# A Riemann-Hilbert Approach to the Multicomponent Kaup-Newell Equation

Authors: Jian-bing Zhang; Ze-xuan Zhang
Journal: Advances in Mathematical Physics
 (2020)

---

## Abstract

A Riemann-Hilbert approach is developed to the multicomponent Kaup-Newell equation. The formula is presented ofN-soliton solutions through an identity jump matrix related to the inverse scattering problems with reflectionless potential.


---

## Body

## 1. Introduction

Many nonlinear partial differential equations especially soliton equations have exact solutions [1–3]. There are a lot of methods to solve soliton equations such as the Hirota bilinear method [2–6], Wronskian technique (Casoratian technique) [5–9], and Darboux transformation [10, 11]. The inverse scattering transformation (IST) is one of the most powerful tools and closely connected with those methods mentioned above [1, 3]. It is also called nonlinear Fourier transform for its procedure to solve the nonlinear equations is similar to the linear Fourier transform. One advantage of the IST is that it can be applied to the whole soliton hierarchies [3]. Recently, researches show that the IST can solve not only classic soliton hierarchies but also soliton equations with self-consistent sources [12], nonisospectral soliton hierarchies [13], hierarchies mixed with isospectral and nonisospectral ones [14], and nonlocal soliton hierarchies [15]. Furthermore, it can generate both soliton and general matrix exponent solutions [16, 17].The Riemann-Hilbert (RH) approach is another effective method to solve soliton equations. It actually shares a close relationship with the IST [18–20]. Both of them start from same matrix spectral problems which possess bounded eigenfunctions analytically extendable to the upper or lower half-plane. To get scattering data, we must consider the asymptotic conditions at infinity on real axis by the IST to solve soliton equations. In fact, the considered conditions are used as the solutions to the corresponding RH problems. When the jump matrix is an identity matrix, the RH problem is equivalent to the IST with reflectionless potentials, and N-soliton solutions can be generated [21–23]. Recently, Ma has already used the method to solve multicomponent soliton equations such as multicomponent AKNS integrable hierarchies and a coupled mKdV equation [24–26].It is known to us all that the three famous derivative nonlinear Schrödinger equations, the Chen-Lee-Liu (CLL) equation [17, 27], Kaup-Newell (KN) equation [28], and Gerdjikov-Ivanov (GI) equation [29, 30], can be reduced from the Kundu equation by choosing different value of the arbitrary parameter [31, 32]. Many properties of them have been researched such as exact solutions [30, 33], conservation laws [34], multi-Hamilton structure [31], and τ-symmetry algebra [32, 34].In this paper, we will present the multicomponent KN equation with itsn+1×n+1 matrix Lax pairs. To formulate an RH problem of the equation, we consider a modify matrix Lax pairs. The formula of generating the N-soliton solutions to multicomponent KN equation will be obtained through taking the identify jump matrix.The paper is organized as follows. In Section2, we will introduce the multicomponent isospectral KN equation and its Lax pairs. In Section 3, we will construct a multicomponent RH problem to the equation introduced in the previous section. In Section 4, the expression of N-soliton solutions will be obtained. We conclude the paper in Section 5.


## 2. The Multicomponent KN Equation

In this section, we will present the isospectral multicomponent KN equation from an+1×n+1 matrix spectral problem by the zero-curvature representation. To our knowledge, there is another powerful method to build soliton equation hierarchies through Kac-Moody algebra and principal gradation [35].Suppose thatpj and qj are smooth functions of variables x and tj=1,2,⋯n, T denotes the transpose of matric, and In is an n×n identity matrix. Let us consider the following Lax pairs
(1a)−iϕx=Uϕ,U=α1λ2λpλqα2Inλ2=λ2Λ+λP,(1b)−iϕt=Vϕ,V=α1λ4−1αpqλ2pλ3−iαpx+2α2pqpλqλ3+iαqx−2α2pqqλα2Inλ4+1αqpλ2,whereα1 and α2 are two real constants; α=α1−α2, p=p1,p2,⋯,pn, and q=q1,q2,⋯,qnT are potential functions; λ is a spectral parameter; and
(2)Λ=α100α2In,P=0pq0.Obviously,p and q are smooth component functions of variables t and x. Assume that p,q, and their derivatives of any order with respect to x vanish rapidly as x⟶∞.The compatibility condition of (1), i.e., the zero curvature equation(3)Ut−Vx+iU,V=0,generates the multicomponent KN soliton equation
(4)pTqt=1α−ipxxT−2αpqpTxiqxx−2αpqqx.For example, whenn=2, the spectral problem (1a) becomes
(5)−iϕ^x=U^ϕ^,U^=α1λ2λp1λp2λq1α2λ20λq20α2λ2=λ2Λ^+λP^,where
(6)Λ^=α1000α2000α2,P^=0p1p2q100q200.Its time evolution is(7)−iϕ^t=V^ϕ^,V^=V11V12V13V21α2λ4+1αq1p1λ21αq1p2λ2V311αq2p1λ2α2λ4+1αq2p2λ2,with
(8)V11=α1λ4−1α<p^,q^>λ2,V12=p1λ3−iαp1x+2α2<p^,q^>p1λ,V13=p2λ3−iαp2x+2α2<p^,q^>p2λ,V21=q1λ3+iαq1x−2α2<p^,q^>q1λ,V31=q2λ3+iαq2x−2α2<p^,q^>q2λ,<p^,q^>=p1q1+p2q2.The 4-component KN equation is(9)p1t=−iαp1xx−2α2p1q1+p2q2p1x,p2t=−iαp2xx−2α2p1q1+p2q2p2x,q1t=iαq1xx−2α2p1q1+p2q2q1x,q2t=iαq2xx−2α2p1q1+p2q2q2x.


## 3. The RH Problem to the Multicomponent KN Equation

In this section, we will build the RH problem to the multicomponent KN equation (5). Here, we only focus on the positive flows. Constructing the RH problem from negative symmetry flows have already appeared in [36] for the homogeneous Am-hierarchy and its g^lm+1,ℂ.Setting(10)V=λ4Λ+Q,it is obvious that the trace ofQ is zero, where
(11)Q=−1αpqλ2pλ3−iαpx+2α2pqpλqλ3+iαqx−2α2pqqλ1αqpλ2.Thus, the equation (5) has the following Lax pairs
(12)−iϕx=λ2Λϕ+λPϕ,−iϕt=λ4Λϕ+Qϕ.Next, we will present the scattering and inverse scattering methods for the multicomponent KN equation (5) by the RH approach. The resulting results will lay the groundwork for N-soliton solutions in the next section. Suppose that all the potentials rapidly vanish when x⟶±∞ or t⟶±∞ and satisfy
(13)∫−∞∞∫−∞∞xmtn∑j=1n∣pj∣+∣qj∣dxdt<∞,m,n≥0.In the RH approach, we treatϕ in the spectral problem (1a) as a fundament matrix. From (12), we note, under (13), one has the asymptotic behavior: ϕ~Eg=eiλ2Λx+iλ4Λt. This motivate us to introduce the variable transformation
(14)ψ=ϕe−iλ2Λx−iλ4Λt,to have the canonical normalization for the associated RH problem:
(15)ψ⟶In+1,whenx,t⟶±∞,whereIn+1 is the n+1×n+1 identity matrix. This way, the spectral problems in (12) equivalently lead to
(16a)ψx=iλ2Λ,ψ+λP˘ψ,(16b)ψt=iλ4Λ,ψ+Q˘ψ,whereP˘=iP and Q˘=iQ. Noticing trP˘=trQ˘=0, we have
(17)detψ=1,by the Abel’s formula.Let us now consider the formulation of an associated RH problem with the variablex. In the scattering problem, we first introduce the matrix solutions ψ±x,λ of (16a) with the asymptotics conditions
(18)ψ±⟶In+1,whenx⟶±∞,respectively. The subscripts above refer to which end of thex-axis the boundary conditions are required. Then, by (17), we have the determinant detψ±=1 for all x∈ℝ. Since ϕ±=ψ±E are both solutions of (12), they must be linearly related, and so, we have
(19)ψ−E=ψ+ESλ,λ2∈ℝ,where
(20)E=eiλ2Λx,Sλ=s1,1s1,2⋯s1,n+1s2,1s2,2⋯s2,n+1⋮⋮⋱⋮sn+1,1sn+1,2⋯sn+1,n+1,λ2∈ℝ,is the scattering matrix. Note thatdetSλ=1 since detψ±=1. Using the method of variation in parameters as well as the boundary condition (19), we can turn the x-part of (12) into the following Volterra integral equation for ψ±:
(21a)ψ−λ,x=In+1+∫−∞xeiλ2Λx−yP˘yψ−λ,yeiλ2Λy−xdy,(21b)ψ+λ,x=In+1−∫x∞eiλ2Λx−yP˘yψ+λ,yeiλ2Λy−xdy.Thus,ψ± allows analytical continuations off the real axis λ2∈ℝ as long as the integrals on their right hand sides converge. Taking α=α1−α2, it is direct to see that the integral equation for the first column of ψ− contains only the exponential factor e−iαλ2x−y. When λ is in the first or third quadrant, i.e., Imλ2>0, let λ2=r+is,s>0. Then, e−iαλ2x−y=e−iαrx−y+αsx−y due to y<x in the integral decays as α<0, and the integral equation for the last n columns of ψ+ contains only the exponential factor eiαλ2x−y, which due to y>x in the integral, also decays when λ2 is in the upper half-plane ℂ+. Thus, these n+1 columns can be analytically continued to the first or third quadrants. Similarly, we find that the last n columns of ψ− and the first column of ψ+ can be analytically continued to the second and fourth quadrants. Let us express
(22)ψ±=ψ1±,ψ2±,⋯,ψn+1±,whereψk± stands for the kth column of ψ±1≤k≤n+1. Then, the matrix solution
(23)P+=P+x,λ=ψ1−,ψ2+,⋯,ψn+1+=ψ−H1+ψ+H2,is analytic in the first and third quadrants ofλ, and the matrix solution
(24)ψ1+,ψ2−,⋯,ψn+1−=ψ+H1+ψ−H2,is analytic in the second and fourth quadrants ofλ, where H1=diag1,0,⋯,0 and H2=diag0,1,⋯,1. In addition, from the Volterra integral equation (21), we know that
(25)P+x,λ⟶In+1,whenλ2∈ℂ+⟶∞,ψ1+,ψ2−,⋯,ψn+1−⟶In+1,whenλ2∈ℂ−⟶∞.Next, we construct the analytic counterpart ofP+ in the second and fourth quadrants of λ. Note that the adjoint equation of the x-part of (12) and the adjoint equation of (16) read as
(26)iϕ~x=ϕ~λ2Λ+λP,iψ~x=λ2ψ~,Λ−iλψ~P˘.It is easy to see that the inverse matricesϕ~±=ϕ±−1 and ψ~±=ψ±−1 solve these adjoint equations, respectively. If we express ψ~± as follows:
(27)ψ~±=ψ~±,1ψ~±,2⋮ψ~±,n+1,whereψ~±,k is the kth row of ψ~±1≤k≤n+1. Then, by similar arguments, we can show that the adjoint matrix solution
(28)P−=ψ~−,1ψ~+,2⋮ψ~+,n+1=H1ψ~−+H2ψ~+=H1ψ−−1+H2ψ+−1,is analytic whenλ is in the second or fourth quadrants, and the other matrix solution
(29)ψ~+,1ψ~−,2⋮ψ~−,n+1=H1ψ~++H2ψ~−=H1ψ+−1+H2ψ−−1,is analytic forλ in first and third quadrants. In the same way, we see that
(30)P−x,λ⟶In+1,whenλ2∈ℂ−⟶∞,ψ~+,1ψ~−,2⋮ψ~−,n+1⟶In+1,whenλ2∈ℂ+⟶∞.Now, we have constructed two matrix functionsP+ and P−, which are analytic in the first or third quadrants and second or fourth quadrants, respectively. Defining
(31)G+x,λ=P+x,λ,λ∈firstorthirdquadrant,G−−1x,λ=P−x,λ,λ∈secondorfourthquadrant,we can easily find that ifλ on the real axis or imaginary axis, the two matrix functions P+ and P− are related by
(32)P−x,λP+x,λ=Gx,λ,λ2∈ℝ,where
(33)Gx,λ=G−−1x,λG+x,λ=EH1+H2SλH1+S−1λH2E−1=E1s^1,2s^1,3⋯s^1,n+1s2,110⋯0s3,101⋯0⋮⋮⋮⋱⋮sn+1,100⋯1E−1,(34)S−1λ=Sλ−1=s^j,ln+1,n+1,j,l=1,2,⋯,n+1.Eq. (32) and Eq.(33) are exactly the associated matrix RH problem we wanted to present. The asymptotic conditions
(35a)P+x,λ⟶In+1,whenλinthefirstorthirdquadrant,(35b)P−x,λ⟶In+1,whenλinthesecondorfouthquadrant,provide the canonical normalization condition for the established RH problem.To finish the direct scattering transform, we take the derivative of (19) with time t and use the vanishing conditions of the potentials; we can show that S satisfies
(36)St=iλ4Λ,S,which gives the time evolution of the scattering coefficients:
(37)s1,j=s1,j0,λeiαλ4t,sj,1=sj,10,λe−iαλ4t,0≤j≤n+1,and the other scattering data do not depend on timet.


## 4.N-Soliton Solutions

The RH problems with zeros can generate soliton solutions. The uniqueness of the associated RH problem (32) does not hold unless the zeros of det P+ in the first or third quadrants and detP− in the second or fourth quadrants are specified and kernel structures of P± at these zeros are determined. Following the definitions of P± as well as the scattering relation between ψ+ and ψ−, we find that
(38)detP+x,λ=detψ−H1+ψ+H2,=s1,1λ,where we have used the factdetψ+=1. Similarly, we have detP−x,λ=s^1,1λ and
(39)s^1,1λ=S−111=s2,2s2,3⋯s2,n+1s3,2s3,3⋯s3,n+1⋮⋮⋱⋮sn+1,2sn+1,3⋯sn+1,n+1.Suppose thats1,1λ has zeros λk,λk2∈ℂ+,1≤k≤2N, and s^1,1 has zeros λ^k,λ^k2∈ℂ−,1≤k≤2N. For simplicity, we assume that all these zeros, λk and λ^k,1≤k≤2N, are simple. Then, each of kerP+λk contains only a single column vector, denoted by vk, and each of kerP+λ^k contains a row vector, denoted by v^k:
(40)P+λkvk=0,v^kP−λ^k=0,1≤k≤2N.The RH problem (32) with the canonical normalization condition (35) and the zero structure (40) can be solved explicitly, and thus, one can readily reconstruct the potential P as follows. Note that P+ is a solution to the spectral problem (16). Therefore, as long as we expand P+ at large λ as
(41)P+x,λ=In+1+1λP1+x+O1λ2,λ⟶∞,inserting this expansion into (16) and comparingO1 terms lead to
(42)P˘=−iΛ,P1+,which implies that
(43)P=−Λ,P1+=0−αP1+12−αP1+13⋯−αP1+1,n+1αP1+2100⋯0αP1+3100⋯0⋮⋮⋮⋱⋮αP1+n+1,100⋯0,where P1+=P1+kl1≤k,l≤n+1. Further, the potentials pk and qk, k=1,2,⋯,n can be computed as
(44)pj=−αP1+1,j+1,qj=αP1+j+1,1j=1,2,⋯n.To obtain soliton solutions, we setG=In+1 in the RH problem (32). This can be achieved if we assume s1,j=sj,1=0,2≤j≤n+1, which means that there is no reflection in the scattering problem. The solutions to this specific RH problem can be given as follows [24, 25]
(45)P+λ=In+1−∑k,l=1NvkM−1klv^lλ−λ^l,P−λ=In+1+∑k,l=1NvkM−1klv^lλ−λl,where M=MklN×N is a square matrix whose entries read
(46)Mkl=v^kvlλl−λ^k,1≤k,l≤2N.Noting that the zerosλk and λ^k are constants, i.e., space and time independent, we can easily find the spatial and temporal evolutions for the vectors, vkx,t and v^kx,t,1≤k≤N. For example, let us take the x-derivative of both sides of the equation P+λkvk=0. By using (16) and P+λkvk=0, we get
(47)P+λk,xdvkdx−iλk2Λvk=0,1≤k≤2N,which implies
(48)dvkdx=iλk2Λvk,1≤k≤2N.The time dependence ofvk:
(49)dvkdt=iλk4Λvk,1≤k≤2N,can be determined similarly through an associated RH problem with the variablet. Summing up, we obtain
(50a)vkx,t=eiλk2Λx+iλk4Λtvk,0,1≤k≤2N,(50b)v^kx,t=v^k,0e−iλ∧k2Λx−iλ∧k4Λt,1≤k≤2N,where vk,0 and v^k,0,1≤k≤2N, are arbitrary constant vectors. Finally, from (45), we get
(51)P1+=−∑k,l=1NvkM−1klv^l,and thus by (44), the N-soliton solution to the system of multicomponent KN equations (4):
(52)pj=α∑k,l=1Nvk,1M−1klv^l,j+1,qj=−α∑k,l=1Nvk,j+1M−1klv^l,1,where vk=vk,1,vk,2,⋯,vk,n+1T and v^k=v^k,1,v^k,2,⋯,v^k,n+1,1≤k≤2N, are arbitrary.


## 5. Conclusions

In general, we construct the RH problem for the multicomponent KN equation in this paper. To build the special RH problem with the identity jump matrix, we introduced a variable transformation to canonical normalization spectral problem. By recombining the solutions of the canonical spectral problem and its adjoint spectral problem, a general jump matrix to the special RH problem was constructed. Letting the general jump matrix to be identity jump matrix, the RH problem was solved. Finally, we obtained the expression of theN-soliton solutions through power series expansion of the spectral parameter in the canonical normalization spectral problem.In this method, the jump matrix is corresponding to the scattering matrix, and the identity jump matrix is equivalent to reflectionless coefficient of the IST. It is well known that there are not only soliton solutions to soliton equations but also rational solutions, Matveev solutions, complexiton solutions, and so on. Recently, there have been active studies on lumps and their interaction solutions with solitons [37, 38]. It would be very interesting to generalize this method to (2+1)-dimensional equations and consider their lumps and interaction solutions. These will be our future projects.



---

*Source: 8844078-2020-11-18.xml*