Broer-Kaup System with Corrections via Inverse Scattering Transform
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.
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 . 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  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:
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
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:
One may find that the BK system with corrections reduces to the BK system as . Under this limitation, and , then from (59),
Let and , then from (59), we have which reduces to the corresponding results in the BK system as .
4.2. Time Dependence of and
The energy in the bound state is discrete. For the bound state of the BK system with corrections, the eigenvalues depend on time. For the bound state, we still have
These conditions mean that the corrections are too small to change the bound state solution of the scattering problem. In other words, the soliton solutions of the nonlinear problem can still be found. Let in (59), using (67) and (42), we have which can be further reduced to in terms of (9). These equations give the time evolution of parameter for the BK system with correction terms. When , (69) become and ; these are the results of the BK system.
4.3. Time Dependence of and
To consider the time evolution of the normalization factors, we introduce another set of functions then the two sets of functions satisfy the following equation:
Substituting the above equation into (42), we get
Integrating (76), we find
It is verified that and admit where
When , (87) are the results of the BK system without corrections.
5. Perturbation Corrections of the Conservation Laws
The BK system has an infinite number of conservation laws, which can be derived by considering the linear spectral problem in (3), that is,
To this end, we eliminate from system (89) and introduce a new function by then we get a Riccati equation
For the BK system, and are independent of , then we obtain the following conserved densities:
Hence, there exists an associated flux that satisfies and the BK system (1); we find
For the BK system with corrections, we still have
Since, depends on , (97) is not valid. In fact, the evolution of densities is an -order term which denotes perturbation correction of the conservation laws. To find the evolution, we rewrite the BK system with corrections
Making use of the functional derivative, we get
We note that the integrand in the first integral is a divergence term which makes the integral varnish. Hence, (103) reduces to
6. Adiabatic Approximate Solutions
In this section, we consider the adiabatic approximate solution of the BK system with corrections. Since the linear spectral problem is valid, from (15) to (30) and in the case of reflectionless potentials, we still have where
Due to where
For the BK system and , and , using and , where
For the BK system with corrections, depends on and its time evolution is -order. has -order’s correction besides the previous time evolution. In this case, the Jost functions , , , and will be denoted by , , , and and the expressions and denoted by and which give the adiabatic approximate solutions of the BK system with correction terms. We note that , , , and satisfy the same equations as (105) and (106). In addition,
It is important to note that the form of the adiabatic approximate solutions is similar to one soliton solution of the BK system without correction terms, while the scattering data follow the time evolution discussed in Section 4. Hence, and are neither the solution of the BK system nor the solution of the BK system with correction terms; they give a part of the -order approximate solutions to the BK system with correction terms. In Section 8, we will discuss the other part of the -order approximate solutions.
7. Slow Variations of the Spectral Parameters
To discuss the slow variations of the spectral parameters, we rewrite and in (117) as
In this case,
In addition, for reflectionless case and , the Jost functions take the followingform:
These equations can be derived by taking and substituting (108) into (29). It is remarked that , , , and have a similar form to those in (127) for the BK system with corrections. In fact, the eigenfunctions in (69) are , , , and . Substituting (127) into (69), we get or