Figure 4:

One soliton-like solutions for v via Expression (26) with the the same parameters as Figure 3, accordingly.

Figures 5 and 6 display the interaction of the two soliton-like solutions. Figures 5 and 6 exhibit that the potential u and v both have the fusion phenomena, when the parameters are taken as above.

Figure 5:

Two soliton-like solutions for u and v via Expressions (17 and 18) with the parameters k₁=1.4, k₂=0.9, Q₁=Q₂=1, P₁=P₂=R=σ=δ₁=δ₂=0, and (a) of u; (b) of v.

Figure 6:

Two soliton-like solutions for u and v via Expressions (17 and 18) with the parameters k₁=1.4, k₂=0.9, Q₁=Q₂=1, σ=2, P₁=P₂=R=δ₁=δ₂=0, and (a) of u; (b) of v.

3.3 Propagation and Interaction of the Special Case III

Figure 7 shows that the potential u is the anti-bell shape while potential v is a bell-shape soliton. Figures 1(a), 3, and 7(a) exhibit that the shape of the potential u changes with the values of Q₁, P₁, and R, while the potential v is a bright soliton in Figures 1(b), 4, and 7(b).

Figure 7:

One-soliton solutions for u and v via Expressions (27 and 28) with the parameters k₁=1.6, σ=0, and (a) of u; (b) of v.

Figure 8 exhibits that the interaction of the two solitons are elastic, i.e. the velocity and direction are invariable before and after interaction.

Figure 8:

Two-soliton solutions for u and v via Expressions (21 and 22) with the parameters n=2, k₁=1.6, k₂=0.5, Q₁=Q₂=P₁=P₂=1, σ=R=δ₁=δ₂=0, and (a) of u; (b) of v.