Discrete Soliton Solutions Research Articles

Discrete solutions and interaction dynamics for coupled nonlocal discrete nonlinear Schrödinger equations in an inhomogeneous optical fiber

The nonlinear Schrödinger (NLS) equation has made great achievement, however the coupled nonlocal discrete nonlinear Schrödinger (CNDNLS) equations have less works. In this paper, we investigate the Darboux transformation (DT) method to obtain discrete soliton solutions of CNDNLS equations. We present two different kinds of solutions through choosing different seed solutions, and analysis the relations among them. Further many novel discrete 1-soliton and 2-soliton are derived with the zero and nonzero seed solutions. Meanwhile, the elastic interaction dynamic of two discrete solitons is displayed, it is shown that the amplitudes keep unchanged after the interactions in long time.

The existence of discrete solitons for the discrete coupled nonlinear Schrödinger system

In this paper, we investigate the nonlinear coupled discrete Schrödinger equations with unbounded potentials. We find simple sufficient conditions for the existence of discrete soliton solution by using the Nehari manifold approach and the compact embedding theorem. Furthermore, by comparing the value of the action functional at the discrete soliton solution with those at nonzero solutions of one component zero, we demonstrate that both components of the discrete soliton solution are nontrivial.

From a generalized discrete NLS equation in discrete alpha helical proteins to the fourth-order NLS equation

In this paper, we consider a generalized integrable discrete nonlinear Schrödinger (NLS) equation, which can describe the dynamics of discrete alpha helical proteins with higher-order excitations and lead to the higher-order NLS equation in the continuum limit. The Darboux transformation (DT) and the soliton solutions of this generalized discrete NLS equation are implemented. It is shown that the integrable properties of the generalized discrete NLS equation, including the discrete Lax pair, the DT and the discrete soliton solutions, give rise to their continuous counterparts as the discrete space step tends to zero.

Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

Under investigation is the discrete modified Korteweg-de Vries (mKdV) equation, which is an integrable discretization of the continuous mKdV equation that can describe some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Through constructing the discrete generalized m , N − m -fold Darboux transformation for this discrete system, the various discrete soliton solutions such as the usual soliton, rational soliton, and their mixed soliton solutions are derived. The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically. The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique. Numerical simulations are used to display the dynamical behaviors of some soliton solutions. The results given in this paper might be helpful for better understanding the physical phenomena in plasma and nonlinear optics.

Discrete nonlocal N-fold Darboux transformation and soliton solutions in a reverse space-time nonlocal nonlinear self-dual network equation

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of [Formula: see text]-fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal [Formula: see text]-symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals.

Discrete soliton solutions of the fractional discrete coupled nonlinear Schrödinger equations: Three analytical approaches

The fractional discrete coupled nonlinear Schrödinger equations are solved on account of the modified Riemann–Liouville fractional derivative and Mittag–Leffler function. By using the fractional generalized Riccati method, generalized Mittag–Leffler function method and fractional generalized tanh–sech function method, some new analytical discrete solutions constructed by generalized trigonometric and hyperbolic functions are obtained, including discrete fractional bright soliton, dark soliton, combined soliton, and periodic solutions. In order to illustrate the effect of fractional order parameter on dynamics of fractional discrete soliton, some results in this study are illustrated graphically. Results indicate that although the combined wave is made up of singular coth function, it does not exhibit the singular property in the discrete case. Moreover, two kinds of scalar soliton solutions are found. These results could be of great significance to further study complex nonlinear discrete physical phenomena.

Modulational instability and dynamics of discrete rational soliton and mixed interaction solutions for a higher-order nonlinear self-dual network equation

In this paper, the higher-order nonlinear self-dual network equation is investigated. Firstly, an integrable lattice hierarchy associated with this equation is constructed from a discrete matrix spectral problem without the denominator. Secondly, the condition of modulational instability for this equation is given. Thirdly, the infinitely many conservation laws are constructed on the basis of its new Lax representation. Finally, the discrete generalised $$(m, N-m)$$ -fold Darboux transformation (DT) with non-zero constant as seed solution is used to derive new rational soliton (RS) and mixed interaction solutions. As an application, RS solutions can be obtained by the discrete generalised $$(1, N-1)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=1$$ ), and mixed interaction solutions of the usual sech-type soliton (US) and RS can be derived by the discrete generalised $$(2, N-2)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=2$$ ). We also perform the numerical simulations for such soliton solutions to explore their dynamical behaviours. Results given in this paper may have some prospective applications for understanding the propagation of electrical signals.

Discrete solitons dynamics in $$\mathscr {PT}$$-symmetric oligomers with complex-valued couplings

We consider an array of double oligomers in an optical waveguide device. A mathematical model for the system is the coupled discrete nonlinear Schrödinger equations, where the gain-and-loss parameter contributes to the complex-valued linear coupling. The array caters to an optical simulation of the parity-time (\(\mathscr {PT}\))-symmetry property between the coupled arms. The system admits fundamental bright discrete soliton solutions. We investigate their existence and spectral stability using perturbation theory analysis. These analytical findings are verified further numerically using the Newton–Raphson method and a standard eigenvalue-problem solver. Our study focuses on two natural discrete modes of the solitons: single- and double-excited-sites, also known as onsite and intersite modes, respectively. Each of these modes acquires three distinct configurations between the dimer arms, i.e., symmetric, asymmetric, and antisymmetric. Although both intersite and onsite discrete solitons are generally unstable, the latter can be stable, depending on the combined values of the propagation constant, horizontal linear coupling coefficient, and gain–loss parameter.

Discrete fractional soliton dynamics of the fractional Ablowitz-Ladik model

A time-fractional Ablowitz-Ladik model is investigated by using the fractional exponential function approach, and bright and dark discrete soliton solutions, exponential discrete solutions and discrete unusual wave solutions are found. Dynamical evolutions of these fractional discrete solitons including bright and dark discrete solitons and two kinds of unusual waves are studied. When fractional order equals to 1, all fractional solutions obtained in this paper can degenerate into those in the literature.

Soliton dynamics based on exact solutions of conformable fractional discrete complex cubic Ginzburg–Landau equation

The Jacobian elliptic function expansion method is modified by considering conformable fractional derivative and applying a lot of relations among Jacobian elliptic functions. This modified method is applied to derive the discrete fractional soliton solutions of conformable fractional discrete complex cubic Ginzburg–Landau equation, including Jacobian elliptic function solutions and some special soliton and triangular function solutions. The singularity like tan and sec solutions ‘between sites’ can be avoided in the lattice. Based on these analytical solutions, the influence of the linear dissipation and filter parameters on wave amplitude modulations and optical explosive excitations is studied.

Some discrete soliton solutions and interactions for the coupled Ablowitz–Ladik equations with branched dispersion

The coupled Ablowitz-Ladik lattice equations are the integrable discretizations of the Schrödinger equation, which can be used to model the propagation of an optical field in a tight binding waveguide array. In this paper, the discrete N-fold Darboux transformation(DT) is used to derive the discrete breather and bright soliton solutions of coupled Ablowitz–Ladik equations. Soliton interaction structures of obtained solutions are shown graphically. Based on 4 × 4 discrete Lax pairs, the transformation matrix T of DT is constructed. Then, we derive novel discrete one-soliton and two-soliton with the zero and nonzero seed solutions. And the dynamic features of breather and bright solutions are displayed, some soliton interaction phenomena are shown in the coupled Ablowitz-Ladik lattice equations. These results may be useful to explain some nonlinear wave phenomena in certain electrical and optical systems.

Integrable discretizations of AB system and multi-soliton solutions

Different integrable discretizations of AB system are proposed in this paper. Discrete AB (dAB) system and two different versions of semi-discrete AB (sdAB) systems are obtained by discretization of associated Lax pair of continuous integrable AB system. Discrete multiple soliton solutions are constructed for dAB system and sdAB systems by applying Darboux transformation. In order to explain our construction we also computed discrete single and double soliton solutions for dAB and sdAB systems. Dynamics of one discrete bright and dark soliton solution and interaction of two discrete bright-bright and dark-dark soliton solutions for dAB system have been presented.

Exact solutions of semi-discrete sine-Gordon equation

In this paper, we study a semi-discrete sine-Gordon (sd-SG) equation and compute various types of solutions analytically. We apply Darboux transformation to the associated spectral problem and construct N-soliton solutions of sd-SG equation in terms of ratio of ordinary determinants. In addition, we also construct explicit expressions of discrete one-kink, two-kink, kink-antikink, breather and degenerate soliton solutions of sd-SG equation in zero background.

Some novel soliton solution, breather solution and Darboux transformation for a generalized coupled Toda soliton hierarchy

A few of discrete integrable coupling systems(DICSs) of previous papers are linear discrete integrable couplings(LDICS). We take a special matrix Lie algebra system(non-semisimple) to construct the Lax pairs, and establish a method for deriving the nonlinear discrete integrable coupling systems(NDICS). From the Lax pairs of the generalized Toda(G-Toda) spectral problem, we can derive a novel NDICS, which is a real NDICS. For the obtained lattice integrable coupling equation, we establish a Darboux transformation (DT) with 4 × 4 Lax pairs, and apply the gauge transformation to a specific equation, then the explicit solutions of the lattice integrable coupling equation are given, which contains discrete soliton solution, breather solution and rogue wave solution. Furthermore, we can derive the discrete explicit solutions with free parameters to depict their dynamic behaviors.

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.

Dynamics and elastic interactions of the discrete multi-dark soliton solutions for the Kaup–Newell lattice equation

Under consideration in this paper is the Kaup–Newell (KN) lattice equation which is an integrable discretization of the KN equation. Infinitely, many conservation laws and discrete N-fold Darboux transformation (DT) for this system are constructed and established based on its Lax representation. Via the resulting N-fold DT, the discrete multi-dark soliton solutions in terms of determinants are derived from non-vanishing background. Propagation and elastic interaction structures of such solitons are shown graphically. Overtaking interaction phenomena between/among the two, three and four solitons are discussed. Numerical simulations are used to explore their dynamical behaviors of such multi-dark solitons. Numerical results show that their evolutions are stable against a small noise. Results in this paper might be helpful for understanding the propagation of nonlinear Alfvén waves in plasmas.

Discrete soliton solutions for a generalized discrete nonlinear Schrödinger equation with variable coefficients via discrete N -fold Darboux transformation

Under investigations in this paper is a discrete generalized nonlinear Schrödinger equation with variable coefficients. Through symbolic computation the discrete spectral problems are analyzed, the discrete N -fold Darboux transformation (DT) is constructed, discrete one and two-soliton solutions in the form of Vandermonde-like determinants are derived and some dynamic behaviors of those solitons are discussed graphically.

N-fold Darboux transformation, conservation laws and modulation instability for the semi-discrete coupled nonlinear Schrödinger equation

In this paper, a semi-discrete coupled nonlinear Schrödinger equation is investigated via the Darboux transformation (DT) method. Based on the Lax pair, the N-fold DT and conservation laws are constructed. Some new discrete soliton solutions under the vanishing and nonvanishing backgrounds are also derived. The dynamic features of those solutions are displayed through figures. The modulation instability (MI) is analyzed and the dispersion relation is derived.

Nonlocal nonlinear Schrödinger equation and its discrete version: Soliton solutions and gauge equivalence

In this paper, we try to understand the geometry for a nonlocal nonlinear Schrödinger equation (nonlocal NLS) and its discrete version introduced by Ablowitz and Musslimani, Phys. Rev. Lett. 110, 064105 (2013); Phys. Rev. E 90, 042912 (2014). We show that, under the gauge transformations, the nonlocal focusing NLS and the nonlocal defocusing NLS are, respectively, gauge equivalent to a Heisenberg-like equation and a modified Heisenberg-like equation, and their discrete versions are, respectively, gauge equivalent to a discrete Heisenberg-like equation and a discrete modified Heisenberg-like equation. Although the geometry related to the nonlocal NLS and its discrete version is not very clear, from the gauge equivalence, we can see that the properties between the nonlocal NLS and its discrete version and NLS and discrete NLS have significant difference. By constructing the Darboux transformation for discrete nonlocal NLS equations including the cases of focusing and defocusing, we derive their discrete soliton solutions, which differ from the ones obtained by using the inverse scattering transformation.

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.