Abstract

The Broer-Kaup system with corrections is considered. Based on the inverse scattering transform, we extend the perturbation theory to discuss the adiabatic approximate solution and -order approximate solution of one soliton to the Broer-Kaup system with corrections.

1. Introduction

The coupled integrable system or the Broer-Kaup (BK) system [1–5] is related to one of the Boussinesq systems by variable transformation. The BK system is used to simulate the bidirectional propagation of long waves in shallow water. Here, we assume that and decay rapidly as .

Kaup first proposed the perturbation theory based upon the inverse scattering transform [6]. In addition to the perturbation theory based upon the inverse scattering transform [7–15], there are many other methods, such as the direct soliton perturbation theory [16–30] and the normal form method [31–34]. For more detail about the perturbation theory, see [11, 35] and references therein.

In this paper, we consider the BK system with corrections through perturbation theory based upon the inverse scattering transform [15] and investigate the adiabatic approximate solution and -order approximate solution of one soliton to the BK system with corrections. In order to carry out the inverse scattering transform to discuss the BK system with corrections, we keep the first Lax equation but give up the second one. For this reason, the analyticity and asymptotic behaviors of Jost functions are the same as the BK system. In this case, the scattering data are all dependent on time, and the time evolution of scattering data is discussed in detail.

This paper organized as follows. In Section 2, we discuss the spectral analysis and construct the Riemann-Hilbert problem of the BK system. In Section 3, we present the BK system with corrections. In Section 4, we consider the time evolution of the scattering coefficients, the discrete spectrum, and the normalization factors. In Section 5, the conservation laws of the Broer-Kaup system without corrections and its perturbation corrections are given. In Section 6, we obtain the adiabatic approximate solution of one soliton, find the slow variations of spectral parameters, and discuss the -order approximate solution of one soliton. In Section 7, we give some short conclusions.

2. Spectral Analysis and Riemann-Hilbert Problem

Equation (1) is the compatibility condition of the following Lax pair: with

Now we introduce the Jost functions and , which satisfy the following boundary problem: where

For simplicity, let the Jost functions have the following form:

There exists a scattering matrix which satisfies then we obtain where denotes the Wronskian. With the help of the Neumann series, we find thatandare analytic inand thatandare analytic in. The asymptotic behaviors of the Jost functions and take the following form: where

If and are confined to be real functions, we obtain the symmetry condition , which implies where the star denotes the complex conjugate. Since is analytic in , if is a zero of , then is also a zero. Similarly, has the simple zeroes , . There are two cases for the eigenvalues: and .

In this paper, we only consider the first case. Assume that has simple zeros (), has simple zeros ().

Next, introducing the first sectional matrices we get the jump condition where the matrix is

From the previous asymptotic behaviors, we find that admits the following normalization condition:

Introducing the second sectional matrices we get another jump condition where the matrix is

In a similar way, has the following normalization condition:

From (11), (15), and (20), we get

Solving the above Riemann-Hilbert problem, we have where and the dot denotes the derivative with respect to . Here, we have used the Cauchy projectors over the real axis

3. BK System with Corrections

Consider the BK system with corrections whereandare functionals ofand, andis a real parameter. When , (33) reduces to the BK system without corrections.

To carry out the inverse scattering transform to the BK system with corrections, we keep the first Lax equation

In this way, the analyticity and the asymptotic behavior of Jost functions are the same as those of the BK system without corrections.

Next, we give up the second Lax equation

And introduce the new functions

Here, functions and satisfy and . Computing and and using (33), we get where

We note that the BK system with corrections (33) is equivalent to (42). We note that if and satisfy the BK system with corrections (33), then (42) will determine the associated time evolution of scattering data. Conversely, if the time evolution of scattering data is determined from (42), the solution of the BK system with corrections can be rebuilt. Here and after, we will start from (3.7) to establish the perturbation theory of the BK system with corrections.

4. Evolution of Scattering Data

4.1. Time Dependence of and

In this section, we discuss the time evolution of the scattering data from (42). For the decay potential, we find, from (37), that

We note that, for the BK system with corrections, , , , and are dependent on . One may find that functions and in (37) take the following asymptotic behaviors as :

It is remarked that (40) are nonhomogeneous equations about and ; thus, they can be expressed as the linear combination of corresponding homogeneous solutions. So, we have the following expressions: where the coefficients and and and are defined by the following equations:

As , (47) and (48) become

Comparing (45) with (53), we find

Substituting (49) into (55), we have

One may find that the BK system with corrections reduces to the BK system as . Under this limitation, and , then from (59),