Non-linear evolution equations (NLEEs) have been used to represent non-linear physical phenomena in several fields, such as fluid mechanics, plasma physics, optical fibres, and solid-state physics, and it is well known that solitary wave solutions of NLEEs play an important role in the non-linear science field, especially in non-linear physical science, because they can provide a lot of physical information and better discover the physical aspects of the problem and have deeper applications . The picosecond optical pulse propagation in the single-mode optical fibre has been described by the non-linear Schrödinger (NLS) equation , . The necessity of research on the coupled NLS equations has been verified due to the different frequencies or polarisations in the multi-mode fibres or fibre arrays . Additionally, coupled NLS equations, which contain both self-phase modulation and cross-phase modulation, also have some applications in such fields as plasma physics , .
Integrable or non-integrable coupled NLS equations have attracted much attention. In the context of physics, many problems, such as the light propagation in a non-linear birefringent optical fibre , the evolution of two surface wave packets in deep water , and Bose–Einstein condensates , , are governed by the coupled NLS equations.
Soliton equations are non-linear partial differential equations (PDEs) described by infinite dimensional integrable systems and are important models describing non-linear phenomena that occur in nature. Some methods have been proposed to solve the PDEs in , , , e.g. the Darboux transformation (DT) , inverse scattering transformation , Bäcklund transformation (BT) , , Painlevé test , and Hirota method . Among those methods, the BT can also be used to obtain a non-trivial solution from a seed solution in , . DT is a powerful method to construct the soliton solution for the integrable equations. There are different methods to derive the DT, for instance the operator decomposition method , gauge transformation , , loop group method , and Riemann–Hilbert method . The DT can be used for constructing multi-soliton and localised coherent structure solutions of non-linear integrable equations in both (1+1) and (2+1) dimensions , , , . The DT of Lax pair of the integrable coupling system composed by triangular system has been discussed .
Recently, some new and important scientific studies of the NLS equation models are derived. Wang et al. considered a variable-coefficient NLS equation with higher-order effects and showed the breather solution in . The higher-order generalised NLS equation describing the propagation of ultra-short optical pulse in optical fibres was solved, and some breather, rogue wave, and semirational solutions were studied in . The modulational instability, higher-order localised wave structures, and non-linear wave interactions for a non-autonomous Lenells–Fokas equation in inhomogeneous fibres is investigated in . Some breather interactions and higher-order non-autonomous rogue waves for the inhomogeneous NLS Maxwell–Bloch equations are presented via the n-fold variable-coefficient modified DT in .
As we know, the complex soliton integrable equation hierarchies with 3×3 Lax pairs are rarely discussed. In order to reveal the analytic soliton solutions of the coupled Schrödinger (CNLS) equation (CNLSE), we will employ the DT method, which is an effective computerisation procedure and has been widely used to construct soliton-like solutions for a class of NLEEs. In this paper, we construct explicit solutions for the coupled NLS equation.
This paper is organised as follows: in Section 2, we prove that the new matrixes and have the same types with U and V, which mean that they have the same structures of U and V transformed into of and In Section 3, we apply the DTs (4) and (33) to construct exact soliton solutions of (1). Similarly, the process can be done continually, and many soliton solutions of the coupling equation (1) are obtained by applying DT once again.
2 DT for CNLSE
In this section, we establish a DT to a CNLSE. The CNLSE can be used to describe the evolution of localised waves in a two-mode non-linear fibre, two-component Bose–Einstein condensate, and other coupled non-linear systems , . The CNLSE is considered as follows:
which governs the simultaneous propagation of two orthogonal components of an electric field in optical fibre  and is also important in describing the effects of averaged random birefringence on an orthogonally polarised pulse in fibres .
Consider the isospectral problem of (1); the Lax pairs of the system is in the form of
Here, q1(x, t), q2(x, t) are potentials, λ is a spectral parameter, and φ=(φ1, φ2, φ3)T is a column vector solution of (2) and (3) associated with an eigenvalue λ.
The aim of this section is to construct DT for the CNLSEs (2) and (3), which is satisfied with the 3×3 matrix transformation on φ, and Now, we recommend a gauge transformation of the CNLSEs (2) and (3):
If the fact that and U, V have the same types, the system (4) is called DT of CNLSE.
Let ψ=(ψ1, ψ2, ψ3)T, ϕ=(ϕ1, ϕ2, ϕ3)T, and X=(X1, X2, X3)T are three basic solutions of the CNLSEs (2) and (3); thus, we give the following linear algebraic system:
where λj and should choose appropriate parameters, so that the determinants of coefficients for (7) are non-zero.
Defining a 3×3 matrix T, and the T is of the following form:
where N is a natural number and are the functions of n and t. Through calculations, we find
which proves that λj (j=1≤j≤3N) are 3N roots of detT. Based on these conditions, we will prove that and have the same forms with U and V, respectively.
Proposition 1. The matrix defined by (5) has the same type as U; that is
in which the transformation formulae between old and new potentials are shown as
The transformation (33) is used to get a DT of the spectral problem (5).
Proof. By assuming and
It is easy to verify that Bsl (1≤s, l≤3) are 3N-order or 3N+1-order polynomials in λ. Hence, by (2), (7), and (8), we see that
Through an accurate calculation, λj (1≤j≤3) is the root of Bsl (1≤s, l≤3). Equation (13) has the following structure:
and (m, n=1, 2, k=0, 1) satisfy the functions without λ. So, the following (15) is obtained:
Through comparing the coefficients of λ in (17), we see
In the following section, we assume that the new matrix has the same type with U, which means that they have the same structures only of U transformed into of After careful calculation, we compare the ranks of λN and get the objective equations as follows:
From (11) and (33), we know that The proof is completed.
Proposition 2. Under the transformation (19), the matrix defined by (6) has the same form as V, that is
Proof. We assume the new matrix also have the same form with V. If we obtain the similar relations between of V and of like (33), we can prove that the gauge transformation under T turns the Lax pairs U, V into new Lax pairs with the same types.
By assuming and
It is easy to verify that Esl (1≤s, l≤3) are 3N+1-order or 3N+2-order polynomials in λ. Based on (3), (7), and (8), we see that
Through an accurate calculation, λj (j=1≤j≤3) is the root of Esl (s, l=1≤j≤3). Thus, (21) has the following structure:
and (m, n=1, 2, k=0, 1) satisfy the functions without λ. So, the following (23) is obtained:
Through comparing the coefficients of λ in (25), we get the objective equations as follows:
In the following section, we assume the new matrix has the same type with V, which means that they have the same structures only of V transformed into of From (33) and (20), we know that The proof is completed.
3 Application of DT and Breather Solutions
Propositions 1 and 2 show that the transformations (4) and (33) are DTs connecting CNLSE (1). In what follows, we can apply the above DTs (4) and (33) to construct exact solutions of (1). Firstly, we give a set of seed solutions and substitute the solutions into (2) and (3). We will get three basic solutions for these equations:
Taking (27) into (8), we obtain
with (1≤i≤2, 1 ≤j≤3N).
In order to calculate, we consider N=1 in (9) and (10). We obtain
From (30), we get
Depending on (7) and (8), we can obtain that
The analytic soliton solutions of CNLSE are obtained by the DT method as follows:
To illustrate the wave propagation of the obtained soliton solutions (33), we can choose these free parameters in the form λ1, λ2, λ3, (m=1, 2, 3, k=1, 2, 3), and the evolution of the intensity distribution for the breather solutions given by (33) is illustrated in Figure 1. The evolution of the intensity distribution for the exact solutions given by (33) is illustrated in Figure 2.
Similarly, the process can be done continually, and many soliton solutions for the coupling equation (1) are received by applying the DT method once again. Then, we obtain the breather solutions for the integrable CNLS system. The breather solutions arise during evolution due to modulation instability. Once they have appeared, they may collide just as can happen with solitons. Collisions of two or more breather solutions with transversal frequencies close to zero may create structures similar to higher-order rational solutions in Figure 1.
Figure 2 presents the evolution of solutions (33) with λ1, λ2, and λ3 given by (33) for different values of the parameter. This figure shows that the intensity of the solitary wave decreases (Fig. 2a) or increases (Fig. 2b) while propagating through an optical medium for different signs of the parameter λi. In all these figures, the time shift and the group velocity of the solitary wave are changing while the solitary wave keeps its shape in propagating along the fibre.
The exact soliton solutions (33) including shape-changing intensity redistributions, amplitude-dependent phase shifts, and related separation distances have been stated in Figures 1 and 2. In system (1), because the ratio between the parameters of self-phase modulation and cross-phase modulation is considered to be equal, this type of integrable coupled NLS system is physically valid only for the pulse propagation in a special kind of optical fibre system. Thus, studies of soliton solutions are of fundamental importance.
In this paper, we used 3×3 Lax pairs to obtain some soliton solutions with the DT. We know that the Darboux matrix for a 3×3 spectral problem is more complex than for a 2×2 spectral problem, and more new solitary solutions can be associated with DT repeated applications. and U, V have the same types gets its establishment, which calls the CNLS system (1) the DT. The method is also appropriate for more non-linear soliton equations in physics and mathematics. We will further consider how to use the DT method with higher-order Lax pairs in the future.
This work was supported by the Natural Science Foundation of Liaoning Province, China (grant no. 201602678).
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Published Online: 2016-12-21
Published in Print: 2017-01-01