On soliton solutions of multi-component semi-discrete short pulse equation

The short pulse (SP) equation is an integrable equation. Multi-component generalizations of the SP equation are important for describing the polarization or anisotropic effects in optical fibers. An integrable semi-discretization of multi-component SP equation via Lax pair and Darboux transformation (DT) has been presented. We derive a Lax pair representation for the multi-component semi-discrete short pulse (sdSP) equation in the form of a block matrices by generalizing the 2 × 2 Lax pair matrices to the case of 2 N × 2 N . A DT is studied for the multi-component sdSP equation and is used to compute soliton solutions of the system. Further, by expanding quasideterminants, we compute cuspon-soliton, smooth-soliton and loop-soliton solutions of the complex sdSP equation.


Introduction
Integrable discretization of nonlinear ordinary and partial differential equations and their multi-component as well as non-abelian matrix generalizations have attracted a great deal of interest in nonlinear dynamics. Such discrete or lattice systems appear in quantum mechanics, quantum field theory, statistical mechanics, mathematical biology, economics etc. They also provide numerical schemes for the differential equations. Different systematic methods have been used to obtain an integrable discretization for a given integrable system. One approach to get a discrete analog of a given integrable system is the Ablowitz-Ladik method. In this method, a proper discretization is performed on the linear system of equations (also known as Lax pair) associated with a given integrable system so that its integrability structure is not spoiled. Various examples such as Korteweg-de Vries (KdV), modified Korteweg-de Vries ( (mKdV), nonlinear Scrödinger equation and Toda lattice etc. have been discretized by Ablowitz-Ladik method. All these discrete equations preserve their integrability properties such as existence of zero-curvature representation, soliton solutions and the existance of infinite sequence of conserved quantities etc [1][2][3][4]. On the other hand, in Hirota method an integrable discretization of a given integrable system can be obtained via bilinearization approach [5,6].
The SP equation for a scalar function u u x t , = ( )is given by u u u 1 6 .
1 . 1 x t xx 3 ¶ ¶ = + ¶ ( ) ( ) Equation (1.1) was initially studied by Shafer-Wayne for the propagation of ultra-short waves in nonlinear optics [7,8]. A hodograph transformation from independent variables x t , ( ) to new variables X T , ( ) given by The set of equations (1.4), (1.5) are known as complex SP equations and can be reduced to complex generalization of equation (1.1) by hodograph transformation [15,16]. Along with continuous model, fully discrete and semi-discrete (or lattice) SP equation and its generalizations have been studied in the articles such as [18][19][20][21]. The integrability properties of such equations are exhibited by the existence of Lax pair representations, Bäcklund transformation, Hirota bilinear method and soliton solutions [18][19][20][21]. In the present work, we define a Lax pair for the complex semi-discrete short-pulse (sdSP) equations (1.4), (1.5) which is a set of differential-difference equations where the space coordinate is taken as one-dimensional lattice and time is taken as continuous. The Lax pair representation of complex sdSP equation is given by with the matrices A j and B j given by where x j is a real and u j is a complex function.  The paper is organized as follows. In section 2, we present Lax pair representations of the multi-component sdSP equation. From the zero-curvature condition, we obtain multi-compenent generalization of sdSP equation. In section 3, a DT is defined on the solutions to the Lax pair and on the solutions to multi-component sdSP equation. Further, the solutions are expressed in terms of quasideterminants. In section 4, one-and twocuspon-soliton, smooth-soliton and loop-soliton solutions are obtained for the complex sdSP equation by expanding the quasideterminants. In the last section, we make concluding remarks.

Lax pair representation
We start with the Lax pair representation of the multi-component sdSP equation. The Lax pair is given by where j in the subscript is the discrete index and , , , we obtain the matrix sdSP equation as By substituting the expression of j  and j  from (2.3) into (2.4), we obtain By substituting the expressions of (2.7)-(2.9) into the set of equations Similarly, N-component complex sdSP equations is given by the following set of equations In general, the 2 2 By substituting the expressions (2.16)-(2.18) into the set of equations (2.5), (2.6), one can obtain the equations (2.14), (2.15) by a straightforward computation. The set of equations (2.14), (2.15) represents an integrable semi-discrete analog of the multi-component SP equation. Now we write the equations (2.14), (2.15) in a more general form. For this, let us write the equations (2.14), (2.15) in vector notation as where c c mn nm = and c 0 nn = . The set of equations (2.23), (2.24) have also been obtained in [20]. In order to recover the equations (1. 10 where δ is the lattice parameter in space-direction. Applying this to the set of equations (2.14), (2.15), we obtain

Darboux transformation
DT is one of solution generating technique in soliton theory that allow us to express the solutions for a given integrable equation in simple explicit form [28][29][30][31][32][33]. In this section, we construct the DT of the multicomponent .
In the present case,  is a 2 2 so that the Lax pair (2.1), (2.2) with the particular matrix j  as solution to (2.1), (2.2), can be written as It appears to be more convenient to express the equation (3.21) in the following way

Explicit soliton solutions
In order to generate soliton solutions, let us take matrix valued seed solution as where ε is a non-zero real constant, so the matrix valued solution j W to the Lax pair (2.1), (2.2) can be written as To generate soliton solutions we proceed as follows: The Here in the present case, K  ( ) are K 2 2 and , 2´matrices respectively. From the equations (3.22) with (4.6) and (4.3), the K-fold DT on the scalar fields x j and u j are given by The two-fold DT on the scalar fields x j and u j are given by The two-soliton solutions (4.26), (4.27) has been depicted in figures 4-9.
We have shown the interaction process between two-cuspon, two-soliton and two-loop solutions with their respective particular values of the parameters in figures 4-9. For example, figures 4 and 5 describes the interaction of two individual cuspons i.e., each cuspon has its own lump of energy and velocity which remains unchanged after interaction with other cuspons. And by a thrice iterated solutions, we get three-cuspon soliton solutions. The configuration of three cuspons represents elastic scattering of three cuspons with their original shapes and velocities. Similarly, by iterated quasideterminant Darboux matrix solutions to K-times, one can obtain multisoliton solutions of the complex sdSP equation.

Concluding remarks
In this paper, we have studied the multi-component or matrix generalization of the sdSP equation. We applied DT in order to calculate multi-cuspon, multi-loop and multi-soliton solutions of the system. The K-time iteration of DT has been carried out with the help of quasideterminants. The profiles of various analytic solutions have been plotted in figures. The solutions obtained in this paper have wide applications in physics and engineering because the SP equation is a physical model of describing the ultra-SPs in nonlinear media.    Moreover, semi-discrete analog of the multi-component SP equation proposed in this paper can be served as an integrable numerical scheme, i.e. self-adaptive moving mesh method used for the numerical simulation of the multi-component SP equation. Further, the kind of techniques used in this paper can also be used to calculate analytical solutions of other semi-discrete and fully discrete integrable systems.