# Broer-Kaup System with Corrections via Inverse Scattering Transform

Authors: Xueru Wang; Junyi Zhu
Journal: Advances in Mathematical Physics
 (2020)

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## 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 andε-order approximate solution of one soliton to the Broer-Kaup system with corrections.


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## Body

## 1. Introduction

The coupled integrable system or the Broer-Kaup (BK) system [1–5]
(1)tt=12v2+2ω−vxx,(2)ωt=vω+12ωxx,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 Section2, 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.


## 2. Spectral Analysis and Riemann-Hilbert Problem

Equation (1) is the compatibility condition of the following Lax pair:
(3)Fx=LF,Ft=MF,with
(4)Lx,λ=−iλσ3+Ux,Mx,λ=λ2σ3+Vx,Ux=v21−ω−v2,Vx=−14vx−v2iλ+v2−iλω−12ωx+ωv14vx−v2.Now we introduce the Jost functionsΦx,λand ψx,λ, which satisfy the following boundary problem:
(5)∂xΦx,λ=Lx,λΦx,λ,Φx,λ⟶Ae−iλxσ3,x⟶−∞,∂xΨx,λ=Lx,λΨx,λ,ψx,λ⟶Ae−iλxσ3,x⟶∞,where
(6)A=112iλ01.For simplicity, let the Jost functions have the following form:(7)Φx,λ=ϕx,λ,ϕ~x,λ,Ψx,λ=ψ~x,λ,ψx,λ.There exists a scattering matrixTλ which satisfies
(8)Φx,λ=Ψx,λTλ,Tλ=aλ−b~λbλa~λ,then we obtain
(9)aλ=Wϕx,λ,ψx,λ,bλ=Wψ~x,λ,ϕx,λ,(10)a~λ=Wψ~x,λ,ϕ~x,λ,−b~λ=Wϕ~x,λ,ψx,λ,where W∙,∙ denotes the Wronskian. With the help of the Neumann series, we find thatϕx,λandψx,λare analytic inD+=λ∈ℂ∣Imλ>0and thatϕ~x,λandψ~x,λare analytic inD−=λ∈ℂ∣Imλ<0. The asymptotic behaviors of the Jost functions Φx,λ and Ψx,λ take the following form:
(11)Φx,λeiλxσ3=eη−1+12iλ∫−∞xωydy12iλe−η−12iλeη−ωe−η−1−12iλ∫−∞xωydy+O1λ2,(12)Ψx,λeiλxσ3=eη+1+12iλ∫∞xωydy12iλe−η+12iλeη+ωeη+1−12iλ∫∞xωydy+O1λ2,where
(13)η±=12∫±∞xvydy.Ifv and ω are confined to be real functions, we obtain the symmetry condition T∗−λ∗=Tλ, which implies
(14)a∗−λ∗=aλ,a~∗−λ∗=aλ,where the star denotes the complex conjugate. Sinceaλ is analytic in D+, if λn∈D+ is a zero of aλ, then −λn∗ is also a zero. Similarly, a~λ has the simple zeroes λ~n, −λ~n∗∈D−. There are two cases for the eigenvalues: λn=−λn∗ and λn≠−λn∗.In this paper, we only consider the first case. Assume thataλ has N simple zeros λn=iνn (νn>0), a~λ has N simple zeros λn=iν~n (ν~n<0).Next, introducing the first sectional matrices(15)M+x,t,λ=ϕx,t,λeiλxaλ,ψx,t,λe−iλxaλ,(16)M−x,t,λ=ψ~x,t,λeiλx,ϕ~x,t,λe−iλx,we get the jump condition
(17)M−x,t,λ=M+x,t,λI−Hx,t,λ,λ∈ℝ,where the matrix Hx,t,λ is
(18)Hx,t,λ=0b~λe−2iλxbλe2iλx0.From the previous asymptotic behaviors, we find thatMx,t,λ admits the following normalization condition:
(19)M±x,t,λ=B+O1λ,λ⟶∞,B=eη+00e−η−.Introducing the second sectional matrices(20)M~+x,t,λ=ϕx,t,λeiλx,ψx,t,λe−iλx,(21)M~−x,t,λ=ψ~x,t,λeiλxa~λ,ϕ~x,t,λe−iλxa~λ,we get another jump condition
(22)M~+x,t,λ=M~−x,t,λI+H~x,t,λ,λ∈ℝ,where the matrix H~x,t,λ is
(23)H~x,t,λ=0b~λe−2iλxbλe2iλx0.In a similar way,M~±x,t,λ has the following normalization condition:
(24)M~±x,t,λ=B~+O1λ,λ⟶∞,B~=eη−00e−η+.From (11), (15), and (20), we get
(25)M±x,t,λ=B+12iλB1+O1λ2,λ⟶∞,(26)M~±x,t,λ=B~+12iλB~1+O1λ2,λ⟶∞,(27)B1=eη+∫∞xωydye−η−eη+−e−η−∫−∞xωydy,(28)B~1=eη−∫−∞xωydye−η+eη−−e−η+∫∞xωydy.Solving the above Riemann-Hilbert problem, we have(29)M±x,t,λ=B+∑n=1NResM+,λnλ−λn+P±M+H,M~±x,t,λ=B~+∑n=1NResM~−,λ~nλ−λ~n+P±M−H,where
(30)ResM+,λn=ϕx,t,λneiλnxȧλn,ψx,t,λne−iλnxȧλn,(31)ResM~−,λ~n=ψ~x,t,λneiλnxa~̇λn,ϕ~x,t,λne−iλnxa~̇λn,and the dot denotes the derivative with respect to λ. Here, we have used the Cauchy projectors P± over the real axis
(32)P±fk=12πi∫−∞+∞fll−k±i0dl.


## 3. BK System with Corrections

Consider the BK system with corrections(33)vt−12v2+2ω−vxx=εqx2,(34)ωt−vω+12ωxx=εpx,whereqxandpxare 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(35)∂xϕx,λ=Lx,λϕx,λ.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(36)∂tϕx,λ=Mx,λϕx,λ.And introduce the new functions(37)ϕˇx,λ=∂t−Mhλϕx,λ,(38)ϕ^x,λ=∂t−Mh~λϕ~x,λ.Here, functionshλ and h~λ satisfy
(39)htλ=λ2hλ,h~tλ=−λ2h~λ,and λt≠0. Computing ∂t−Lϕˇx,λ and ∂t−Lϕ^x,λ and using (33), we get
(40)∂t−Lϕˇx,λ=Gx,λhλϕx,λ,(41)∂t−Lϕ^x,λ=Gx,λh~λϕ~x,λ,where
(42)Gx,λ=−iλtσ3+εQx,(43)Qx=qx20−px−qx2.We note that the BK system with corrections (33) is equivalent to (42). We note that if vx and ωx 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 ofaλ and a~λ

In this section, we discuss the time evolution of the scattering data from (42). For the decay potential, we find, from (37), that
(44)ϕˇx,λ⟶∂t−λ2hλ10e−iλx,x⟶−∞,ϕ^x,λ⟶∂t+λ2hλ10eiλx,x⟶−∞.We note that, for the BK system with corrections,aλ, a~λ, bλ, and b~λ are dependent on t. One may find that functions ϕˇx,λ and ϕ^x,λ in (37) take the following asymptotic behaviors as x⟶∞:
(45)ϕˇx,λ⟶hλatλ10e−iλx+hλbtλ+λ2bλ12iλ1eiλx,(46)ϕ^x,λ⟶h~λb~tλ−λ2b~λ10e−iλx+h~λa~tλ12iλ1eiλx.It is remarked that (40) are nonhomogeneous equations about ϕˇx,λ and ϕ^x,λ; thus, they can be expressed as the linear combination of corresponding homogeneous solutions. So, we have the following expressions:
(47)ϕˇx,λ=αx,λψx,λ+βx,λϕx,λ,(48)ϕ^x,λ=α~x,λψ~x,λ+β~x,λϕ~x,λ,where the coefficients α and β and α~ and β~ are defined by the following equations:
(49)αx,λ=ihλaλ∫−∞xϕTy,λσ2Gy,λϕy,λdy,(50)βx,λ=−ihλaλ∫−∞xψTy,λσ2Gy,λϕy,λdy,(51)α~x,λ=−ih~λa~λ∫−∞xϕ~Ty,λσ2Gy,λϕ~y,λdy,(52)β~x,λ=ih~λa~λ∫−∞xψ~Ty,λσ2Gy,λϕ~y,λdy.Asx⟶∞, (47) and (48) become
(53)ϕˇx,λ⟶β∞,λaλ10e−iλx+α∞,λ+β∞,λbλ12iλ1eiλx,(54)ϕ^x,λ⟶α~∞,λ−β~∞,λb~λ10e−iλx+β~∞,λa~λ12iλ1eiλx.Comparing (45) with (53), we find
(55)hλatλ=β∞,λaλ,(56)hλbtλ+λ2bλ=α∞,λ+β∞,λbλ,(57)h~λa~tλ=β~∞,λa~λ,(58)−h~λb~tλ−λ2b~λ=α~∞,λ−β~∞,λb~λ.Substituting (49) into (55), we have
(59)atλ=−i∫−∞∞ψTx,λσ2Gx,λϕx,λdx,(60)btλ+λ2bλ=−ia−1λ∫−∞∞ϕx,λ−bλψx,λTσ2Gx,λϕx,λdx,(61)a~tλ=i∫−∞∞ψ~Tx,λσ2Gx,λϕ~x,λdx,(62)b~tλ−λ2b~λ=ia~−1λ∫−∞∞ϕ~x,λ+b~λψ~x,λTσ2Gx,λϕ~x,λdx.One may find that the BK system with corrections reduces to the BK system asε⟶0. Under this limitation, λt=0 and Gx,λ⟶0, then from (59),
(63)atλ=0,btλ+λ2bλ=0,a~tλ=0,b~tλ−λ2b~λ=0.Letρλ=bλ/aλ and ρ~λ=b~λ/a~λ, then from (59), we have
(64)ρtλ+λ2ρλ=iεa−2λ∫−∞∞ϕTx,λσ2Qx,λϕx,λdx,ρ~tλ−λ2ρ~λ=iεa~−2λ∫−∞∞ϕ~Tx,λσ2Qx,λϕ~x,λdx,which reduces to the corresponding results in the BK system as ε⟶0.


### 4.2. Time Dependence ofλn and λ~n

The energy in the bound state is discrete. For the bound state of the BK system with corrections, the eigenvaluesλn depend on time. For the bound state, we still have
(65)aλn=0,a~λ~n=0,(66)ϕx,λn=dntψx,λn,ϕ~x,λ~n=d~ntψ~x,λ~n.We note that equations (65) and (66) are valid for any t; hence,
(67)atλn=0,a~tλ~n=0.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. Letλ⟶λn in (59), using (67) and (42), we have
(68)2λn,t∫−∞∞ϕ1x,λnϕ2x,λndx+ε∫−∞∞ϕTx,λnσ2Qxϕx,λndx=0,2λ~n,t∫−∞∞ϕ~1x,λ~nϕ~2x,λ~ndx+ε∫−∞∞ϕ~Tx,λ~nσ2Qxϕ~x,λ~ndx=0,which can be further reduced to
(69)ȧλnλn,t=−εdn−1t∫−∞∞qxϕ1x,λnϕ2x,λndx+pxϕ12x,λndx,(70)a~̇λ~nλ~n,t=εd~n−1t∫−∞∞qxϕ~1x,λ~nϕ~2x,λ~ndx+pxϕ~12x,λ~ndx,in terms of (9). These equations give the time evolution of parameter λn for the BK system with correction terms. When ε⟶0, (69) become λn,t=0 and λ~n,t=0; these are the results of the BK system.


### 4.3. Time Dependence ofdnt and d~nt

To consider the time evolution of the normalization factors, we introduce another set of functions(71)ψˇx,λ=∂t−Mh−1λψx,λ,ψ^x,λ=∂t−Mh~−1λψ~x,λ,then the two sets of functions satisfy the following equation:
(72)ϕˇx,λn=h2λndntψˇx,λn+hλndn,tt+2λn2dntψx,λn,(73)ϕ^x,λ~n=h~2λ~nd~ntψ^x,λ~n+h~λ~nd~n,tt+2λ~n2d~ntψ~x,λ~n.In view of the analyticity ofϕx,λ and ψx,λ, let λ=λn in (47) and λ~=λ~nin (48); we have
(74)ϕˇx,λn=ζx,λnϕx,λn,(75)ϕ~x,λ~n=ζ~x,λ~nϕ~x,λ~n,in terms of (66).Substituting the above equation into (42), we get
(76)ζxx,λnȧλn=ihλndn−1tϕ̇x,λn−dntψ̇x,λnTσ2Gx,λnϕx,λn,(77)ζ~xx,λ~na~̇λ~n=−ih~λnd~n−1tϕ~̇x,λn−d~ntψ~̇x,λ~nTσ2Gx,λ~nϕ~x,λ~n.Integrating (76), we find
(78)ζx,λn=ihλnȧλndnt∫−∞xϕ̇y,λn−dntψ̇y,λnTσ2Gy,λnϕy,λndy,(79)ζ~x,λ~n=−ih~λ~na~̇λ~nd~nt∫−∞xϕ~̇y,λn−d~ntψ~̇y,λ~nTσ2Gy,λ~nϕ~y,λ~ndy.It is verified thatψˇx,λn and ψˇx,λ~n admit
(80)∂x−Lx,λnψˇx,λn=Gx,λnh−1λnψx,λn,(81)∂x−Lx,λ~nψ^x,λ~n=Gx,λ~nh~−1λ~nψ~x,λ~n,(82)ψˇx,λn=γx,λnψx,λn,(83)ψ^x,λ~n=γ~x,λ~nψ~x,λ~n,where
(84)γx,λn=−iȧλnhλn∫x∞ϕ̇y,λn−dntψ̇y,λnTσ2Gy,λnψy,λndy,(85)γ~x,λ~n=ia~̇λ~nh~λ~n∫x∞ϕ~̇y,λn−d~ntψ~̇y,λ~nTσ2Gy,λ~nψ~y,λ~ndy.Substituting (74) and (78) and (82) and (84) into (72), we get the evolution of dnt and d~nt(86)dn,tt+2λn2dnt=iλn,tȧλndnt∫−∞∞ϕ̇x,λn−dntψ̇x,λnTσ1ϕx,λndx+iεȧλn∫−∞∞ϕ̇x,λn−dntψ̇x,λnTσ2Qxϕx,λndx,d~n,tt−2λ~n2d~nt=−iλ~n,tȧλndnt∫−∞∞ϕ~̇x,λ~n−d~ntψ~̇x,λ~nTσ1ϕ~x,λ~ndx−iεa~̇λ~n∫−∞∞ϕ~̇x,λ~n−d~ntψ~̇x,λ~nTσ2Qxϕ~x,λ~ndx,which reduce to
(87)dn,tt+2λn2dnt=iεȧλn∫−∞∞ϕ̇x,λn−dntψ̇x,λnTσ2Qxϕx,λndx,(88)d~n,tt−2λ~n2d~nt=−iεa~̇λ~n∫−∞∞ϕ~̇x,λ~n−d~ntψ~̇x,λ~nTσ2Qxϕ~x,λ~ndx.Whenε⟶0, (87) are the results of the BK system without corrections.


## 4.1. Time Dependence ofaλ and a~λ

In this section, we discuss the time evolution of the scattering data from (42). For the decay potential, we find, from (37), that
(44)ϕˇx,λ⟶∂t−λ2hλ10e−iλx,x⟶−∞,ϕ^x,λ⟶∂t+λ2hλ10eiλx,x⟶−∞.We note that, for the BK system with corrections,aλ, a~λ, bλ, and b~λ are dependent on t. One may find that functions ϕˇx,λ and ϕ^x,λ in (37) take the following asymptotic behaviors as x⟶∞:
(45)ϕˇx,λ⟶hλatλ10e−iλx+hλbtλ+λ2bλ12iλ1eiλx,(46)ϕ^x,λ⟶h~λb~tλ−λ2b~λ10e−iλx+h~λa~tλ12iλ1eiλx.It is remarked that (40) are nonhomogeneous equations about ϕˇx,λ and ϕ^x,λ; thus, they can be expressed as the linear combination of corresponding homogeneous solutions. So, we have the following expressions:
(47)ϕˇx,λ=αx,λψx,λ+βx,λϕx,λ,(48)ϕ^x,λ=α~x,λψ~x,λ+β~x,λϕ~x,λ,where the coefficients α and β and α~ and β~ are defined by the following equations:
(49)αx,λ=ihλaλ∫−∞xϕTy,λσ2Gy,λϕy,λdy,(50)βx,λ=−ihλaλ∫−∞xψTy,λσ2Gy,λϕy,λdy,(51)α~x,λ=−ih~λa~λ∫−∞xϕ~Ty,λσ2Gy,λϕ~y,λdy,(52)β~x,λ=ih~λa~λ∫−∞xψ~Ty,λσ2Gy,λϕ~y,λdy.Asx⟶∞, (47) and (48) become
(53)ϕˇx,λ⟶β∞,λaλ10e−iλx+α∞,λ+β∞,λbλ12iλ1eiλx,(54)ϕ^x,λ⟶α~∞,λ−β~∞,λb~λ10e−iλx+β~∞,λa~λ12iλ1eiλx.Comparing (45) with (53), we find
(55)hλatλ=β∞,λaλ,(56)hλbtλ+λ2bλ=α∞,λ+β∞,λbλ,(57)h~λa~tλ=β~∞,λa~λ,(58)−h~λb~tλ−λ2b~λ=α~∞,λ−β~∞,λb~λ.Substituting (49) into (55), we have
(59)atλ=−i∫−∞∞ψTx,λσ2Gx,λϕx,λdx,(60)btλ+λ2bλ=−ia−1λ∫−∞∞ϕx,λ−bλψx,λTσ2Gx,λϕx,λdx,(61)a~tλ=i∫−∞∞ψ~Tx,λσ2Gx,λϕ~x,λdx,(62)b~tλ−λ2b~λ=ia~−1λ∫−∞∞ϕ~x,λ+b~λψ~x,λTσ2Gx,λϕ~x,λdx.One may find that the BK system with corrections reduces to the BK system asε⟶0. Under this limitation, λt=0 and Gx,λ⟶0, then from (59),
(63)atλ=0,btλ+λ2bλ=0,a~tλ=0,b~tλ−λ2b~λ=0.Letρλ=bλ/aλ and ρ~λ=b~λ/a~λ, then from (59), we have
(64)ρtλ+λ2ρλ=iεa−2λ∫−∞∞ϕTx,λσ2Qx,λϕx,λdx,ρ~tλ−λ2ρ~λ=iεa~−2λ∫−∞∞ϕ~Tx,λσ2Qx,λϕ~x,λdx,which reduces to the corresponding results in the BK system as ε⟶0.


## 4.2. Time Dependence ofλn and λ~n

The energy in the bound state is discrete. For the bound state of the BK system with corrections, the eigenvaluesλn depend on time. For the bound state, we still have
(65)aλn=0,a~λ~n=0,(66)ϕx,λn=dntψx,λn,ϕ~x,λ~n=d~ntψ~x,λ~n.We note that equations (65) and (66) are valid for any t; hence,
(67)atλn=0,a~tλ~n=0.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. Letλ⟶λn in (59), using (67) and (42), we have
(68)2λn,t∫−∞∞ϕ1x,λnϕ2x,λndx+ε∫−∞∞ϕTx,λnσ2Qxϕx,λndx=0,2λ~n,t∫−∞∞ϕ~1x,λ~nϕ~2x,λ~ndx+ε∫−∞∞ϕ~Tx,λ~nσ2Qxϕ~x,λ~ndx=0,which can be further reduced to
(69)ȧλnλn,t=−εdn−1t∫−∞∞qxϕ1x,λnϕ2x,λndx+pxϕ12x,λndx,(70)a~̇λ~nλ~n,t=εd~n−1t∫−∞∞qxϕ~1x,λ~nϕ~2x,λ~ndx+pxϕ~12x,λ~ndx,in terms of (9). These equations give the time evolution of parameter λn for the BK system with correction terms. When ε⟶0, (69) become λn,t=0 and λ~n,t=0; these are the results of the BK system.


## 4.3. Time Dependence ofdnt and d~nt

To consider the time evolution of the normalization factors, we introduce another set of functions(71)ψˇx,λ=∂t−Mh−1λψx,λ,ψ^x,λ=∂t−Mh~−1λψ~x,λ,then the two sets of functions satisfy the following equation:
(72)ϕˇx,λn=h2λndntψˇx,λn+hλndn,tt+2λn2dntψx,λn,(73)ϕ^x,λ~n=h~2λ~nd~ntψ^x,λ~n+h~λ~nd~n,tt+2λ~n2d~ntψ~x,λ~n.In view of the analyticity ofϕx,λ and ψx,λ, let λ=λn in (47) and λ~=λ~nin (48); we have
(74)ϕˇx,λn=ζx,λnϕx,λn,(75)ϕ~x,λ~n=ζ~x,λ~nϕ~x,λ~n,in terms of (66).Substituting the above equation into (42), we get
(76)ζxx,λnȧλn=ihλndn−1tϕ̇x,λn−dntψ̇x,λnTσ2Gx,λnϕx,λn,(77)ζ~xx,λ~na~̇λ~n=−ih~λnd~n−1tϕ~̇x,λn−d~ntψ~̇x,λ~nTσ2Gx,λ~nϕ~x,λ~n.Integrating (76), we find
(78)ζx,λn=ihλnȧλndnt∫−∞xϕ̇y,λn−dntψ̇y,λnTσ2Gy,λnϕy,λndy,(79)ζ~x,λ~n=−ih~λ~na~̇λ~nd~nt∫−∞xϕ~̇y,λn−d~ntψ~̇y,λ~nTσ2Gy,λ~nϕ~y,λ~ndy.It is verified thatψˇx,λn and ψˇx,λ~n admit
(80)∂x−Lx,λnψˇx,λn=Gx,λnh−1λnψx,λn,(81)∂x−Lx,λ~nψ^x,λ~n=Gx,λ~nh~−1λ~nψ~x,λ~n,(82)ψˇx,λn=γx,λnψx,λn,(83)ψ^x,λ~n=γ~x,λ~nψ~x,λ~n,where
(84)γx,λn=−iȧλnhλn∫x∞ϕ̇y,λn−dntψ̇y,λnTσ2Gy,λnψy,λndy,(85)γ~x,λ~n=ia~̇λ~nh~λ~n∫x∞ϕ~̇y,λn−d~ntψ~̇y,λ~nTσ2Gy,λ~nψ~y,λ~ndy.Substituting (74) and (78) and (82) and (84) into (72), we get the evolution of dnt and d~nt(86)dn,tt+2λn2dnt=iλn,tȧλndnt∫−∞∞ϕ̇x,λn−dntψ̇x,λnTσ1ϕx,λndx+iεȧλn∫−∞∞ϕ̇x,λn−dntψ̇x,λnTσ2Qxϕx,λndx,d~n,tt−2λ~n2d~nt=−iλ~n,tȧλndnt∫−∞∞ϕ~̇x,λ~n−d~ntψ~̇x,λ~nTσ1ϕ~x,λ~ndx−iεa~̇λ~n∫−∞∞ϕ~̇x,λ~n−d~ntψ~̇x,λ~nTσ2Qxϕ~x,λ~ndx,which reduce to
(87)dn,tt+2λn2dnt=iεȧλn∫−∞∞ϕ̇x,λn−dntψ̇x,λnTσ2Qxϕx,λndx,(88)d~n,tt−2λ~n2d~nt=−iεa~̇λ~n∫−∞∞ϕ~̇x,λ~n−d~ntψ~̇x,λ~nTσ2Qxϕ~x,λ~ndx.Whenε⟶0, (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,
(89)ϕ1,x=−iλ+v2ϕ1+ϕ2,(90)ϕ2,x=−ωϕ1+iλ−v2ϕ2.To this end, we eliminateϕ2 from system (89) and introduce a new function ϕ^ by
(91)ϕ1=e−iλx+η−+ϕ∧,then we get a Riccati equation
(92)ϕ^xx+ϕ^x2−2iλϕ^x+vϕ^x+ω=0.Assume thatϕ^ has the following expansion:
(93)ϕ^=∑j=0∞ϕ^j2iλj,then by substituting (93) into (92), we find
(94)ϕ^1=∫−∞xωydy,ϕ^2=∫−∞xωyy+vyωydy,ϕ^3=∫−∞xωyyy+ω2y+v2yωy+vωyy+vyωyydy.For the BK system,aλ and bλ are independent of t, then we obtain the following conserved densities:
(95)I1=∫−∞∞ωxdx,I2=∫−∞∞vxωxdx,(96)I3=∫−∞∞ω2x+v2xωx+vxωxxdx,(97)ddtIj=0,Ij=∫−∞∞Tjxdx.Hence, there exists an associated fluxXjx that satisfies
(98)∂tTjx+∂xXjx=0,and the BK system (1); we find
(99)T1=ω,X1x=−vω+12ωx.For the BK system with corrections, we still have(100)ϕ^∞,λ=∑j=0∞Ij2iλj.Since,aλ depends on t, (97) is not valid. In fact, the evolution of densities Ij is an ε-order term which denotes perturbation correction of the conservation laws. To find the evolution, we rewrite the BK system with corrections
(101)vt=S+εqx,ωt=R+εpxS=12v2+2ω−vxx,R=vω+12ωxx.Making use of the functional derivative, we get(102)ddtIj=∫−∞∞δIjδvxddtvx+δIjδωxddtωxdx,(103)ddtIj=∫−∞∞δIjδvxS+δIjδωxRdx+ε∫−∞∞δIjδvxqx2+δIjδωxpxdx.We note that the integrand in the first integral is a divergence term which makes the integral varnish. Hence, (103) reduces to
(104)ddtIj=ε∫−∞∞δIjδvxqx+δIjδωxpxdx.


## 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
(105)ψ~1x,λ=eη+e−iλx+∑j=1Nicjλ−λjψ1x,λeiλjxe−iλx,ψ~2x,λ=∑j=1Nicjλ−λjψ2x,λeiλjxe−iλx,(106)ψ1x,λ=∑j=1Nic~jλ−λ~jψ~1x,λ~jeiλ~jxeiλx,ψ2x,λ=e−η+eiλx+∑j=1Nic~jλ−λ~jψ~2x,λ~jeiλ~jxeiλx,where
(107)cj=djiȧλj,c~j=d~jia~̇λ~j.The solutions of this system can be expressed in a closed form. For example, forN=1, let λ=λ~1 in (105) and λ=λ1 in (106), and using Cramer’s rule, we find
(108)ψ~1x,λ~1=1Δeη+e−iλ~1x,(109)ψ1x,λ1=−f~1Δeη++iλ1x,(110)ψ~2x,λ~1=f1Δe−η+−iλ~1x,(111)ψ2x,λ1=1Δe−η++iλ1x,where
(112)Δ=1+f1f~1,f1=ic1λ~1−λ1e2iλ1x,f~1=ic~1λ~1−λ1e−2iλ~1x.Due to(113)aλ=eηλ−λ1λ−λ~1,a~λ=eηλ−λ~1λ−λ1,where
(114)η=η−−η+=12∫±∞∞vxdx.Hence,(115)f1=−d1te−ηe−2iλ1x,f~1=d~1te−ηe−2iλ~1x.For the BK system andN=1, λ1=iυ1 and λ~1=iυ~1, using d1,tt+2λ12d1t=0 and d~1,tt−2λ~12d~1t=0,
(116)f1=e−z,f~1=e−z~,where
(117)Z=η+2ν1x−2ν12t+Z10,(118)Z~=−η−2ν~1x−2ν~12t+Z~10.Comparing (25) and (29), we obtain
(119)eη+ω=−2c1ψ2x,λ1eiλ1x,e−η+=−2c~1ψ~1x,λ~1e−iλ~1x,which gives rise to the solution of the BK system
(120)v=−ν1+ν~1+tanhF,ω=ν1−ν~12sech2F,where
(121)F=12Z+Z~.For the BK system with corrections,λ1 depends on t and its time evolution is ε-order. d1t has ε-order’s correction besides the previous time evolution. In this case, the Jost functions ϕx,λ, ϕ~x,λ, ψ~x,λ, and ψx,λ will be denoted by ϕax,λ, ϕ~ax,λ, ψ~ax,λ, and ψax,λ and the expressions eη+ω and e−η+ denoted by eη+ωa and e−η+a which give the adiabatic approximate solutions of the BK system with correction terms. We note that ϕax,λ, ϕ~ax,λ, ψ~ax,λ, and ψax,λ satisfy the same equations as (105) and (106). In addition,
(122)eη+ωa=−2c1ψ2ax,λ1eiλ1x,(123)e−η+a=−2c1ψ~1ax,λ~1e−iλ~1x.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 Section4. Hence, eη+ωa and e−η+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.


## 7. Slow Variations of the Spectral Parameters

To discuss the slow variations of the spectral parameters, we rewriteZandZ~ in (117) as
(124)Z=η+2ν1x+δ,dδdt=−2ν12,(125)Z~=−η−2ν~1x+δ~,dδ~dt=−2ν~12.In this case,(126)d1t=eδ,d~1t=eδ~.In addition, for reflectionless case andN=1, the Jost functions take the followingform:
(127)ψ1x,λ=12λ~1−λ1λ−λ~1eη+eiλxf1eFsechF,(128)ψ2x,λ=eiλxe−η+λ−λ~1λ−12λ1+λ~1+12λ1−λ~1tanhF,(129)ψ~1x,λ=e−iλxeη+λ−λ1λ−12λ1+λ~1−12λ1−λ~1tanhF,(130)ψ~2x,λ=12λ~1−λ1λ−λ1e−η+e−iλxf~1eFsechF,(131)ϕ1x,λ=e−iλxeη−λ−λ~1λ−12λ1+λ~1−12λ1−λ~1tanhF,(132)ϕ2x,λ=12λ~1−λ1λ−λ~1eηe−η+e−iλxf~1eFsechF,(133)ϕ~1x,λ=12λ~1−λ1λ−λ1e−ηeη+eiλxf1eFsechF,(134)ϕ~2x,λ=eiλxeη−λ−λ1λ−12λ1+λ~1+12λ1−λ~1tanhF.These equations can be derived by takingN=1 and substituting (108) into (29). It is remarked that ψax,λ, ψ~ax,λ, ϕax,λ, and ϕ~ax,λ have a similar form to those in (127) for the BK system with corrections. In fact, the eigenfunctions in (69) are ψax,λ, ψ~ax,λ, ϕax,λ, and ϕ~ax,λ. Substituting (127) into (69), we get
(135)λ1,t=−14iε∫−∞∞sech2qx−pxe2η+e−Z~dF,λ~1,t=−14iε∫−∞∞sech2qx+pxe2η+eZdF,or
(136)v1,t=−14ε∫−∞∞sech2qx−pxe2η+e−Z~dF,v~1,t=−14ε∫−∞∞sech2qx+pxe2η+eZdF.Next, we investigate the variation of parameter ind1t and d~1t. To this end, combining (127) and (66), we find
(137)d1t=−e−2iλ1xe−Zeη,d~1t=e2iλ~1xe−Z~e−η.For the BK system with corrections, (126) is not valid and (124) implies
(138)d1,tt+2λ12d1t=d1tδt−2ν12,(139)d~1,tt−2λ~12d~1t=d~1tδ~t+2ν~12.Using relevant integral equations in (87), we find from (138) that
(140)δt=2ν12−ε2∫−∞∞xsech2Fqx−pxe2η+e−Z~+sechFν1−ν~1qxsinF−pxe2η+eZe−FdF,δ~t=−2ν~12+ε2∫−∞∞xsech2Fqx+pxe2η+e−Z−sechFν~1−ν1qxcosF−pxe2η+e−Z~eFdF.Now, for one soliton solution, the formulas of spectral parameters are given.


## 8. Corrections of the Adiabatic Solutions

In this section, we will discuss the other part of theε-order approximate solutions to the BK system with correction terms. In the case of N=1,
(141)ψ~1x,λeiλx=eη++ic1λ−λ1ψ1x,λ1eiλ1x+J1x,λ,(142)ψ~2x,λeiλx=ic1λ−λ1ψ2x,λ1eiλ1x+J2x,λ,(143)ψ1x,λe−iλx=ic~1λ−λ~1ψ~1x,λ~1e−iλ~1x+J~1x,λ,(144)ψ2x,λe−iλx=e−η++ic~1λ−λ~1ψ~2x,λ~1e−iλ~1x+J~2x,λ,where
(145)Jjx,λ=12iπ∫−∞∞ρkk−λ+i0ψjx,keikxdk,j=1,2,J~jx,λ=12iπ∫−∞∞ρ~kk−λ−i0ψ~jx,ke−ikxdk,j=1,2.To solve system (141) to ε-order, we introduce
(146)ψx,λ=ψax,λ+δψx,λ,ψ~x,λ=ψ~ax,λ+δψ~x,λ,and note that ψax,λ and ψ~ax,λ satisfy
(147)ψ~1ax,λ=eη+e−iλx+ic1λ−λ1ψ1ax,λ1eiλ1xe−iλx,ψ~2ax,λ=ic1λ−λ1ψ2ax,λ1eiλ1xe−iλx,ψ1ax,λ=ic~1λ−λ~1ψ~1ax,λ~1e−iλ~1xeiλx,ψ2ax,λ=e−η+eiλx+ic~1λ−λ~1ψ~2ax,λ1e−iλ~1xeiλx,which are called the adiabatic approximate equations of system (141), then equations to the ε-order from (141) take the following form:
(148)δψ~1x,λeiλx=ic1λ−λ1δψ1x,λ1eiλ1x+J1ax,λ,(149)δψ~2x,λeiλx=ic1λ−λ1δψ2x,λ1eiλ1x+J2ax,λ,(150)δψ1x,λe−iλx=ic~1λ−λ~1δψ~1x,λ~1e−iλ~1x+J~1ax,λ,(151)δψ2x,λe−iλx=ic~1λ−λ~1δψ~2x,λ~1e−iλ~1x+J~2ax,λ,(152)Jjax,λ=12iπ∫−∞∞ρkk−λψjax,keikxdk,j=1,2,(153)J~jax,λ=12iπ∫−∞∞ρ~kk−λψ~jax,ke−ikxdk,j=1,2.Using Cramer’s rule, we find, from (148), that
(154)δψ2x,λ1=1Δeiλ1xJ~2ax,λ1+J2ax,λ~1ic~1λ−λ~1e−2iλ~1x,(155)δψ~1x,λ~1=1Δe−iλ~1xJ1ax,λ~1+J~1ax,λ1ic1λ~1−λ1e2iλ1x.Hence,ε-order approximate solutions of the BK system with correction terms can be written as
(156)eη+ω=eη+ωa+δeη+ω,(157)e−η+=e−η+a+δe−η+,In addition, from (122) and (156), we know that
(158)δeη+ω=−2c1δψ2x,λ1eiλ1x+1π∫−∞∞ρkψ2ax,kdk,(159)δe−η+=−2c~1δψ~1x,λ~1e−iλ~1x−1π∫−∞∞ρ~kψ~1ax,kdk.Substituting (154) into (158), we get
(160)δeη+ω=eη+aπ∫−∞∞ρ~kψ~2ax,k2−ρkψ2ax,k2dk,δe−η+=e−η+aπ∫−∞∞ρkψ1ax,k2−ρ~kψ~1ax,k2dk.


## 9. Conclusions and Discussions

In this work, we discussed the perturbation theory of the Broer-Kaup equation with corrections which is a nearly integrable system. The technique based on the inverse scattering transform is extended to obtain the evolutions of the scattering data, including the discrete spectrums, the scattering coefficients, and the normalized factors. These evolution equations of the scattering data are determined by integral representations of corrections and eigenfunctions. Hence, the adiabatic dynamic of one soliton can be figured out by the time-dependent scattering data, if the corrections are given.In general, the perturbations by adding corrections can be dissipative terms.



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