A Five-Component Generalized mKdV Equation and Its Exact Solutions

: In this paper, a 3 × 3 spectral problem is proposed and a ﬁve-component equation that consists of two different mKdV equations is derived. A Darboux transformation of the ﬁve-component equation is presented relating to the gauge transformations between the Lax pairs. As applications of the Darboux transformations, interesting exact solutions, including soliton-like solutions and a solution that consists of rational functions of e x and t , for the ﬁve-component equation are obtained.


Introduction
In the study of nonlinear partial differential equations, the theory on solitons is an indispensable part [1]. A lot of systematic methods, such as the inverse scattering method [2,3], are developed to find exact explicit solutions for soliton equations. In the past few decades, as the soliton theory grew vigorously, solitons has been observed in solid physics, fluid physics, plasma physics, laser physics, condensed matter physics, etc. The research on soliton is an important topic in many physics laboratories. Many published articles are about soliton equations and integrability [4], Hamiltonian structures [5][6][7], various forms of solutions and their properties [8][9][10] and so on. Among the many ways to study soliton equations [11][12][13][14][15][16][17], the Darboux transformation method [18][19][20][21][22][23][24][25] is one of the most effective and fruitful tools. Darboux transformations can be used to obtain the rogue-wave solution [26,27], solutions on periodic backgrounds [28][29][30], and so on.
Inspired by the coupled mKdV equation [31,32], the complex modified Korteweg-de Vries equation [33], the coupled nonlinear Schrödinger equation [34,35], and so on, we propose a new five-component nonlinear integrable equation in this paper, where x, t ∈ R. Different from the traditional two-component mKdV equation, the five-component Equation (1) contains two different mKdV equations. That is, when v = w = r = q = 0 and when u = w = r = q = 0, the five-component Equation (1) is reduced to, respectively, and Of course these are related by the reflection x = −x that replaces right-traveling waves by left-traveling waves. Resorting to the gauge transformations between the Lax pairs, we derive Darboux transformations and exact solutions for (1). As a result, we present several different type of solitons (including soliton solutions and a solution that is a rational function of e x and t) for (1). The structure of the paper is as follows. In Section 2, a Lax pair associated with the five-component Equation (1) is proposed, and a Darboux transformation is constructed. In Section 3, as applications of the Darboux transformations, some example solutions for (1) are constructed and illustrated.

Darboux Transformation
The five-component nonlinear integrable Equation (1) is associated with the spectral problem where λ ∈ C is the spectral parameter independent of x and t, and The soliton Equation (1) is yielded by the zero-curvature equation whereÛ = (T x + TU)T −1 ,V = (T t + TV)T −1 . In order to ensure thatÛ,V and U, V are the same form except that the old potentials u, v, w, q, r are replaced by the new potentialsû,v,ŵ,q,r respectively, we suppose that T has the following form, where It is easy to see that where In view of (4) and (5) when λ = λ 1 , and denote Computing Equation (13), the expressions of a, b, c are obtained, Substituting (14) into (9), the expressions of d, e, f can be written as Theorem 1. Let the following conditions be satisfied: 1. u, v, w, q, r is a known solution of the five-component Equation (1); 2. λ 1 ∈ R is a fixed constant; and 3. Φ 1 = (φ 1 , ψ 1 , ϕ 1 ) T is a nonzero solution of the Lax pair (4) and (5) when λ = λ 1 .
Denote β 1 = ψ 1 /φ 1 and γ 1 = ϕ 1 /φ 1 . Then the new potentialsû,v,ŵ,q,r given by the Darboux transformation is a new solution of the five-component Equation (1), and the corresponding Darboux matrix is T. In the above equations, the quantities a, b, c, d, f , g are given by (14) and (15).
Proof. First of all, we proveÛT = T x + TU, whereÛ = U| u=û,v=v . Denote Now we prove that expressions (17) and (18) are true. A direct computation shows that From the first two expressions of the Darboux transformation (16), it is easy to verify that expression (17) holds.
Next, we prove this expression (18) is valid. From Equation (9), it is immediate that According to Equations (14) and (15), we derive the first derivatives of a, b, c, d, g, f with respect to x.
Then, we arrive at Hence, Equation (18) is true, and thenÛT = T x + TU is valid.
In the second place, we proveVT = T t + TV, whereV = V| u=û,v=v . Denote Comparing the coefficients of λ j inVT = T t + TV, we haveV So we prove that expressions (23)-(26) are true. It is easy to see that
u v w r q Figure 2. A solution that is a rational function of e x and t.

Conclusions
In this paper, we propose a new five-component nonlinear integrable equation associated with a