Darboux Transformation and New Multi-Soliton Solutions of the Whitham-Broer-Kaup System

Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters is constructed. Then some new explicit solutions for the Whitham-Broer-Kaup system are obtained via the given Darboux transformation.

Many solutions have been obtained for the WBK system (1), such as the analytical solution, the soliton-like solution, the soliton solutions, the periodic solution, the rational solution, and so on [15]- [19].
In this paper, through a proper transformation r q q r cr c c v c r q q r r q q r c c r the WBK system ( 1) is transformed into the parameter Levi system ( ) Based on the obtained Lax pair, we construct the N-fold DT of the parameter Levi system (3) and then get the N-fold DT of the WBK system (1).Resorting to the obtained DT, we get new multi-soliton solutions of the WBK system.
The paper is organized as follows.In Section 2, we construct the N-fold DT of the Levi system and the WBK system.In Section 3, DT will be applied to generate explicit solutions of the WBK system (1).

Darboux Transformation
In this section, we first construct the N-fold DT of the parameter Levi system, and then get explicit solutions of the WBK system.
We consider the following spectral problem corresponding to the Levi system (3) and its auxiliary problem where λ is a spectral parameter and ( ) = which leads to the Levi system (3) by a direct computation.Now we introduce a transformation of (4) and ( 5) where T is defined by , .
x t

T TU UT T TV VT
Then the Lax pair (4) and ( 5) are transformed into where U , V have the same form as U , V , except replacing q , r , x q , x r with q , r , x q , x r , re- spectively.
In order to make the Lax pair (4) and ( 5) invariant under the transformation (6), it is necessary to find a matrix T .
Let the matrix T in (6) be in the form of ( ) ( ) ( ) ( ) ( ) with ( ) ( ) ( ) ( ) be two basic solutions of the spectral problem (4) and use them to define a linear algebraic system ( ) where the constants j λ , ( ) are suitably chosen such that the determinant of the coefficients of (11) are nonzero.If we take , , , 0 1 are uniquely determined by (11).From (10), we have We note that (11) can be written as a linear algebraic system and ( ) which implies that ( ) where γ is independent of λ .From the above facts, we can prove the following propositions.Proposition 1.Let α satisfy the following first-order differential equation Then the matrix U determined by Equation (7) is the same form as U : ( ) , where the transformations from the old potentials q , r to q , r are given by where T * denotes the adjoint matrix of T .It is easy to see that ( ) ( ) f λ are 2N th-order polynomials in λ , while ( ) f λ are (2N − 1)th-order polynomials in λ .From ( 4) and ( 12), we get ( ) By using ( 16) and ( 20), we can prove that ( ) . From (15), we have

kj T f k j λ =
Hence, together with (19), we have that is ( ) are independent of λ .By comparing the coefficients of Substituting ( 17) into ( 24)-( 26) yields From ( 7) and ( 22), we find that ( ) r q q A D r q A r q r q D r q r q r A D r q A r q r q D r q Let the basic solution ( ) j ϕ λ , ( ) j ψ λ of (4) satisfy (5) as well.Through a similar way as Proposition 1, we can prove that V has the same form as V under the transformation ( 6) and (18).We get the following propo- sition.
Proposition 2. Suppose α satisfy the following equation Then the matrix V defined by (9) has the same form as V , that is where q and r are given by (18).The proof of Proposition 2 is similar with Proposition 1, but it is much more tedious and then we omit the proof for brevity.For the similar proof we can also refer to [20] [21].
According to Proposition 1 and 2, the Lax pair ( 4) and ( 5) is transformed into the Lax pair ( 8) and ( 9), then the transformation ( 6) and ( 18): ( ) ( ) is called the DT of the Lax pair ( 4) and ( 5).The Lax pair leads to the parameter Levi system (3) and then the transformation ( 6) and ( 18): ( ) ( ) ; , ; , q r q r ϕ ϕ → is also called DT of the parameter Levi system (3).On the other hand, together with the transformation (2), the parameter Levi system (3) is transformed into the WBK system (1), then we get the solutions of the WBK system (1).
From the transformation (2), we find that r q q r cr c c v c r q q r r q q r c c r is another solution of the WBK system (1), where ( ) , q r is determined by (30).Then the transformation

New Solutions
In this section, we take a trivial solution ( ) ( ) , 0,1 q r = as the "seed" solution, to obtain multi-soliton solutions of the WBK system (1).
Substituting ( ) ( ) , 0,1 q r = into the Lax pair ( 4) and ( 5), the two basic solutions are with ( ) According to (12), we get For simplicity, we discuss the following two cases, i.e. 1 N = and 2 N = .As 1 N = , let 1 λ λ = , solving the linear algebraic system (11) and ( 13), we have according to (28), we ) Substituting ( 35) into (31), we obtain the solution the WBK system (1) as ), we find that [ ] , together with ( 11) and ( 13), we have  Then we get another solution of the WBK system (1) by using of (31 is a three-M-type-soliton solution with two overtaking solitons and one head-on soliton (see Figure 1(b)).We note that by the obtained DT, we can get ( ) soliton solutions which are different from those in [19] which are 2N -soliton solutions.
t =is the field of the horizontal velocity, and
-soliton solution with two overtaking solitons and one head-on soliton (see Figure1(a)) and [ ]