Broer-Kaup System with Corrections via Inverse Scattering Transform

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 v and ω decay rapidly as ∣x ∣⟶∞. 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 ε-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 ε -order approximate solution of one soliton. In Section 7, we give some short conclusions.


Introduction
The coupled integrable system or the Broer-Kaup (BK) system [1][2][3][4][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 v and ω decay rapidly as |x | ⟶∞.
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 ε-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 ε -order approximate solution of one soliton. In Section 7, we give some short conclusions. with Now we introduce the Jost functions Φðx, λÞand ψðx, λÞ, which satisfy the following boundary problem: where For simplicity, let the Jost functions have the following form: There exists a scattering matrix TðλÞ which satisfies then we obtain where Wð•, •Þ denotes the Wronskian. With the help of the Neumann series, we find thatϕðx, λÞandψðx, λÞare analytic inD + = fλ ∈ ℂ | Imλ > 0gand that e ϕðx, λÞandψðx, λÞare analytic inD − = fλ ∈ ℂ | Imλ < 0g. The asymptotic behaviors of the Jost functions Φðx, λÞ and Ψðx, λÞ take the following form: where If v and ω are confined to be real functions, we obtain the symmetry condition T * ð−λ * Þ = TðλÞ, which implies where the star denotes the complex conjugate. Since aðλÞ is analytic in D + , if λ n ∈ D + is a zero of aðλÞ, then −λ * n is also a zero. Similarly,ãðλÞ has the simple zeroesλ n , − λ * n ∈ D − . There are two cases for the eigenvalues: λ n = − λ * n and λ n ≠ −λ * n . In this paper, we only consider the first case. Assume that aðλÞ has N simple zeros λ n = iν n (ν n > 0),ãðλÞ has N simple zeros λ n = iν n (ν n < 0).
Next, introducing the first sectional matrices we get the jump condition where the matrix Hðx, t, λÞ is From the previous asymptotic behaviors, we find that Mðx, t, λÞ admits the following normalization condition: 2 Advances in Mathematical Physics Introducing the second sectional matrices we get another jump conditioñ where the matrixHðx, t, λÞ is In a similar way,M ± ðx, t, λÞ 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 λ. Here, we have used the Cauchy projectors P ± over the real axis

BK System with Corrections
Consider the BK system with corrections whereqðxÞandpðxÞare functionals ofvandω, andεis a real parameter. When ε ⟶ 0, (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

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Here, functions hðλÞ andhðλÞ satisfy and λ t ≠ 0. Computing f∂ t − Lgϕˇðx, λÞ and f∂ t − Lg b ϕðx, λÞ and using (33), we get where We note that the BK system with corrections (33) is equivalent to (42). We note that if vðxÞ and ωðxÞ 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.

Evolution of Scattering Data
4.1. Time Dependence of aðλÞ andãðλÞ. 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, aðλÞ, aðλÞ, bðλÞ, andbðλÞ are dependent on t. One may find that functions ϕˇðx, λÞ and b ϕðx, λÞ in (37) take the following asymptotic behaviors as x ⟶ ∞: It is remarked that (40) are nonhomogeneous equations about ϕˇðx, λÞ and b ϕðx, λÞ; thus, they can be expressed as the linear combination of corresponding homogeneous solutions. So, we have the following expressions: where the coefficients α and β andα andβ are defined by the following equations: As x ⟶ ∞, (47) and (48) become b Comparing (45) with (53), we find Advances in Mathematical Physics Substituting (49) into (55), we have One may find that the BK system with corrections reduces to the BK system as ε ⟶ 0. Under this limitation, λ t = 0 and Gðx, λÞ ⟶ 0, then from (59), Let ρðλÞ = bðλÞ/aðλÞ andρðλÞ =bðλÞ/ãðλÞ, then from (59), we have which reduces to the corresponding results in the BK system as ε ⟶ 0.

Time
Dependence of λ n andλ n . The energy in the bound state is discrete. For the bound state of the BK system with corrections, the eigenvalues fλ n g depend on time. For the bound state, we still have a λ n ð Þ = 0,ãλ n = 0, ð65Þ We note that equations (65) and (66) are valid for any t; hence, a t λ n ð Þ = 0,ã tλn = 0: ð67Þ 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 λ ⟶ λ n 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 λ n for the BK system with correction terms. When ε ⟶ 0, (69) become λ n,t = 0 andλ n,t = 0; these are the results of the BK system.
then the two sets of functions satisfy the following equation: b ϕ x,λ n =h In view of the analyticity of ϕðx, λÞ and ψðx, λÞ, let λ = λ n in (47) andλ =λ n in (48); we have e ϕ x,λ n =ζ x,λ n e ϕ x,λ n , ð75Þ in terms of (66). Substituting the above equation into (42), we get Integrating (76), we find It is verified that ψˇðx, λ n Þ and ψˇðx,λ n Þ admit where Substituting (74) and (78) and (82) and (84) into (72), we get the evolution of d n ðtÞ andd n ðtÞ Advances in Mathematical Physics which reduce to When ε ⟶ 0, (87) are the results of the BK system without corrections.

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 ϕ 2 from system (89) and introduce a new function b ϕ by then we get a Riccati equation Assume that b ϕ has the following expansion: then by substituting (93) into (92), we find For the BK system, aðλÞ and bðλÞ are independent of t, then we obtain the following conserved densities: Hence, there exists an associated flux X j ðxÞ that satisfies and the BK system (1); we find For the BK system with corrections, we still have Since, aðλÞ depends on t, (97) is not valid. In fact, the evolution of densities I j is an ε-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

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We note that the integrand in the first integral is a divergence term which makes the integral varnish. Hence, (103) reduces to

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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, ðe η + ωÞ a and ðe −η + Þ a are neither the solution of the BK system nor the solution of the BK system with correction terms; they give a part of the ε-order approximate solutions to the BK system with correction terms. In Section 8, we will discuss the other part of the ε-order approximate solutions.