# New Lax Pairs and Darboux Transformation and Its Application

Authors: Huizhang Yang; Weiguo Rui
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

New Lax pairs of a shallow water wave model of
generalized KdV equation type are presented. According to this Lax
pair, we constructed a new spectral problem. By using this
spectral problem, we constructed Darboux transformation with the
help of a gauge transformation. Applying this Darboux
transformation, some new exact solutions including double-soliton solution of the shallow
water wave model of generalized KdV equation type are obtained. In
order to visually show dynamical behaviors of these double soliton
solutions, we plot graphs of profiles of them and discuss their
dynamical properties.


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## Body

## 1. Introduction

It is well known that the Lax pair and Darboux transformation can be employed to obtain multisoliton solution of nonlinear evolution equations. Darboux transformations provide us with purely algebraic, powerful method to construct solutions for systems of nonlinear equations. In recent years, more and more researchers used the Lax pair and Darboux transformation to investigate soliton solutions of classical nonlinear wave equations and some new soliton equations which were generated by new spectral problems; see [1–30] and references cited therein. In general, a systematical theory on such Darboux transformation even for n×n matrix spectral problem and the resulting zero curvature equation has a beautiful algebraic structure for associated evolution equations, which tells symmetry algebras of the obtained evolution equations; see [31, 32] and references cited therein. Sometimes, it is found that there are many infinity symmetries from the adopted zero curvature equation.In this paper, we will investigate the Lax pairs, Darboux transformation, and double soliton solutions of the following famous shallow water wave model of generalized KdV equation type:(1)ut-h023uxxt+c0ux+3α2uux-c0h026uxxx=αh023uxuxx+αh026uuxxx,
which appeared in [33], where 0<α≪1 is a small-amplitude parameter. Only dropping the right-hand side of (1) gives BBM equation. Dropping the right-hand side of (1) and replacing the term uxxt by the term -c0uxxx give KdV equation [34]. Thus, (1) can be seen as a BBM equation extended by retaining higher order terms in an asymptotic expansion in terms of the small-amplitude parameter α. Dropping the term -(c0h02/6)uxxx from (1) and letting h0=3,c0=2κ,α=2, (1) becomes the celebrated Camassa-Holm equation [33] as follows:
(2)ut+2κ-uxxt+3uux=2uxuxx+uuxxx,
where u is the fluid velocity in the x direction (or equivalent to the height of the water’s free surface above a flat bottom), κ is a constant related to the critical shallow water wave speed, and subscripts denote partial derivatives. Letting c0=1,ν=-(1/3)h02,γ=(3/2)α,β=-(1/6)h02c0, (1) can be rewritten as
(3)ut+ux+νuxxt+βuxxx+γuux+13γν(uuxxx+2uxuxx)=0,
which comes from physical considerations via the methodology introduced by Fuchssteiner and Fokas in [35, 36]. The Lax pairs of (3) with γ=1 are given by [37, 38] as follows:
(4)Ψxx=[k2(1-v)+v4ν]Ψ=0,σv=u+νuxx,σ=β-ν2ν,(5)Ψt+(-12+c)Ψ+(u+βν+2σ4νk2-1)Ψx=0,
where c is an arbitrary constant. It is a pity that the [37] is not a formal publication, and it is only preprint paper, so we cannot know its reality contents whether the authors have obtained soliton solutions of (3) by the Lax pairs (4) and (5). In fact, (3) has been studied by many authors in recent years; see the following brief introductions.In [39], by using the bifurcation theory of dynamic system, some subsection-function and implicit function solutions such as compactons, solitary waves, smooth periodic waves, and nonsmooth periodic waves with peaks as well as the existence conditions have been presented by Bi. By using the same method, Li and Zhang [40] studied a generalization form of the modified KdV equation, which is more complex than (3). In [40], the existence of solitary wave, kink and antikink wave solutions, and uncountably many smooth and nonsmooth periodic wave solutions are discussed. By using the improved method named integral bifurcation method [41], Rui et al. [42] obtain all kinds of soliton-like or kink-like wave solutions, periodic wave solutions with loop or without loop, smooth compacton-like periodic wave solutions, and nonsmooth periodic cusp wave solutions for (3). In [43], Long and Chen discussed the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave, and compactons were for (3). From the above references, (1) (i.e., (3)) is a very important water wave model.The rest of this paper is organized as follows. In Section2, we will derive new Lax pair and Darboux transformation of (1). In Section 3, by using this Darboux transformation, we will investigate soliton solutions of (1) and discuss the dynamic properties of these soliton solutions.


## 2. Lax Pair and Darboux Transformation of (1)

Through a series of tedious computation, we obtain Lax pairs of (1) as follows:
(6)ϕxx=(-c04λα+34h02-u-(1/3)h02uxx2λ)ϕ,(7)ϕt=-(c02+3λα2h02+α2u)ϕx+α4uxϕ.
Obviously, the Lax pairs (6) and (7) are different from the Lax pairs (4) and (5) under c0=1,ν=-(1/3)h02,γ=(3/2)α,β=-(1/6)h02c0. They are new Lax pairs which we obtained. By using the new Lax pairs (6) and (7), we will construct a Darboux transformation for obtaining soliton solutions of (1).First, we consider the following spectral problems:(8)ϕx=Mϕ,ϕt=Nϕ,
with(9)M=(01-c04λα+34h02-u-(1/3)h02uxx2λ0),N=(α4ux-(c02+3λα2h02+α2u)α4uxx-(c02+3λα2h02+α2u)(-c04λα+34h02-u-(1/3)h02uxx2λ)-α4ux),where α is a constant, λ is a spectral parameter, and u is a potential function. From compatibility, condition ϕxxt=ϕtxx yields a zero curvature equation Mt-Nx+[M,N]=O. Substituting M,N into the zero curvature equation, by a direct calculation, (1) is obtained successfully.Next, we will construct a Darboux Transformation (DT) of the spectral problems (8). In fact, the DT is actually a gauge transformation
(10)ϕ¯=Tϕ
of the spectral problems (8). It is required that ϕ¯ also satisfies the same form of spectral problems
(11)ϕ¯x=M¯ϕ¯,M¯=(Tx+TM)T-1,(12)ϕ¯t=N¯ϕ¯,N¯=(Tt+TN)T-1.It means that we have to find a matrixT such that the old potential u is replaced by the new one u¯.Suppose(13)T=T(λ)=(A(λ)B(λ)C(λ)D(λ)),
where
(14)A(λ)=AN(λN+∑k=0N-1‍Akλk),B(λ)=AN(∑k=0N-1‍Bkλk),(15)C(λ)=1AN(∑k=0N-1‍Ckλk),D(λ)=1AN(λN+∑k=0N-1‍Dkλk),
and AN,Ak,Bk,Ck, and Dk(0≤k≤N-1) are functions of x and t.Letϕ(λj)=(ϕ1(λj),ϕ2(λj))T,ψ(λj)=(ψ1(λj),ψ2(λj))T be two basic solutions of (8). From (10), there exist constants rj(0≤j≤N-1), which satisfy
(16)(A(λj)ϕ1(λj)+B(λj)ϕ2(λj))-rj(A(λj)ψ1(λj)+B(λj)ψ2(λj))=0,(C(λj)ϕ1(λj)+D(λj)ϕ2(λj))-rj(C(λj)ψ1(λj)+D(λj)ψ2(λj))=0.
Further, (16) can be written as a linear algebraic system
(17)A(λj)+δjB(λj)=0,C(λj)+δjD(λj)=0.
That is
(18)∑k=0N-1‍(Ak+δjBk)λjk=-λjN,∑k=0N-1‍(Ck+δjDk)λjk=-δjλjN,
where
(19)δj=ϕ2(λj)-rjψ2(λj)ϕ1(λj)-rjψ1(λj),1≤j≤2N,
and the constants λj(λk≠λs as  k≠s),rj are suitably chosen such that determinant of coefficients for (18) is nonzero. Therefore, AN,Ak,Bk,Ck, and Dk(0≤k≤N-1) are uniquely determined by (18).Equations (14) and (15) show that the detT(λ) is a 2Nth-order polynomial in λ, and
(20)detT(λj)=A(λj)D(λj)-B(λj)C(λj).
On the other hand, from (17), we have A(λj)=-δjB(λj),C(λj)=-δjD(λj). Thus we have
(21)detT(λ)=β∏j=12N-1‍(λ-λj),
where β is independent of λ. Equation (21) implies that λj(1≤j≤2N) are 2N roots of detT(λ).Second, we prove the following theory of Darboux transformation for special variable.Theorem 1.
LetAN satisfy
(22)AN2=1.
Then the matrix M¯ determined by (11) has the same form as M; that is,
(23)M¯=(01-c04λα+34h02-u¯-(1/3)h02u¯xx2λ0),
where the transformation from the old potential u into new one u¯ is given by
(24)u¯=u+6h02AN-1,(25)AN-1,x=CN-1-34h02BN-1,BN-1,x=DN-1-AN-1,CN-1,x=34h02(AN-1-DN-1)+c04α+u-(1/3)h02uxx2,DN-1,x=34h02BN-1-CN-1.Proof.
LetT-1=T*/detT and
(26)(Tx+TM)T*=(f11(λ)f12(λ)f21(λ)f22(λ)).
It is easy to see that f11(λ) and f22(λ) are 2Nth-order polynomials in λ, f12(λ) and f21(λ) are (2N-1)th-order polynomials in λ. From (19) and (8), we find
(27)δjx=-c04λα+34h02-u-(1/3)h02uxx2λ-δj2.
Through direct calculation, all λj(0≤j≤2N) are roots of fns(n,s=1,2). Together with (21) and (26), we get
(28)(Tx+TM)T*=(detT)P(λ),
with
(29)P(λ)=(p11(0)p12(0)p21(0)p22(0)),
where pns(0)(n,s=1,2) are independent of spectral parameter λ. Indeed, (28) can be written as
(30)Tx+TM=P(λ)T.
Comparing the coefficients ofλN in (30), we find
(31)p11(0)=-p22(0)=∂xlnAN,p12(0)=AN2,p21(0)=1AN234h02.
Substituting (22) into (31) yields
(32)p11(0)=-p22(0)=0,p12(0)=AN2=1.
From (22), (24), (25), and (31) and noticing u¯ in (23), we get
(33)p21(0)=-c04λα+34h02-u¯-(1/3)h02u¯xx2λ.
Thus P(λ)=M¯. The proof of Theorem 1 is completed.Finally, by using same way to Theorem1, we prove that N¯ in (12) has the same form as N under the transformation (10) and (24); see the following theory and its proof.Theorem 2.
The matrixN¯ defined by (12) has the same type as N, in which the old potential u is mapped into u¯ via the same DT (24).Proof.
LetT-1=T*/detT and
(34)(Tt+TN)T*=(g11(λ)g12(λ)g21(λ)g22(λ)).
It is easy to see that g11(λ) and g22(λ) are 2Nth-order polynomials in λ, g12(λ) and g21(λ) are (2N+1)th-order polynomials in λ. By using (19) and (8), we obtain
(35)δjt=α4uxx-(c02+3λα2h02+α2u)×(-c04λα+34h02-u-(1/3)h02uxx2λ)-α2uxδj+(c02+3λα2h02+α2u)δj2.
Through direct calculation, all λj(0≤j≤2N) are roots of gns(n,s=1,2). Together with (21) and (34), we get
(36)(Tt+TN)T*=(detT)Q(λ),
with
(37)Q(λ)=(q11(0)q12(1)λ+q12(0)q21(1)λ+q21(0)q22(0)),
where qns(l)(n,s=1,2,l=0,1) are independent of spectral parameter λ. Equation (37) can be written as
(38)Tt+TN=Q(λ)T.
Comparing the coefficients ofλN+1 and λN in (38) leads to
(39)q12(1)=-3α2h02AN2,q21(1)=-1AN23α2h0234h02,(40)q11(0)=∂tlnAN+α4ux-3α2h0234h02BN-1+3α2h02CN-1,(41)q12(0)=-(c02+α2u)AN2-3α2h02AN2AN-1+3α2h02AN2DN-1,(42)q21(0)=1AN2α4uxx-1AN234h02(c02+α2u)-1AN23α2h0234h02DN-1+1AN23α2h02(c04α+u-(1/3)h02uxx2)+AN-1AN23α2h0234h02,(43)q22(0)=-∂tlnAN-3α2h02CN-1-α4ux+3α2h0234h02BN-1.
Substituting (22), (24), and (25) into (40) to (43), we can get
(44)q11(0)=α4ux-3α2h0234h02BN-1+3α2h02CN-1=α4ux-3α2h02AN-1,x=α4u¯x,q12(1)λ+q12(0)=-3α2h02λ-(c02+α2u)-3α2h02AN-1+3α2h02DN-1=-3α2h02λ-(c02+α2u)-3αh02+3α2h02(AN-1+DN-1)=-(c02+3λα2h02+α2u¯),q22(0)=-3α2h02CN-1-α4ux+3α2h0234h02BN-1=-α4ux-3α2h02AN-1,x=-α4u¯x,q21(1)λ+q21(0)=-3α2h0234h02λ+α4uxx-34h02(c02+α2u)-3α2h0234h02DN-111+3α2h02(c04α+u-(1/3)h02uxx2)+3α2h0234h02AN-1=α4uxx-34h02(c02+3λα2h02+α2u)+3α2h02CN-1,x.
On the other hand, from (24), (25), and (33), we have
(45)α4u¯xx-(c02+3λα2h02+α2u¯)×(-c04λα+34h02-u¯-(1/3)h02u¯xx2λ)=α4u¯xx-(c02+3λα2h02+α2u¯)34h02=α4uxx+3α2h02AN-1,xx-(c02+3λα2h02+α2u)34h02-6α2h0234h02AN-1=α4uxx-(c02+3λα2h02+α2u)34h02+3α2h02CN-1,x.
From (44) and (45), obviously we have
(46)q21(1)λ+q21(0)=α4u¯xx-(c02+3λα2h02+α2u¯)×(-c04λα+34h02-u¯-(1/3)h02u¯xx2λ).
Thus Q(λ)=N¯, and this completes the proof of Theorem 2.


## 3. Exact Soliton Solution of (1) and Its Dynamic Properties

In this section, we will construct the explicit solutions of the integrable shallow water wave (1) by using the Darboux transformation (24).Choosingu=u0 (u0 is an arbitrary constant) as a seed solution of (1), and substituting u=u0 into the spectral problems (8). Then we get two basic solutions of (8) as follows:
(47)ϕ(λj)=(coshμj-c04λjα+34h02-u02λjsinhμj),ψ(λj)=(sinhμj-c04λjα+34h02-u02λjcoshμj)
with
(48)μj=-c04λjα+34h02-u02λj[x-(c02+3λjα2h02+u0α2)t],
where λj is a nonzero constant and 0<j≤2N.According to (19), we have
(49)δj=-c04λjα+34h02-u02λjtanhμj-rj1-rjtanhμj,
where rj(0<j≤2N) are nonzero constants.Using the Cramer rule to solve the linear algebraic system (18), we obtain
(50)AN-1=ΔAN-1Δ,
where(51)Δ=|1δ1λ1δ1λ1⋯λ1kδ1λ1k⋯λ1N-1δ1λ1N-11δ2λ2δ2λ2⋯λ2kδ2λ2k⋯λ2N-1δ2λ2N-1⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮1δ2Nλ2Nδ2Nλ2N⋯λ2Nkδ2Nλ2Nk⋯λ2NN-1δ2Nλ2NN-1|,ΔAN-1=|1δ1λ1δ1λ1⋯λ1kδ1λ1k⋯-λ1Nδ1λ1N-11δ2λ2δ2λ2⋯λ2kδ2λ2k⋯-λ2Nδ2λ2N-1⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮1δ2Nλ2Nδ2Nλ2N⋯λ2Nkδ2Nλ2Nk⋯-λ2NNδ2Nλ2NN-1|.As examples, we will investigate exact solutions of (1) in two simple cases N=1 and N=2. When N=1, solving the linear algebraic system (18) leads to
(52)A0=δ1λ2-δ2λ1δ2-δ1.
Substituting (49) and (52) into (24), a singular double-soliton solution of (1) is obtained as follows:
(53)u¯[1]=6h02f[-c04λ1α+34h02-u02λ1tanhμ1-r11-r1tanhμ1λ2=6h02fp--c04λ2α+34h02-u02λ2tanhμ2-r21-r2tanhμ2λ1],
where
(54)f=-c04λ2α+34h02-u02λ2tanhμ2-r21-r2tanhμ2--c04λ1α+34h02-u02λ1tanhμ1-r11-r1tanhμ1.
By using program of computer, it is easy to verify that the solution (53) satisfies (1), and this shows that the Darboux transformation (24) which we obtained is correct. In order to show the properties of the above singular double-soliton solutions visually, as an example, we plot the 3-D graphs of solution (53) for some fixed parameters, which are shown in Figures 1 and 2.The 3-D graphs of profiles of the singular double-soliton solution (53) for fixed parameters c0=1.5, α=0.2, λ1=-1, λ2=-0.2, r1=-8.0, r2=2.0, and a=0.1.
(a)
0.3
<
h
0
<
0.5
,
λ
1
<
0
,
λ
2
<
0
(b)
0.2
<
h
0
<
0.3
,
λ
1
<
0
,
λ
2
<
0The 3-D graphs of profiles of the singular double-soliton solution (53) for parameters: (a) h0=0.15, c0=1.5, α=0.2, λ1=-1, λ2=-0.2, r1=-8.0, r2=2.0, and a=0.1; (b) h0=3, c0=-1.5, α=0.2, λ1=1, λ2=0.2, r1=8, r2=2.0, and a=-2.
(a)
0<h0<0.2,λ1<0,λ2<0
(b)
h
0
>
1
,
λ
1
>
0
,
λ
2
>
0WhenN=2, using the Cramer rule to solve the linear algebraic system (18), we obtain
(55)A1=ΔA1Δ,
with
(56)Δ=|1δ1λ1δ1λ11δ2λ2δ2λ21δ3λ3δ3λ31δ4λ4δ4λ4|,ΔA1=|1δ1-λ12δ1λ11δ2-λ22δ2λ21δ3-λ32δ3λ31δ4-λ42δ4λ4|,
where δj(j=1,2,3,4) are given by (49). From (24), an explicit solution of (1) is obtained by the following:
(57)u¯[2]=u+6h02A1,
where A1 is given by (55). Equation (57) is a very complex solution, and it is not soliton solution. In order to show the properties of solution (57), under the fixed parameters λ1=-0.2, λ2=-0.3, λ3=-0.4, λ4=-0.1, a=0, c0=15, α=1/2, h0=0.8, r1=-0.2, r2=-0.3, r3=-0.4, r4=-0.5, t=0.1, we plot its 2-D profile, which is shown in Figure 3.Figure 3
The 2-D graph of profile of the exact soliton solution (57) for fixed parameters.



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*Source: 548690-2013-11-04.xml*