Research Article | Open Access
The Modified Coupled Hirota Equation: Riemann-Hilbert Approach and N-Soliton Solutions
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 determinant and determinant, and the dynamical behaviors of the single-soliton solution are displayed graphically.
The Hirota equationis an important integrable model , where is a real parameter. This equation was initially proposed by Hirota  as a model for describing the ultrashort pulses sufferred from higher-order dispersion and self-steepening effect . 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 :where represent complex field envelope, is a small dimensionless real parameter, and “” denotes complex conjugation. The coupled higher-order Hirota equations were firstly studied in , 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 at any later time for prescribed initial conditions . 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 , the expansion approach is utilized to investigate the dispersive dark optical soliton for the Schrödinger-Hirota equation , the extended trial equation method is used to study the dispersive optical solitons for the Schrödinger-Hirota equation , and the Bäcklund transformation method is adopted to study the optical solitons for the Schrödinger-Hirota equation with power law nonlinearity .
The structure of this paper is arranged as follows. In Section 2, 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 problemwhere is a spectral parameter and is a matrix function. The matrices are defined as follows: within which Moreover, we mention that a nonlocal two-wave interaction system associated with an easier matrix spectral problem is investigated with the aid of the Riemann-Hilbert approach by .
Let us study (2) for the localized solutions; we simply suppose that the potentials u and v decay to zero sufficiently fast as , and is taken to be positive without loss of generality. It is convenient for us to introduce a new matrix spectral functionwhere is a solution of the above spectral equations (3) and (4) at . Hence, (3) and (4) can be rewritten aswhere We point out that the matrix possesses the symmetry condition, i.e.,where the superscript denotes the Hermitian of a matrix.
In the scattering process, the Jost solutions of (10) fulfill the following asymptotic condition:where I denotes the unit matrix and the subscripts in 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.,where denotes the determinant of matrix and represents the trace of a matrix, combined with the fact that , it follows thatfor all . Besides, it is easy to check that the matrix Jost solutions solve the following Volterra integral equations:Hence admit analytical continuations if the above Volterra integrals converge. Let us split into column vectors, i.e., , and then the column vectors can be analytically continued to the upper half plane , while the column vectors can be analytically continued to the lower half plane
Denoting , it is easy to show that both and are fundamental matrix solutions of the spectral problem (3), which indicates that they must be linearly related by a matrix the so-called scattering matrix. That is,with In view of (15) and (17), clearly one obtainsFrom (17), one can easily deducewhich implies that one has to study the analytic properties of before deriving the analytic property about the entries of Let us begin with the adjoint spectral equation of (10):One can simply find that satisfy the above equation (20), where is partitioned into rows in the following form:By similar discussions, we know that the row vectors are analytic in , while other row vectors are analytic for . Thanks to the analytic property of and , it follows that admits analytic continuation to , and can be analytically continued to , while are only defined for. Similarly we have that admits analytic continuation to , and can be analytically continued to , while are only defined for .
2.2. Riemann-Hilbert Problem
Next, we shall construct the Riemann-Hilbert problem. To this end, we introduce the following matrix function:whereTherefore Moreover, in view of (16), it is easy to verify that the large- asymptotics of these analytical functions areFrom the analytic property of , we get that is analytic in To obtain the analytic counterpart of in , we begin with the inverse matrix of , that is,where Then we shall investigate the following matrix function:which is analytic for . Similarly, one has the following asymptotics property:
By now, we have obtained two analytic matrix functions and , which are analytic in and , respectively. On the axes, they are related bywherewhere is followed by the fact that . Equation (28) is just the matrix Riemann-Hilbert problem we needed, and the associated canonical condition is as follows:
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,where the symmetric condition is used. It is easy to verify from (31) that also fulfills the adjoint spectral equation (20). As discussed above, we know that solves (20) and the boundary conditons (30); thus we haveMoreover, in view of (22), (26), and (32), one getsOn the other hand, one simply findsUsing (33) and (34), one has the following symmetric property:
3. Solutions to Riemann-Hilbert Problem
Now we firstly consider the regular Riemann-Hilbert problem (28), i.e., and in their analytic domain. Equation (28) can be rewritten aswhere By Plemelj’s formula , we simply get that the solution to the regular Riemann-Hilbert problem (28) under condition (30) takes the following form:
Next, we turn to investigate the nonregular Riemann-Hilbert problem (28); that is, and possess simple zeros. From the discussion in the last section, we can assume that are simple zeros of ; then we can simply denote the zeros of by . Under this circumstance, since all the zeros are simple ones, then the kernal of is one-dimensional and can be spanned by a single column vector ; therefore one hasBy taking the Hermitian conjugate to (41), one simply deduces thatTaking account of (35), one arrives atFor simplicity, we denote
Moreover, we can derive the explicit expressions for . To this end, let us start from the fact that satisfies the spectral equation (10), that is,By taking the x-derivative to (22), one simply hasand, inserting (45) into (46), we haveTaking the x-derivative to (41), in view of (47), one hasBy similar procedure, we haveTaking account of (48) and (49), it is easy to obtain thatwhere is a constant vector. By (44), we conclude that
In the remaining subsection, we shall construct a matrix function which could cancel all the zeros of Firstly, we define a matrix functionIt turns out that only admits a simple pole singularity at Moreover, it can be shown easily thatandHence, we can construct the following two matrix functions:in whichwith
From the above discussions, we could defineIt follows easily that the functions are analytic in , respectively; moreover, are nonzero in their corresponding analytic zones and as Therefore, the nonregular Riemann-Hilbert problem (28) turns into regular oneand the normalization condition
4. The Inverse Problem
In this section, with the aid of large- expansion of the solutions to the Riemann-Hilbert problem (28), we shall reconstruct the potentials and . To this end, we firstly investigate the following asymptotic expansion:Inserting (61) into (10) and comparing the coefficients of the powers of , we find from which we can deduceTherefore, we havewhere denotes the (i,j)-th entry of the matrix .
To present concise expressions for and , we need to find . To do this, we begin to simplify the expressions of and Since has simple pole singularities at , one can simply setwhere are column vector and row vector related to . In view of , one hasInserting (65) into (66), one simply getswhereBy solving the linear equations (67), and can be rewritten as
Now we are in a position to reconstruct the potentials. By Plemelj’s formula , the nonregular Riemann-Hilbert problem (59) can be solved as follows:where As , one hashenceIt follows from (69) thatIn view of (58), (72), and (73), one findswhere
At the end of the section, we shall establish the temporal evolution of scattering data Let us firstly study the evolution of , since fulfills (11), i.e.,and then we haveIn view of (17), one simply hasThen by (77) and (78), we haveSince the potentials and decay to zero sufficiently fast as , then we know that tends to zero matrix as . Thus by taking the limit to (79), one hasFrom (80) we haveSimilarly, one can deriveNotice that the initial data and are determined by the initial values
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., In this case, , thusTaking account of (64), one hasDenote , then we have and, therefore, we have
5.1. Single-Soliton Solutions
When , the single-soliton solutions for the modified coupled Hirota equations (2) take the following form: with . Denote and then the above single-soliton solutions can be rewritten as follows:The amplitude functions and both admit the shape of a hyperbolic secant with peak amplitude , and their velocities are The phases of the single-soliton solutions and depend linearly on both space and time , and parameters and are the initial location and phase of the solitary waves.
When the data are chosen as before, as displayed in Figures 1 and 2, both the peak amplitudes of and are , the velocities are equal to , and the initial location and phase are determined, respectively, by and . Moreover, the solitary waves propagate from the left to the right.
5.2. N-Soliton Solutions
When , the N-soliton solutions for the modified coupled Hirota equations (2) can be rewritten aswhere the two matrices are defined as follows:
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 and to eliminate the zeros, which transformed the nonregular Riemann-Hilbert problem into regular one, which could be solved directly by applying the Plemeljs formula . 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 determinant and 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.
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this article.
This work was supported by the National Natural Science Foundation of China (Grant Nos.11331008, 11522112, and 11601487).
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