# The Modified Coupled Hirota Equation: Riemann-Hilbert Approach and N-Soliton Solutions

Authors: Siqi Xu
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

The Cauchy initial value problem of the modified coupled Hirota equation is studied in the framework of Riemann-Hilbert approach. The N-soliton solutions are given in a compact form as a ratio of(N+1)×(N+1) determinant and N×N determinant, and the dynamical behaviors of the single-soliton solution are displayed graphically.


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## Body

## 1. Introduction

The Hirota equation(1)iut+12uxx+u2u-iαuxxx-6iαu2ux=0is an important integrable model [1], where α is a real parameter. This equation was initially proposed by Hirota [1] as a model for describing the ultrashort pulses sufferred from higher-order dispersion and self-steepening effect [2]. This Hirota equation is an integrable generalization of the well-known nonlinear Schrödinger equation (NLSE). Subsequently, the multisolitons, breathers, rogue waves, and high-order rogue waves for the Hirota equation (1) were studied by many researchers via generalized Darboux transformation method and other methods [2–6]. For the integrability and other types of solutions of the Hirota model (1), we refer to [7–9].The aim of this paper is to study the modified coupled Hirota equation in the following form [10]:(2)iut+12uxx+u2+v2u+iϵuxxx+6u2+3v2ux+3uv∗vx=0,ivt+12vxx+u2+v2v+iϵvxxx+6v2+3u2vx+3vu∗ux=0,where u=u(x,t),v=v(x,t) represent complex field envelope, ϵ is a small dimensionless real parameter, and “∗” denotes complex conjugation. The coupled higher-order Hirota equations were firstly studied in [11], where the authors used them to describe electronmagnetic pulse propogation in coupled optical waveguides and obtained soliton solutions by the inverse scattering approach. For dark soliton solutions and composite rogue waves for the coupled Hirota equation, we refer to [10, 12–14]. In the present paper, we shall seek the solutions u(x,t),v(x,t) at any later time t for prescribed initial conditions u(x,0),v(x,0). That is, we shall solve an Cauchy problem for the modified coupled Hirota equation (2); actually we are going to construct the multisoliton solutions for this system with the aid of the Riemann-Hilbert approach [15–20]. We mention that there exist many other efficient methods to investigate the exact solition solutions for the nonlinear evolution equations; for instance, the first integral method is used to study the exact 1-soliton solutions for a variety of Boussinesq-like equations [21], the G′/G- expansion approach is utilized to investigate the dispersive dark optical soliton for the Schrödinger-Hirota equation [22], the extended trial equation method is used to study the dispersive optical solitons for the Schrödinger-Hirota equation [23], and the Bäcklund transformation method is adopted to study the optical solitons for the Schrödinger-Hirota equation with power law nonlinearity [24].The structure of this paper is arranged as follows. In Section2, we start with the spectral analysis of the Lax pair of (2) and then we shall formulate the corresponding Riemann-Hilbert problem for this equations. In Section 3, we shall solve the Riemann-Hilbert problem and discuss the spatial and temporal evolutions of scattering data. In Section 4, N-soliton solutions of (2) will be constructed. In Section 5, we graphically show the behavior of single-soliton solutions for (2).


## 2. Riemann-Hilbert Formulation

In this section, we shall study the direct scattering problems for (2) and establish the corresponding Riemann-Hilbert problem.


### 2.1. Direct Scattering Process

The modified coupled Hirota equation (2) is Lax integrable with the linear spectral problem(3)Yx=UY=i12ϵλσ+U~Y,(4)Yt=VY=i192ϵ2iλ3+2λ2σ+V~Y,where λ is a spectral parameter and Y(x,t,λ) is a matrix function. The matrices σ,U~,V~ are defined as follows: (5)σ=-200010001,U~=0-u-vu∗00v∗00,V~=116ϵλ2U~+λV1+V2,with(6)V1=14ie-u2ϵ-iux-v2ϵ-ivxu∗2ϵ-iux∗-iu2-ivu∗v∗2ϵ-ivx∗-iuv∗-iv2,V2=ϵe1+e2+i2eϵe3-i2uxϵe4-i2vx-ϵe3∗-i2ux∗-ϵe1-i2u2ϵe5-i2vu∗-ϵe4∗-i2vx∗-ϵe5∗-i2uv∗-ϵe2-i2v2,in which (7)e=u2+v2,e1=uux∗-u∗ux,e2=vvx∗-v∗vx,e3=uxx+2eu,e4=vxx+2ev,e5=u∗vx-vux∗.Moreover, we mention that a nonlocal two-wave interaction system associated with an easier 3×3 matrix spectral problem is investigated with the aid of the Riemann-Hilbert approach by [25].Let us study (2) for the localized solutions; we simply suppose that the potentials u and v decay to zero sufficiently fast as x→±∞, and ϵ is taken to be positive without loss of generality. It is convenient for us to introduce a new matrix spectral function(8)Jx,t,λ=Yx,t,λE1-1,where(9)E1=ei/12ϵλσx+i/192ϵ2λ3+2λ2σtE1 is a solution of the above spectral equations (3) and (4) at x→±∞. Hence, (3) and (4) can be rewritten as(10)Jx=iλ12ϵσ,J+U~J,(11)Jt=i192ϵ2λ3+2λ2σ,J+V~J,where [σ,J]=σJ-Jσ. We point out that the matrix Q possesses the symmetry condition, i.e.,(12)U~†=-U~,where the superscript † denotes the Hermitian of a matrix.In the scattering process, the Jost solutionsJ±(x,λ) of (10) fulfill the following asymptotic condition:(13)J+x,λ→I,x→+∞,J-x,λ→I,x→-∞,where I denotes the 3×3 unit matrix and the subscripts in J± represent to which end of the x axis the boundary conditions are set. With the aid of an identity from the matrix calculus, i.e.,(14)det⁡Jx=det⁡J·trJxJ-1,where det⁡(J) denotes the determinant of matrix J and tr(·) represents the trace of a matrix, combined with the fact that tr(Q)=0, it follows that(15)det⁡J±=1for all (x,λ). Besides, it is easy to check that the matrix Jost solutions J± solve the following Volterra integral equations:(16)J+x,λ=I-∫x+∞ei/12ϵλσx-ζU~J+ζ,λe-i/12ϵλσx-ζdζ,J-x,λ=I+∫-∞xei/12ϵλσx-ζU~J-ζ,λe-i/12ϵλσx-ζdζ.Hence J±(x,λ) admit analytical continuations if the above Volterra integrals converge. Let us split J± into column vectors, i.e., J±=([J±]1,[J±]2,[J±]3), and then the column vectors [J-]1,[J+]2,[J+]3 can be analytically continued to the upper half plane λ∈C+, while the column vectors [J+]1,[J-]2,[J-]3 can be analytically continued to the lower half plane λ∈C-.DenotingE=ei/12ϵλσx, it is easy to show that both J+E and J-E are fundamental matrix solutions of the spectral problem (3), which indicates that they must be linearly related by a matrix S(λ)- the so-called scattering matrix. That is,(17)J-E=J+ESλ,λ∈R,with S(λ)=(sij(λ))3×3. In view of (15) and (17), clearly one obtains(18)det⁡Sλ=1.From (17), one can easily deduce(19)Sλ=E-1J+-1J-E,λ∈R,which implies that one has to study the analytic properties of J+-1 before deriving the analytic property about the entries of S(λ). Let us begin with the adjoint spectral equation of (10):(20)Kx=i12ϵλσ,K-KU~One can simply find that J±-1 satisfy the above equation (20), where J±-1 is partitioned into rows in the following form:(21)J+-1=J+-11J+-12J+-13,J--1=J--11J--12J--13.By similar discussions, we know that the row vectors [J+-1]1,[J--1]2,[J--1]3 are analytic in λ∈C+, while other row vectors [J--1]1,[J+-1]2,[J+-1]3 are analytic for λ∈C-. Thanks to the analytic property of J+-1 and J-, it follows that s11 admits analytic continuation to C+, and s22,s33,s23,s32 can be analytically continued to C-, while s13,s31,s12,s21 are only defined forλ∈R. Similarly we have that r11 admits analytic continuation to C-, and r22,r33,r23,r32 can be analytically continued to C+, while r13,r31,r12,r21 are only defined for λ∈R.


### 2.2. Riemann-Hilbert Problem

Next, we shall construct the Riemann-Hilbert problem. To this end, we introduce the following matrix function:(22)P+=J-H1+J+H2+J+H3=J+ESλE-1H1+H2+H3=J+ESλH1+H2+H3E-1=J+Es1100s2110s3101E-1where(23)H1=diag1,0,0,H2=diag0,1,0,H3=diag0,0,1.Therefore det(P+)=s11. Moreover, in view of (16), it is easy to verify that the large-λ asymptotics of these analytical functions are(24)P+x,λ→I,λ∈C+→∞.From the analytic property of J±, we get that P+ is analytic in λ∈C+. To obtain the analytic counterpart of P+ in C-, we begin with the inverse matrix of S(λ), that is,(25)Rλ=S-1λ=E-1J--1J+E,where R(λ)=(rij)3×3. Then we shall investigate the following matrix function:(26)P-=J--11J+-12J+-13=H1J--1+H2J+-1+H3J+-1=H1ERλE-1+H2+H3J+-1=Er11r12r13010001E-1J+-1,which is analytic for λ∈C-. Similarly, one has the following asymptotics property:(27)P-x,λ→∞,λ∈C-→∞.By now, we have obtained two analytic matrix functionsP+ and P-, which are analytic in C+ and C-, respectively. On the axes, they are related by(28)P-x,λP+x,λ=Gx,λ,λ∈R,where(29)Gx,λ=EH1Rλ+H2+H3E-1J+-1J+ESλH1+H2+H3E-1=Er11s11+r12s21+r13s31r12r13s2110s3101E-1=1r12e-i/4ϵλxr13e-i/4ϵλxs21ei/4ϵλx10s31ei/4ϵλx01,where r11s11+r12s21+r13s31=1 is followed by the fact that R(λ)S(λ)=I. Equation (28) is just the matrix Riemann-Hilbert problem we needed, and the associated canonical condition is as follows:(30)P±x,λ→I,λ→∞.


### 2.3. Symmetric Properties

In the remaining subsection, we shall investigate the symmetric properties which will be used later. To this end, we firstly take the Hermitian transpose of the spectral equation (10), that is,(31)J±†x,λ∗x=-iλ12ϵJ±†x,λ∗σ-σJ±†x,λ∗+J±†x,λ∗U~†=iλ12ϵσ,J±†x,λ∗-J±†x,λ∗U~,where the symmetric condition U~†=-U~ is used. It is easy to verify from (31) that J±†(x,λ∗)) also fulfills the adjoint spectral equation (20). As discussed above, we know that J±-1(x,λ) solves (20) and the boundary conditons (30); thus we have(32)J±†x,λ∗=J±-1x,λ.Moreover, in view of (22), (26), and (32), one gets(33)P+†x,λ∗=Es11∗s21∗s31∗010001E-1J+†x,λ∗=Es11∗s21∗s31∗010001E-1J+-1x,λ.On the other hand, one simply finds(34)S†λ∗=E-1J-†J+-1†E=E-1J--1J+E=Rλ.Using (33) and (34), one has the following symmetric property:(35)P+†x,λ∗=Es11∗s21∗s31∗010001E-1J+-1x,λ=Er11r12r13010001E-1J+-1x,λ=P-x,λ.Next, we are going to investigate the property ofs11, which matters a lot in later analysis. From (34), one has the following relations:(36)s11∗λ∗=r11λ,sjj∗λ∗=rjjλ,j=2,3skl∗λ∗=rlkλ,k,l=1,2,3,k≠l.By (36), we know that if λ1∈C+ is a zero of s11, then λ1∗∈C- is also a zero of r11.


## 2.1. Direct Scattering Process

The modified coupled Hirota equation (2) is Lax integrable with the linear spectral problem(3)Yx=UY=i12ϵλσ+U~Y,(4)Yt=VY=i192ϵ2iλ3+2λ2σ+V~Y,where λ is a spectral parameter and Y(x,t,λ) is a matrix function. The matrices σ,U~,V~ are defined as follows: (5)σ=-200010001,U~=0-u-vu∗00v∗00,V~=116ϵλ2U~+λV1+V2,with(6)V1=14ie-u2ϵ-iux-v2ϵ-ivxu∗2ϵ-iux∗-iu2-ivu∗v∗2ϵ-ivx∗-iuv∗-iv2,V2=ϵe1+e2+i2eϵe3-i2uxϵe4-i2vx-ϵe3∗-i2ux∗-ϵe1-i2u2ϵe5-i2vu∗-ϵe4∗-i2vx∗-ϵe5∗-i2uv∗-ϵe2-i2v2,in which (7)e=u2+v2,e1=uux∗-u∗ux,e2=vvx∗-v∗vx,e3=uxx+2eu,e4=vxx+2ev,e5=u∗vx-vux∗.Moreover, we mention that a nonlocal two-wave interaction system associated with an easier 3×3 matrix spectral problem is investigated with the aid of the Riemann-Hilbert approach by [25].Let us study (2) for the localized solutions; we simply suppose that the potentials u and v decay to zero sufficiently fast as x→±∞, and ϵ is taken to be positive without loss of generality. It is convenient for us to introduce a new matrix spectral function(8)Jx,t,λ=Yx,t,λE1-1,where(9)E1=ei/12ϵλσx+i/192ϵ2λ3+2λ2σtE1 is a solution of the above spectral equations (3) and (4) at x→±∞. Hence, (3) and (4) can be rewritten as(10)Jx=iλ12ϵσ,J+U~J,(11)Jt=i192ϵ2λ3+2λ2σ,J+V~J,where [σ,J]=σJ-Jσ. We point out that the matrix Q possesses the symmetry condition, i.e.,(12)U~†=-U~,where the superscript † denotes the Hermitian of a matrix.In the scattering process, the Jost solutionsJ±(x,λ) of (10) fulfill the following asymptotic condition:(13)J+x,λ→I,x→+∞,J-x,λ→I,x→-∞,where I denotes the 3×3 unit matrix and the subscripts in J± represent to which end of the x axis the boundary conditions are set. With the aid of an identity from the matrix calculus, i.e.,(14)det⁡Jx=det⁡J·trJxJ-1,where det⁡(J) denotes the determinant of matrix J and tr(·) represents the trace of a matrix, combined with the fact that tr(Q)=0, it follows that(15)det⁡J±=1for all (x,λ). Besides, it is easy to check that the matrix Jost solutions J± solve the following Volterra integral equations:(16)J+x,λ=I-∫x+∞ei/12ϵλσx-ζU~J+ζ,λe-i/12ϵλσx-ζdζ,J-x,λ=I+∫-∞xei/12ϵλσx-ζU~J-ζ,λe-i/12ϵλσx-ζdζ.Hence J±(x,λ) admit analytical continuations if the above Volterra integrals converge. Let us split J± into column vectors, i.e., J±=([J±]1,[J±]2,[J±]3), and then the column vectors [J-]1,[J+]2,[J+]3 can be analytically continued to the upper half plane λ∈C+, while the column vectors [J+]1,[J-]2,[J-]3 can be analytically continued to the lower half plane λ∈C-.DenotingE=ei/12ϵλσx, it is easy to show that both J+E and J-E are fundamental matrix solutions of the spectral problem (3), which indicates that they must be linearly related by a matrix S(λ)- the so-called scattering matrix. That is,(17)J-E=J+ESλ,λ∈R,with S(λ)=(sij(λ))3×3. In view of (15) and (17), clearly one obtains(18)det⁡Sλ=1.From (17), one can easily deduce(19)Sλ=E-1J+-1J-E,λ∈R,which implies that one has to study the analytic properties of J+-1 before deriving the analytic property about the entries of S(λ). Let us begin with the adjoint spectral equation of (10):(20)Kx=i12ϵλσ,K-KU~One can simply find that J±-1 satisfy the above equation (20), where J±-1 is partitioned into rows in the following form:(21)J+-1=J+-11J+-12J+-13,J--1=J--11J--12J--13.By similar discussions, we know that the row vectors [J+-1]1,[J--1]2,[J--1]3 are analytic in λ∈C+, while other row vectors [J--1]1,[J+-1]2,[J+-1]3 are analytic for λ∈C-. Thanks to the analytic property of J+-1 and J-, it follows that s11 admits analytic continuation to C+, and s22,s33,s23,s32 can be analytically continued to C-, while s13,s31,s12,s21 are only defined forλ∈R. Similarly we have that r11 admits analytic continuation to C-, and r22,r33,r23,r32 can be analytically continued to C+, while r13,r31,r12,r21 are only defined for λ∈R.


## 2.2. Riemann-Hilbert Problem

Next, we shall construct the Riemann-Hilbert problem. To this end, we introduce the following matrix function:(22)P+=J-H1+J+H2+J+H3=J+ESλE-1H1+H2+H3=J+ESλH1+H2+H3E-1=J+Es1100s2110s3101E-1where(23)H1=diag1,0,0,H2=diag0,1,0,H3=diag0,0,1.Therefore det(P+)=s11. Moreover, in view of (16), it is easy to verify that the large-λ asymptotics of these analytical functions are(24)P+x,λ→I,λ∈C+→∞.From the analytic property of J±, we get that P+ is analytic in λ∈C+. To obtain the analytic counterpart of P+ in C-, we begin with the inverse matrix of S(λ), that is,(25)Rλ=S-1λ=E-1J--1J+E,where R(λ)=(rij)3×3. Then we shall investigate the following matrix function:(26)P-=J--11J+-12J+-13=H1J--1+H2J+-1+H3J+-1=H1ERλE-1+H2+H3J+-1=Er11r12r13010001E-1J+-1,which is analytic for λ∈C-. Similarly, one has the following asymptotics property:(27)P-x,λ→∞,λ∈C-→∞.By now, we have obtained two analytic matrix functionsP+ and P-, which are analytic in C+ and C-, respectively. On the axes, they are related by(28)P-x,λP+x,λ=Gx,λ,λ∈R,where(29)Gx,λ=EH1Rλ+H2+H3E-1J+-1J+ESλH1+H2+H3E-1=Er11s11+r12s21+r13s31r12r13s2110s3101E-1=1r12e-i/4ϵλxr13e-i/4ϵλxs21ei/4ϵλx10s31ei/4ϵλx01,where r11s11+r12s21+r13s31=1 is followed by the fact that R(λ)S(λ)=I. Equation (28) is just the matrix Riemann-Hilbert problem we needed, and the associated canonical condition is as follows:(30)P±x,λ→I,λ→∞.


## 2.3. Symmetric Properties

In the remaining subsection, we shall investigate the symmetric properties which will be used later. To this end, we firstly take the Hermitian transpose of the spectral equation (10), that is,(31)J±†x,λ∗x=-iλ12ϵJ±†x,λ∗σ-σJ±†x,λ∗+J±†x,λ∗U~†=iλ12ϵσ,J±†x,λ∗-J±†x,λ∗U~,where the symmetric condition U~†=-U~ is used. It is easy to verify from (31) that J±†(x,λ∗)) also fulfills the adjoint spectral equation (20). As discussed above, we know that J±-1(x,λ) solves (20) and the boundary conditons (30); thus we have(32)J±†x,λ∗=J±-1x,λ.Moreover, in view of (22), (26), and (32), one gets(33)P+†x,λ∗=Es11∗s21∗s31∗010001E-1J+†x,λ∗=Es11∗s21∗s31∗010001E-1J+-1x,λ.On the other hand, one simply finds(34)S†λ∗=E-1J-†J+-1†E=E-1J--1J+E=Rλ.Using (33) and (34), one has the following symmetric property:(35)P+†x,λ∗=Es11∗s21∗s31∗010001E-1J+-1x,λ=Er11r12r13010001E-1J+-1x,λ=P-x,λ.Next, we are going to investigate the property ofs11, which matters a lot in later analysis. From (34), one has the following relations:(36)s11∗λ∗=r11λ,sjj∗λ∗=rjjλ,j=2,3skl∗λ∗=rlkλ,k,l=1,2,3,k≠l.By (36), we know that if λ1∈C+ is a zero of s11, then λ1∗∈C- is also a zero of r11.


## 3. Solutions to Riemann-Hilbert Problem

In this section, we shall solve the Riemann-Hilbert problem (28) in both regular and nonreguluar case. Before this, from (15) and (22), we simply see that(37)det⁡P+λ=s11λ,and, similarly, we also have(38)det⁡P-λ=r11λ.Now we firstly consider the regular Riemann-Hilbert problem (28), i.e., det⁡P+=s11≠0 and det⁡P-=r11≠0 in their analytic domain. Equation (28) can be rewritten as(39)P+-1λ-P-λ=G^λP+-1λ,λ∈R,where G^=I-G. By Plemelj’s formula [26], we simply get that the solution to the regular Riemann-Hilbert problem (28) under condition (30) takes the following form:(40)P+-1λ=I+12πi∫-∞+∞G^ξP+-1ξξ-λdξ,λ∈C+.Next, we turn to investigate the nonregular Riemann-Hilbert problem (28); that is, det⁡P+(λ)=s11(λ) and det⁡P-(λ)=r11(λ) possess simple zeros. From the discussion in the last section, we can assume that {λk∈C+,1≤k≤N} are simple zeros of s11(λ); then we can simply denote the zeros of r11(λ) by {λk∗∈C-,1≤k≤N}. Under this circumstance, since all the zeros are simple ones, then the kernal of P+(λk) is one-dimensional and can be spanned by a single column vector vk; therefore one has(41)P+λkvk=0,1≤k≤N,By taking the Hermitian conjugate to (41), one simply deduces that(42)vk†P+†λk=0,1≤k≤N.Taking account of (35), one arrives at(43)vk†P-λk∗=0,1≤k≤N.For simplicity, we denote(44)v^k≔vk†,1≤k≤N.Moreover, we can derive the explicit expressions forvk. To this end, let us start from the fact that J+ satisfies the spectral equation (10), that is,(45)J+,x=iλ12ϵσ,J++U~J+.By taking the x-derivative to (22), one simply has(46)P+,x=J+,xEs1100s2110s3101E-1+J+iλ12ϵσEs1100s2110s3101E-1+J+Es1100s2110s3101E-1-iλ12ϵσ,and, inserting (45) into (46), we have(47)P+,x=iλ12ϵσ,P++U~P+.Taking the x-derivative to (41), in view of (47), one has(48)P+λkvk,x-iλk12ϵσvk=0.By similar procedure, we have(49)P+λkvk,t-iλk3+2λk2192ϵ2σvk=0.Taking account of (48) and (49), it is easy to obtain that(50)vk=eiλk/12ϵσx+iλk3+2λk2/192ϵ2σtvk0,1≤k≤N.where vk0 is a constant vector. By (44), we conclude that(51)v^k=vk0†e-iλk∗/12ϵσx-iλk∗3+2λk∗2σt/192ϵ2,1≤k≤N.In the remaining subsection, we shall construct a matrix functionΥ(x,t;λ) which could cancel all the zeros of P±. Firstly, we define a matrix function(52)Υ1λ=I+λ1∗-λ1λ-λ1∗v1v^1v^1v1.It turns out that Υ1(λ) only admits a simple pole singularity at λ=λ1∗∈C-. Moreover, it can be shown easily that(53)Υ1-1λ=I-λ1∗-λ1λ-λ1v1v^1v^1v1,Υ1λ1v1=0,v^1Υ1-1λ1∗=0,and(54)det⁡Υ1λ=λ-λ1λ-λ1∗,det⁡P+λΥ1-1λλ=λ1≠0,det⁡ΥλP-λλ=λ1∗≠0.Hence, we can construct the following two matrix functions:(55)Υλ=ΥNλΥN-1λ⋯Υ1λ,Υ-1λ=Υ1-1λΥ2-1λ⋯ΥN-1λ,in which(56)Υjλ=I+λj∗-λjλ-λj∗wjw^jw^jwj,Υj-1λ=I-λj∗-λjλ-λjwjw^jw^jwj,j=2,…,N,with(57)wj=Υj-1λjΥj-2λj⋯Υ1λjvj,w^j=Υ1-1λj∗Υ2-1λj∗⋯Υj-1-1λj∗v^j,j=2,…,N.From the above discussions, we could define(58)P+λ=P+λΥ-1λ,P-λ=ΥλP-λIt follows easily that the functions P± are analytic in C±, respectively; moreover, det⁡P± are nonzero in their corresponding analytic zones and P±(λ)→I as λ→∞. Therefore, the nonregular Riemann-Hilbert problem (28) turns into regular one(59)P-λP+λ=ΥλGλΥ-1λ,λ∈R,and the normalization condition(60)P±λ→I,λ→∞.


## 4. The Inverse Problem

In this section, with the aid of large-λ expansion of the solutions to the Riemann-Hilbert problem (28), we shall reconstruct the potentials u and v. To this end, we firstly investigate the following asymptotic expansion:(61)P+=P+,0+1λP+,1+1λ2P+,2+O1λ3.Inserting (61) into (10) and comparing the coefficients of the powers of λ, we find (62)iσ,P+,0=0,iσ,P+,1+U~P+,0=0,from which we can deduce(63)P+,0=I,U~=-i12ϵσ,P+,1.Therefore, we have(64)u=-i4ϵP+,112,v=-i4ϵP+,113,where (P+,1)ij denotes the (i,j)-th entry of the matrix P+,1.To present concise expressions foru and v, we need to find P+,1. To do this, we begin to simplify the expressions of Υ(λ) and Υ-1(λ). Since Υ(λ) has simple pole singularities at λj∗∈C-,j=1,2,…,N, one can simply set(65)Υλ=I+∑j=1Nzjv^jλ-λj∗,Υ-1λ=I-∑j=1Nvjz^jλ-λj,where zj,z^j are column vector and row vector related to vj,v^j. In view of Υ(λ)Υ-1(λ)=I, one has(66)Resλ=λjΥλΥ-1λ=Υλjvj=0,1≤j≤N.Inserting (65) into (66), one simply gets(67)z1,z2,…,zNM=v1,v2,…,vN,where(68)M=MijN×N=v^ivjλi∗-λjN×N.By solving the linear equations (67), Υ(λ) and Υ-1(λ) can be rewritten as(69)Υλ=I+∑i,j=1Nviv^jM-1ijλ-λj∗,Υ-1λ=I-∑i,j=1Nviv^jM-1ijλ-λj.Now we are in a position to reconstruct the potentials. By Plemelj’s formula [26], the nonregular Riemann-Hilbert problem (59) can be solved as follows:(70)P+-1λ=I+12πi∫-∞+∞ΥζG^ζΥ-1ζP+-1ζζ-λdζ,where G^=I-G. As λ→∞, one has(71)P+-1λ→I-12πiλ∫-∞+∞ΥζG^ζΥ-1ζP+-1ζdζ,hence(72)P+λ→I+12πiλ∫-∞+∞ΥζG^ζΥ-1ζP+-1ζdζ.It follows from (69) that(73)Υλ→I+1λ∑i,j=1Nviv^jM-1ij,λ→∞.In view of (58), (72), and (73), one finds(74)P+,1x,t=∑i,j=1Nviv^jM-1ij+12πi∫-∞+∞ΥζG^ζΥ-1ζP+-1ζdζ,where(75)G^x,t;λ=I-G=-E0r12t,λr13t,λs21t,λ00s31t,λ00E-1.At the end of the section, we shall establish the temporal evolution of scattering datas21,s31,r12,r13. Let us firstly study the evolution of s21(λ),s31(λ), since J(x,λ) fulfills (11), i.e.,(76)J-,t=i192ϵ2λ3+2λ2σ,J-+U~J-,and then we have(77)J-Et=i192ϵ2λ3+2λ2σ,J-E+U~J-EIn view of (17), one simply has(78)J-E=J+ESThen by (77) and (78), we have(79)J+ESt=i192ϵ2λ3+2λ2σ,J+ES+U~J+ES.Since the potentials u and v decay to zero sufficiently fast as x→∞, then we know that U~ tends to zero matrix as x→∞. Thus by taking the limit x→∞ to (79), one has(80)St=i192ϵ2λ3+2λ2σ,S.From (80) we have(81)s21t;λ=s210;λexp⁡i64ϵ2λ3+2λ2t,s31t;λ=s310;λexp⁡i64ϵ2λ3+2λ2t.Similarly, one can derive(82)r12t;λ=r120;λexp⁡-i64ϵ2λ3+2λ2t,r13t;λ=r130;λexp⁡-i64ϵ2λ3+2λ2t.Notice that the initial data r12(0,λ),r13(0,λ) and s21(0,λ),s31(0,λ) are determined by the initial values u(0,x),v(0,x).


## 5. Soliton Solutions

In this section, we shall construct a explicit formula of the N-soliton solutions for the modified coupled Hirota equation. As is known to all, the soliton solutions correspond to the vanishing scattering coefficients, i.e.,s21=s31=r12=r13=0. In this case, G^=0, thus(83)P+,1=∑i,j=1Nviv^jM-1ij.Taking account of (64), one has(84)u=-i4ϵ∑i,j=1Nviv^j12M-1ij,v=-i4ϵ∑i,j=1Nviv^j13M-1ij.Denote vk0=(αk,βk,γk)T,θk=iλk/12ϵx+i/192ϵ2(λk3+2λk2)t, then we have (85)vk=αke-2θk,βkeθk,γkeθkT,and, therefore, we have(86)u=-i4ϵ∑i,j=1Nαiβj∗e-2θi+θj∗M-1ij,v=-i4ϵ∑i,j=1Nαiγj∗e-2θi+θj∗M-1ij.


### 5.1. Single-Soliton Solutions

WhenN=1, the single-soliton solutions for the modified coupled Hirota equations (2) take the following form: (87)u=-λ1Iα1β1∗2ϵexp⁡-λ1Ix/4ϵ+λ1I2-3λ1R2-4λ1R/64ϵ2λ1Itexp⁡-iλ1Rx/4ϵ-itλ1R3-3λ1Rλ1I2+2λ1R2-2λ1I2/64ϵ2α12+β12+γ12exp⁡-λ1Ix/2ϵ-3λ1R2-λ1I2+4λ1R/32ϵ2λ1Itv=-λ1Iα1γ1∗2ϵexp⁡-λ1Ix/4ϵ+λ1I2-3λ1R2-4λ1R/64ϵ2λ1Itexp⁡-iλ1Rx/4ϵ-itλ1R3-3λ1Rλ1I2+2λ1R2-2λ1I2/64ϵ2α12+β12+γ12exp⁡-λ1Ix/2ϵ-3λ1R2-λ1I2+4λ1R/32ϵ2λ1Itwith λ1=λ1R+iλ1I,(λ1I)>0. Denote (88)x0=4ϵλ1Iln⁡α1β12+γ12,σ0=λ1R96ϵ2λ1R2-3λ1I2,and then the above single-soliton solutions can be rewritten as follows:(89)ux,t=-λ1I4ϵα1β1α1β12+γ12·sech⁡-λ1I4ϵx+3λ1R2+4λ1R-λ1I216ϵt+x0·exp⁡-iλ1R6ϵx-iλ1R2-λ1I2t48ϵ2-iσ0,vx,t=-λ1I4ϵα1γ1α1β12+γ12·sech⁡-λ1I4ϵx+3λ1R2+4λ1R-λ1I216ϵt+x0·exp⁡-iλ1R6ϵx-iλ1R2-λ1I2t48ϵ2-iσ0.The amplitude functions |u| and |v| both admit the shape of a hyperbolic secant with peak amplitude λ1I/4|ϵ|, and their velocities are -3λ1R2+4λ1R-λ1I2/16ϵ. The phases of the single-soliton solutions u(x,t) and v(x,t) depend linearly on both space x and time t, and parameters x0 and σ0 are the initial location and phase of the solitary waves.Settingα1=β1=i,γ1=-i,λ1R=λ1I=1,ϵ=1/12, we plot the graphics of single-soliton solutions for the modified coupled Hirota equations (2) in Figures 1 and 2.Figure 1
The modula of single-soliton, solutionu in 3D plot.Figure 2
The modula of single-soliton, solutionv in 3D plot.When the data are chosen as before, as displayed in Figures1 and 2, both the peak amplitudes of |u| and |v| are 32/2, the velocities are equal to -4.5, and the initial location and phase are determined, respectively, by x0=-ln⁡2/6 and σ0=-3. Moreover, the solitary waves propagate from the left to the right.


### 5.2. N-Soliton Solutions

WhenN≥2, the N-soliton solutions for the modified coupled Hirota equations (2) can be rewritten as(90)u=i4ϵdet⁡F1det⁡M,v=i4ϵdet⁡F2det⁡M,where the two (N+1)×(N+1) matrices F1,F2 are defined as follows:(91)F1=0β1∗eθ1∗β2∗eθ2∗⋯βN∗eθN∗α1e-2θ1M11M12⋯M1Nα2e-2θ2M21M22⋯M2N⋮⋮⋮⋮⋮αNe-2θNMN1MN2⋯MNN,(92)F2=0γ1∗eθ1∗γ2∗eθ2∗⋯γN∗eθN∗α1e-2θ1M11M12⋯M1Nα2e-2θ2M21M22⋯M2N⋮⋮⋮⋮⋮αNe-2θNMN1MN2⋯MNN.


## 5.1. Single-Soliton Solutions

WhenN=1, the single-soliton solutions for the modified coupled Hirota equations (2) take the following form: (87)u=-λ1Iα1β1∗2ϵexp⁡-λ1Ix/4ϵ+λ1I2-3λ1R2-4λ1R/64ϵ2λ1Itexp⁡-iλ1Rx/4ϵ-itλ1R3-3λ1Rλ1I2+2λ1R2-2λ1I2/64ϵ2α12+β12+γ12exp⁡-λ1Ix/2ϵ-3λ1R2-λ1I2+4λ1R/32ϵ2λ1Itv=-λ1Iα1γ1∗2ϵexp⁡-λ1Ix/4ϵ+λ1I2-3λ1R2-4λ1R/64ϵ2λ1Itexp⁡-iλ1Rx/4ϵ-itλ1R3-3λ1Rλ1I2+2λ1R2-2λ1I2/64ϵ2α12+β12+γ12exp⁡-λ1Ix/2ϵ-3λ1R2-λ1I2+4λ1R/32ϵ2λ1Itwith λ1=λ1R+iλ1I,(λ1I)>0. Denote (88)x0=4ϵλ1Iln⁡α1β12+γ12,σ0=λ1R96ϵ2λ1R2-3λ1I2,and then the above single-soliton solutions can be rewritten as follows:(89)ux,t=-λ1I4ϵα1β1α1β12+γ12·sech⁡-λ1I4ϵx+3λ1R2+4λ1R-λ1I216ϵt+x0·exp⁡-iλ1R6ϵx-iλ1R2-λ1I2t48ϵ2-iσ0,vx,t=-λ1I4ϵα1γ1α1β12+γ12·sech⁡-λ1I4ϵx+3λ1R2+4λ1R-λ1I216ϵt+x0·exp⁡-iλ1R6ϵx-iλ1R2-λ1I2t48ϵ2-iσ0.The amplitude functions |u| and |v| both admit the shape of a hyperbolic secant with peak amplitude λ1I/4|ϵ|, and their velocities are -3λ1R2+4λ1R-λ1I2/16ϵ. The phases of the single-soliton solutions u(x,t) and v(x,t) depend linearly on both space x and time t, and parameters x0 and σ0 are the initial location and phase of the solitary waves.Settingα1=β1=i,γ1=-i,λ1R=λ1I=1,ϵ=1/12, we plot the graphics of single-soliton solutions for the modified coupled Hirota equations (2) in Figures 1 and 2.Figure 1
The modula of single-soliton, solutionu in 3D plot.Figure 2
The modula of single-soliton, solutionv in 3D plot.When the data are chosen as before, as displayed in Figures1 and 2, both the peak amplitudes of |u| and |v| are 32/2, the velocities are equal to -4.5, and the initial location and phase are determined, respectively, by x0=-ln⁡2/6 and σ0=-3. Moreover, the solitary waves propagate from the left to the right.


## 5.2. N-Soliton Solutions

WhenN≥2, the N-soliton solutions for the modified coupled Hirota equations (2) can be rewritten as(90)u=i4ϵdet⁡F1det⁡M,v=i4ϵdet⁡F2det⁡M,where the two (N+1)×(N+1) matrices F1,F2 are defined as follows:(91)F1=0β1∗eθ1∗β2∗eθ2∗⋯βN∗eθN∗α1e-2θ1M11M12⋯M1Nα2e-2θ2M21M22⋯M2N⋮⋮⋮⋮⋮αNe-2θNMN1MN2⋯MNN,(92)F2=0γ1∗eθ1∗γ2∗eθ2∗⋯γN∗eθN∗α1e-2θ1M11M12⋯M1Nα2e-2θ2M21M22⋯M2N⋮⋮⋮⋮⋮αNe-2θNMN1MN2⋯MNN.


## 6. Conclusions

Starting from the spectral analysis of the Lax pair of the modified coupled Hirota equation (2), we managed to construct the corresponding matrix Riemann-Hilbert problem. We mainly discussed the solutions to the general nonregular matrix Riemann-Hilbert problem; after a regularization procedure, we constructed two matrix functions Υ(λ) and Υ-1(λ) to eliminate the zeros, which transformed the nonregular Riemann-Hilbert problem into regular one, which could be solved directly by applying the Plemeljs formula [26]. Subsequently, the N-soliton solutions to the modified coupled Hirota equation (2) were obtained by the reconstruction of potentials, which was displayed in a compact form as a ratio of (N+1)×(N+1) determinant and N×N determinant. In addition, the dynamical behaviors of the single-soliton solutions were shown graphically. We point out that in the present paper we only treat the case when the potentials fulfill the vanishing boundary conditions; for the general case when the potentials do not vanish at the infinity, more general solutions could be obtained, which may be studied in the future.



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