# N-Soliton Solutions of the Coupled Kundu Equations Based on the Riemann-Hilbert Method

Authors: Mengshuang Tao; Huanhe Dong
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The Kundu equation, which can be used to describe many phenomena in physics and mechanics, has crucial theoretical meaning and research value. In previous studies, the single Kundu equation has been investigated by the Riemann-Hilbert method, but few researchers have focused on the coupled Kundu equations. To our knowledge, many phenomena in nature can be only described by coupled equations, such as species competition and signal interactions. In this paper, we discuss N-soliton solutions of the coupled Kundu equations according to the Riemann-Hilbert method. Starting from the spectral problem, the coupled Kundu equations are generated, and the Riemann-Hilbert problem is presented. When the jump matrix of the Riemann-Hilbert problem is the identity matrix, the N-soliton solutions of the coupled Kundu equations can be expressed explicitly.


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## Body

## 1. Introduction

It is well known that Ginzburg-Landau equation [1], a very important physical model, has been widely used in many scientific research fields, such as superconductivity, superfluidity, fluid dynamics, the diffusion reaction equation, nonlinear optics, and quantum field theory. Kundu equation [2], as a special case of the complex Ginzburg-Landau equation, has also attracted increasingly more attention. Many researchers have discussed the integrable properties and exact solutions of the Kundu equation through various methods [3, 4], but we note that the coupled Kundu equations have been rarely studied [5, 6]. To our knowledge, many significant natural science and engineering problems can be reduced to the study of coupled equations, such as species competition and signal interactions. Thus, it is necessary to discuss the solution of the coupled Kundu equations by advanced methods.In the research of nonlinear partial differential equations [7–11] and the study of the properties of nonlinear equations [12–16], constructing a precise solution of a nonlinear equation is an important subject in soliton research [17–20], and many methods can be used to solve nonlinear equation [21–23]. In recent years, with the development of soliton theory [24–29], nonlinear partial differential equations relevant to engineering [30, 31] are playing an important role, with more researchers concentrating on Riemann-Hilbert method [32, 33] to study these important equations; for example, the Sasa-Satsuma equation was discussed by Geng [34], and the Ostrovsky-Vakhnenko equation was researched by Fan [35]. The Riemann-Hilbert method is a general term for a method of studying an integrable equation by using the Riemann-Hilbert problem as a tool [36]. The Riemann-Hilbert method is one of the most useful approaches for investigating N-soliton solutions of integrable equations [37], initial boundary value problems [38], and long-time asymptotic behavior [39, 40].In this paper, based on the Riemann-Hilbert method, we discuss the coupled Kundu equations(1)iqt+qxx+2i2β-1qrqx+i4β-1q2rx+β4β-1q3r2=0,irt-rxx+2i2β-1qrrx+i4β-1r2qx+β4β-1r3q2=0,whereqandrare arbitrary complex functions ofxandt,βis a constant, andi=-1. Supposingr=-q∗, (1) can be read as(2)iqt+qxx-2i2β-1q2qx-i4β-1q2qx∗+β4β-1q4q=0,whereq∗denotes the complex conjugate ofq. Equation (2) is known as the Kundu equation. Whenβtakes different values, we can obtain the following crucial equations.(i)
Whenβ=0, we can deduce the Kaup-Newell equation(3)iqt+qxx+2iq2qx+iq2qx∗=0.(ii)
Whenβ=1/4, we can deduce the Chen-Lee-Liu equation(4)iqt+qxx+iq2qx=0.(iii)
Whenβ=1/2, we can deduce the Gerdjikov -Ivanov equation(5)iqt+qxx-iq2qx∗+12q4q=0.We can easily find that (2)-(5) are special cases for (1). These equations are commonly used in physics and mechanics, such as plasma physics, nonlinear fluids, and quantum physics. Moreover, based on the Riemann-Hilbert method, the coupled Kundu equations are not discussed by other published papers.The structure of this paper is as follows. Based on the Riemann-Hilbert method, we investigate N-soliton solutions of the coupled Kundu equations. In Section2, the coupled Kundu equations and their Hamiltonian structures are generated. With the help of the Lax pair, the spectral problem is analyzed, and the Riemann-Hilbert problem is constructed in Section 3. The solution of the Riemann-Hilbert problem is presented in Section 4. In Section 5, the long-time behavior of the solution and the N-soliton solutions is studied. In addition, we also discuss some special cases of the coupled Kundu equations in Section 6.


## 2. Generation of the Coupled Kundu Equations

In the section, we consider the spectral problem(6)Ψx=W1Ψ=-iλ2-iβqrλqλriλ2+iβqrΨ.By solving the adjoint equation (7)W~2x=W1,W~2=W1W~2-W~2W1,where (8)W~2=abc-a=∑k=0∞akbkck-akλ-k,we can obtain the following recursive relations:(9)a2k+1=-b2k=c2k=0,b2k+1=12ib2k-1x+iqa2k-βqrb2k-1,c2k+1=-12ic2k-1x+ira2k-βqrc2k-1,a2kx=qc2k+1-rb2k+1=-βqrqc2k-1-rb2k-1-12iqc2k-1x+rb2k-1x.Based on (9), we conclude (10)a0=-2i,b1=2q,c1=2r,a2=-iqr,b3=iqx-2β-1q2r,c3=-irx-2β-1qr2,a4=142rqx-qrx+i8β-3q2r2.In the following, we consider the auxiliary spectral problem(11)Ψt=W2nΨ,where (12)W2n=Δn00-Δn+∑k=0na2kλ2n-k+2b2k+1λ2n-k+1c2k+1λ2n-k+1-a2kλ2n-k+2,andΔis a modification. According to the compatibility conditions of (6) and (11) (13)W1t-W2xn+W1W2n=0,we obtain(14)Δnx=-iβqtr+rtq,qt=b2n+1x+2iβqrb2n+1+2qΔn,rt=c2n+1x-2iβqrc2n+1-2rΔn.Based on (14), we deriveΔn=2βa2(m+1)and (15)qtrt=b2n+1x+2iβqrb2n+1+4qβa2m+1c2n+1x-2iβqrc2n+1-4rβa2m+1which can be rewritten as(16)qtrt=Γ3Γ2c2n+1b2n+1=Γ3Γ2Γ1Γ2c2n-1b2n-1=Γ3Γ2Γ1Γ2n2r2q,where (17)Γ3=1-2iβq∂-1r-2iβq∂-1q2iβr∂-1r1+2iβr∂-1q,Γ2=0∂+2iβqr∂-2iβqr0,Γ1=12r∂-1r-i+r∂-1qi+q∂-1rq∂-1q,through the recursive relations of (9).Define(18)V2k+1=c2k+1-2iβra2kb2k+1-2iβqa2k,V1=21-2βr21-2βq,u=qr,and (19)J=Γ3Γ2Γ3∗,Γ=Γ3∗-1Γ2Γ3∗,whereΓ3∗denotes the complex conjugate ofΓ3anda2k=∂-1(qc2k+1-rb2k+1). Then, (16) can be rewritten as(20)ut=JV2n+1=JΓV2n-1=⋯=JΓnV1,where(c2k+1,b2k+1)T=Γ3∗V2n+1.Taking the Killing-Cartan form〈C,D〉astr〈C,D〉, we can easily obtain (21)W2,∂W1∂q=cλ-2iβra,W2∂W1∂r=bλ-2iβqa,W2,∂W1∂λ=-4iaλ+rb+qc.In addition, we have (22)δδu-4iaλ+rb+qc=λ-ε∂∂λλεcλ-irabλ-iqa, relying on the trace identity. The equation can be read as (23)δδu-4ia2n+2+rb2n+1+qc2n+1=2n+εV2n+1. Lettingn=0, we obtainε=0. Thus,(24)V2n+1=δHnδu, where (25)H0=2qr,Hn=4ia2n+2-rb2n+1-qc2n+12n,n≥1. Combining (20) with (24), we acquire the Hamilton structure (26)ut=JδHnδu=JΓδHn-1δu=⋯=JΓnδH0δu,n≥1. Supposingn=1, we obtain the coupled Kundu equations (27)iqt+qxx+2i2β-1qrqx+i4β-1q2rx+β4β-1q3r2=0,irt-rxx+2i2β-1qrrx+i4β-1r2qx+β4β-1r3q2=0.


## 3. Riemann-Hilbert Problem

Based on the above discussion and (6) and (11), the Lax pair of the coupled Kundu equations can be read as(28)Ψx=-iλ2-iβqrλqλriλ2+iβqrΨ,Ψt=ABC-AΨ,where (29)A=-2iλ4-iqrλ2+12β2rqx-qrx+i8β-3q2r2,B=2qλ3+iqx-2β-1rq2λ,C=2rλ3-irx+2β-1qr2λ.Denote(30)σ3=100-1,R=0qr0,Q=Q100Q2,where (31)Q1=12β2rqx-qrx+i8β-3q2r2,Q2=-12β2rqx-qrx+i8β-3q2r2,and we have(32)Ψx=-iλ2σ3Ψ+M1Ψ,Ψt=-2iλ4σ3Ψ+M2Ψ,where (33)M1=Rλ-iβR2σ3,M2=2Rλ3-iR2σ3λ2-iRx+2β-1R3λ+Q.Letting (34)Ψ=Le-iλ2x+2λ4tσ3,the equivalent Lax pair of (32) can be given by(35)Lx=-iλ2σ3,L+M1L,Lt=-2iλ4σ3,L+M2L,which implies that(36)M1†=-M1,M2†=-M2,where†represents the Hermitian of a matrix andr=-q∗. In addition, we can easily deduce that(37)trM1=trM2=0.In terms of a scattering problem, the asymptotic condition ofL±(x,λ)can be given by(38)L±→I,whenx→±∞,whereIdenote the2×2unit matrix and the subscript inL±refers to the positive and negative side of thex-axis. According to (37) and Abel’s formula, we obtain(39)detL±=1.Supposing (40)G=L-E,H=L+E,are both solutions of (32), we know thatGandHare linearly related.Denote(41)E=e-iλ2σ3x,and we have (42)Gx,λ=Hx,λSλ,which implies that(43)L-E=L+ES,λ∈R,where (44)Sλ=s11s12s21s22,λ∈Ris the scattering matrix. By means of using (39) and (43), we deduce (45)detSλ=1.Based on above discussion, we focus on(G,H)satisfying(46)Ψx+iλ2σ3Ψ=M1Ψ.Assuming that theM1Ψterm of (46) is an inhomogeneous term, we know that the solution to the homogeneous equation on the left hand side of (46) isE. We can turn (46) into Volterra integral equations for(G,H)according to (38) and the method of variation of parameters. Thus, we conclude(47)L+x,λ=I-∫x+∞eiλ2σ3y-xM1L+λ,yeiλ2σ3x-ydy,(48)L-x,λ=I+∫-∞xeiλ2σ3y-xM1L-λ,yeiλ2σ3x-ydy,When the integrals on the right sides of (47)-(48) converge, we can find thatL±(x,λ)are analytical continuations off the real axisλ∈R. In addition, we can infer that the first column ofL+(x,λ)and the second column ofL-(x,λ)can be analytically continued to the lower half planeλ∈C-. Similarly, the second column ofL+(x,λ)and the first column ofL-(x,λ)can be analytically continued to the upper half planeλ∈C+Defining the following notation:(49)G=g1,g2,H=h1,h2,then the Jost solutions(50)F+x,λ=g1,h2eiλ2σ3x=L-σ1+L+σ2are analytic inλ∈C+and (51)h1,g2eiλ2σ3x=L-σ2+L+σ1are analytic inλ∈C-, where(52)σ1=diag1,0,σ2=diag0,1.It is easy to find that(53)F+x,λ→I,λ∈C+→∞,and (54)h1,g2eiλ2σ3x→I,λ∈C-→∞,by (47) and (48). With the help of the adjoint scattering equation of (35),(55)Px=-iλ2σ3,P-PM1,we can acquire the analytic counterpart ofF+inC-. It is easy to check thatL±-1meets (43). In addition, depending on (56)0=LL-1x=LxL-1+LL-1x,and (24), we obtain(57)L-1x=-iλ2σ3,L-1-L-1M1.Therefore, we find thatL±-1(x,λ)meets adjoint equation (55).In the following, we expressG-1andH-1in the following form: (58)G-1=g~1g~2,H-1=h~1h~2.Similarly, it is easy to see that the adjoint Jost solutions(59)F-=e-iλ2σ3xg~1h~2=σ1L--1+σ2L+-1are analytic inλ∈C-and (60)e-iλ2σ3xh~1g~2=σ1L+-1+σ2L--1are analytic inλ∈C+. In addition,(61)F-x,λ→I,λ∈C-→∞,and (62)e-iλ2σ3xh~1g~2→I,λ∈C+→∞.From the above equations, we obtain(63)G=g1+,g2-,H=h1-,h2+,G-1=g~1-g~2+,H-1=h~1+h~2-,where the superscripts±refer to the half plane of analyticity. In fact, these analytic properties of the Jost solutions have a direct influence on the scattering matrixS(λ)(64)S=H-1G=h~1+h~2-g1+,g2-,S-1=G-1H=g~1-g~2+h1-,h2+.It is easy to see that the analytic properties ofSandS-1can be read as (65)S-1=s~11-s~12s~21s~22+,S=s11+s12s21s22-, where the superscripts±indicate which scattering elements are analytic in the half ofλplane. In addition, we find (66)s~11=s22,s~22=s11,s~12=-s12,s~21=-s21, where the analytic properties ofS-1could be obtained byS.On the real line, based on (43), (50), and (38), we can obtain the matrix Riemann-Hilbert problem(67)F-x,λF+x,λ=Kx,λ,λ∈R,where (68)K=Eσ1+σ2Sσ1+S-1σ2E-1=E1s~12s211E-1.We derive the normalization condition for this Riemann-Hilbert problem (69)F±x,λ→I,λ→∞,by using (53) and (61). Based on (50), (59), and (43), we have(70)detF+=s~22=s11,detF-=s22=s~11.Supposing that we can solve this Riemann-Hilbert problem based on(s21,s~12), we can reconstructRaccording to the asymptotic expansion of its solution. In addition,F+andF-are solutions of (35) and (57), respectively.By defining(71)F±x,λ=I+F1±λ+F2±λ2+Oλ-3,λ→∞,and substituting (71) into (35), we have (72)Fx+=-iλ2σ3,F++λRF+-iβR2σ3F+,Fx-=-iλ2σ3,F-+λRF+-iβR2σ3F-.According to a comparison of the coefficient ofλ, we obtain(73)R=-iσ3,F1+=-iσ3,F1-,and the solutionqcan be given by (74)q=2iF1+12=-2iF1-12.We have(75)Lx†=-iλ2σ3,L†-L†M1,by taking the Hermitian conjugate of the spectral equation of (35) and (36).It is not difficult to notice thatL†(x,λ∗)orL±†(x,λ∗)meets (55). In addition,L±-1(x,λ)meets (57). Hence,L±†(x,λ∗)) andL±-1(x,λ)must be linearly dependent. Resorting to (38), inx→±∞, we infer thatL±†(x,λ∗)) andL±-1(x,λ)have the same boundary conditions. In addition, they have the same solutions as (55). Thus,(76)L±†x,λ∗=L±-1x,λ.We obtain thatF±has the following property:(77)F+†λ∗=F-λ, by using (75), (50), and (59). In addition, resorting to (43), we obtain(78)S†λ∗=S-1λ.


## 4. The Solution of the Riemann-Hilbert Problem

In order to obtain soliton solutions, we assumeK=Iin (67). From Riemann-Hilbert problem (67), it is easily noticed thatKrelies onλandxin the asymptotic conditionsF±(x,λ)→I,λ→∞. Thus, we can regardβas a constant that is independent of the solutions of the Riemann-Hilbert problem. We assume thats~22has zeros{λk∈C+,1≤k≤N}and thats22has zeros{λk∈C-,1≤k≤N}. In short, all the zeros of(λ~k,λk),k=1,…,Nare simple zeros of(s~22,s22). The kernels ofF+(λk)andF-(λ~k)include only a single column vectorvkand row vectorv¯k, respectively. We have(79)v~kF-λ~k=0,F+λkvk=0.Based on (78), we have(80)λ~k=λk∗.Taking the Hermitian conjugate of (79) and using (80), we obtain(81)v~k†F-λ~k=0.We infer that(vk,v~k)satisfy(82)v~k=vk†,by comparing (81) with the first equation of (79).The solutions of this special Riemann-Hilbert problem can be written as(83)F+λ=I+∑j,k=1NvjM-1jkv~kλ-λ~,where the matrixMcan be defined as(84)Mjk=v~jvkλj∗-λk.The zerosλkandλ~kare independent of time and space. For the sake of seeking spatial and temporal evolutions for the vectorsvk(x,t), we take the derivative of (79) with respect tox. We have(85)F+λ~k,xdvkdx+iλk2σ3vk=0,depending on (35).Hence, we can write(86)dvkdx+iλk2σ3vk=βkxvk,whereβk(x)is a scalar function. The solution of (86) is(87)vkx=eiλk2σ3xvk0e∫x0xβkydy,wherevk0=vk|x=0. Substituting (87) into (83)-(84), we find that thee∫x0x‍βk(y)dyterm cancels out and does not affect the results. Hence, lettingβk=0, we have (88)vkx=e-iλk2σ3xvk0.Similarly, we obtain(89)v~kx=v~k0eiλ~k2σ3x.In the same way, the time dependencies ofvkandv~kare derived. As a result, we derive(90)vkx=e-iλk2σ3x-2iλk4σ3tvk0,(91)v~kx=v~k0eiλ~k2σ3x+2iλ~k4σ3t.


## 5. Long-Time Behavior of the Solution and N-Soliton Solutions

Because the regular matrix Riemann-Hilbert problem(92)F~-λF~+λ=ΥλGλΥ-1λ,λ∈R,where(93)F+λ=F~+λΥλ,F-λ=Υ-1λF~-λ,(94)Υλ=I+∑j,k=1NvjM-1jkv~kλ-λ~,Υ-1λ=I-∑j,k=1NvjM-1jkv~kλ-λ~is not explicitly solvable, the expression for the solution of the coupled Kundu equations at a later time is not available in a general localized initial condition. However, we can obtain the long-time asymptotic state of the coupled Kundu equations. Then, we can express the solution of (92) as follows:(95)F~+-1λ=I+12πi∫-∞∞ΥζG~ζΥ-1ζF~+-1ζζ-λdζ,where(96)G~=I-G=-E0-s~12s210E-1.Asλ→∞, we have(97)F~+-1λ→I-12πi∫-∞∞ΥζK~ζΥ-1ζF~+-1ζdζ,and(98)F~+λ→I+12πi∫-∞∞ΥζK~ζΥ-1ζF~+-1ζdζ.Asλ→∞, we can easy find that(99)Υλ→I+1λ∑j,k=1NvjM-1jkv~k,by using (94). Thus, relying on (93), (98), and (99), we findFin the expansion(100)F1+x,t=∑j,k=1NvjM-1jkv~k+12πi∫-∞∞ΥζKζΥ-1ζF+-1ζdζ.Depending on(101)s~12t,λ=s~120,λe8iλk4t,s21t,λ=s210,λe8iλk4tandE=e-iλ2σ3, (96) can be written as (102)G~λ;x,t=0-s~120,λe4iλk2x+8iλk4t-s210,λe4iλk2x+8iλk4t0.Thus, it is easy to infer that(103)F1+λ→∑j,k=1NvjM-1jkv~k,t→∞.q(x,t)andr(x,t)can be expressed by(104)qx,t→2i∑j,k=1NvjM-1jkv~k12,rx,t→-2i∑j,k=1NvjM-1jkv~k21.In the following, we discuss the N-soliton solutions of (28). WhenK=I,K~=0, based on (100) and (73), we obtain(105)F+λ=∑j,k=1NvjM-1jkv~k,and(106)qx,t=2i∑j,k=1NvjM-1jkv~k12,rx,t=-2i∑j,k=1NvjM-1jkv~k21,wherevkis given by (90), andv~k=vk†andMare given by (84).In addition, we introduce the notation(107)δk=-iλk2x-2iλk4t.Then, the solutions ofqandrcan be written out explicitly as(108)qx,t=2i∑j,k=1Ncjeδj-δk∗M-1jk,rx,t=-2i∑j,k=1Ncjeδj-δk∗M-1jk,where the elements of theN×NmatrixMare given by(109)Mjk=1λj∗-λke-δk+δj∗+cj∗cke-δk+δj∗,andvk0=(ck,1)Twithout loss of generality.


## 6. Discussion

In the following, supposingN=1in (108), we obtain the single-soliton solution(110)qx,t=2iλ1∗-λ1c1eδ1-δ1∗e-δ1+δ1∗+c12eδ1+δ1∗,rx,t=-2iλ1∗-λ1c1eδ1-δ1∗eδ1+δ1∗+c12e-δ1+δ1∗.Suppose (111)λ=a+bi,whereaandbare the real and imaginary parts ofλ, respectively. We obtain (112)δ1+δ1∗=4abx+16aba2-b2t,δ1-δ1∗=-2ia2-b2x-4ia2-b22-4a2b2t.Thus, the single-soliton solutions can be rewritten as(113)qx,t=4bsech4abx+16aba2-b2te-2ia2-b2x-4ia2-b22-4a2b2t,rx,t=-4bsech4abx+16aba2-b2te-2ia2-b2x-4ia2-b22-4a2b2t,which are shown in Figure 1.Figure 1
The single-soliton solutions|q|and|r|are represented by (113), wherea=0.05,b=0.8.AssumingN=2in (108), we obtain the two-soliton solution(114)qx,t=-2i-c1eδ1-δ1∗M22+c2eδ2-δ2∗M12Mc1eδ1-δ2∗M21-c2eδ2-δ2∗M11M,rx,t=2i-c1eδ1-δ1∗M22+c2eδ2-δ2∗M12Mc1eδ1-δ2∗M21-c2eδ2-δ2∗M11M,where (115)M11=e-δ1+δ1∗+c12eδ1+δ1∗λ1∗-λ1,M12=e-δ2+δ1∗+c1∗c2eδ2+δ1∗λ1∗-λ2,M21=e-δ1+δ2∗+c2∗c1eδ1+δ2∗λ2∗-λ1,M22=e-δ2+δ2∗+c22eδ2+δ2∗λ2∗-λ2.In essence, we obtain the expression of the N-soliton solutions in (108) based on the Riemann-Hilbert method. In addition, as an example, the single-soliton solution is precisely given in (113), and the two-soliton solution is expressed in (114). Similarly, according to (108), we can obtain three-soliton solutions, four-soliton solutions, and so on.



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