Discrete Darboux Transformation Research Articles

Birth-death chains on a spider: Spectral analysis and reflecting-absorbing factorization

We consider discrete-time birth-death chains on a spider, i.e. a graph consisting of N discrete half lines on the plane that are joined at the origin. This process can be identified with a discrete-time quasi-birth-death process on the state space N0×{1,2,…,N}, represented by a block tridiagonal transition probability matrix. We prove that we can analyze this process by using spectral methods and obtain the n-step transition probabilities in terms of a weight matrix and the corresponding matrix-valued orthogonal polynomials (the so-called Karlin-McGregor formula). We also study under what conditions we can get a reflecting-absorbing factorization of the birth-death chain on a spider which can be seen as a stochastic UL block factorization of the transition probability matrix of the quasi-birth-death process. With this factorization we can perform a discrete Darboux transformation and get new families of “almost” birth-death chains on a spider. The spectral matrix associated with the Darboux transformation will be a Geronimus transformation of the original spectral matrix. Finally, we apply our results to the random walk on a spider, i.e. with constant transition probabilities.

Darboux transformation and multi-soliton solutions of the discrete sine-Gordon equation

AbstractWe study the discrete Darboux transformation and construct multi-soliton solutions in terms of the ratio of determinants for the integrable discrete sine-Gordon equation. We also calculate explicit expressions of single-, double-, triple-, and quadruple-soliton solutions as well as single- and double-breather solutions of the discrete sine-Gordon equation. The dynamical features of discrete kinks and breathers are also illustrated.

Discrete Darboux transformation for Ablowitz–Ladik systems derived from numerical discretization of Zakharov–Shabat scattering problem

The numerical discretization of the Zakharov–Shabat Scattering problem using integrators based on the implicit Euler method, trapezoidal rule and the split-Magnus method yield discrete systems that qualify as Ablowitz–Ladik systems. These discrete systems are important on account of their layer-peeling property which facilitates the differential approach of inverse scattering. In this paper, we study the Darboux transformation at the discrete level by following a recipe that closely resembles the Darboux transformation in the continuous case. The viability of this transformation for the computation of multisoliton potentials is investigated and it is found that irrespective of the order of convergence of the underlying discrete framework, the numerical scheme thus obtained is of first order with respect to the step size.

Modulational instability and dynamics of rational soliton solutions for the coupled Volterra lattice equation associated with $$4\times 4$$ Lax pair

The coupled Volterra lattice equation associated with $$4\times 4$$ Lax pair is under investigation, which is an integrable discrete form of a coupled KdV equation applied widely in fluids, Bose–Einstein condensation and atmospheric dynamics. First, we explore the conditions for modulational instability (MI) of the constant seed background for this equation. Secondly, we present the discrete Darboux transformation (DT) and generalised DT based on the new $$4\times 4$$ Lax pair. Through the resulting discrete DT, the bell-shaped and anti-N-shaped soliton solutions of the coupled Volterra lattice equation are derived. Moreover, we derive the M-shaped and N-shaped rational solitons and bell-shaped and N-shaped semirational soliton solutions of the coupled Volterra lattice equation via the discrete generalised DT. Finally, we numerically study the dynamical behaviours of such soliton solutions and find that the rational and semirational soliton solutions have better numerical stability than the usual soliton solution, although three types of solutions are robust against a small noise. The results may be helpful for understanding the two-layered fluid waves near ocean shores described by the coupled Korteweg–de Vries (KdV) equation.

Soliton solutions, conservation laws and modulation instability for the discrete coupled modified Korteweg–de Vries equations

In this paper, the discrete coupled modified Korteweg–de Vries equations are systematically investigated. Based on the Lax pair, N-fold discrete Darboux transformation, discrete soliton solutions, conservation laws and modulation instability are analyzed and presented.

Modulation instability, conservation laws and soliton solutions for an inhomogeneous discrete nonlinear Schrödinger equation

In this paper, an inhomogeneous discrete nonlinear Schrodinger equation is analytically investigated. The modulation instability condition and conservation laws are derived. By virtue of the discrete Darboux transformation, two types of explicit solutions on the vanishing and non-vanishing backgrounds are generated. Those results might be useful in the study of solitons propagation in discrete optical fibers.

Discrete Hirota equation: discrete Darboux transformation and new discrete soliton solutions

Under investigation in this paper is the discrete Hirota equation which is a combination of discrete NLS and discrete complex modified KdV equations. The discrete spectral problem analysis has been made, and discrete Darboux transformation(DT) has been constructed based on \(2\times 2\) discrete Lax pair for system (1.3). In addition, we have derived new discrete one-soliton solutions by using the obtained discrete DT for system (1.3). Figures have been plotted to display the dynamic features of discrete solitons.

Discrete Darboux transformation for discrete polynomials of hypergeometric type

Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn).