Modulational instability and dynamics of multi-rogue wave solutions for the discrete Ablowitz-Ladik equation

Abstract

The higher order discrete rogue waves (RWs) of the integrable discrete Ablowitz-Ladik equation are reported using a novel discrete version of generalized perturbation Darboux transformation. The dynamical behaviors of strong and weak interactions of these RWs are analytically and numerically discussed, which exhibit the abundant wave structures. We numerically show that a small noise has the weaker effect on strong-interaction RWs than weak-interaction RWs, whose main reason may be related to main energy distributions of RWs. The interaction of two first-order RWs is shown to be non-elastic. Moreover, we find that the maximal number (Smax) of the possibly split first-order ones of higher order RWs is related to the number (Pmax) of peak points of their strongest-interaction cases, that is, Smax = (Pmax + 1)/2. The results will excite to further understand the discrete RW phenomena in nonlinear optics and relevant fields.

Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours

“…Proof. Similarly to refs [29,30], based on the usual N-fold DT [65] and the N-order Darboux matrix G n (t; λ s ) with the obtained new functions A…”

“…Similarly to refs [29,30], we assume that equation (2.1) admits the formal solutions P n = F e gt+iωn , Q n = G e gt+iωn , where F, G are real-valued constant amplitudes, g denotes the gain of MI and ω the real-valued wavenumber. The assumed solutions can make equation (2.1) yield the dispersion relation for the perturbations (i.e.…”

“…It should be pointed out that the RSs vary more slowly than the usual solitons such that the RSs may be more easily stable. Most of the previous studies focused on the continuous nonlinear wave models (see and references therein), except for a few works on discrete RWs of some nonlinear lattice equations (NLEs) (see [25][26][27][28][29][30] and references therein). Moreover, RWs appearing as the collisions of solitons were observed in nonlinear optics [31,32] and in the discrete DNA models [33], and were demonstrated in the discrete non-integrable NLS models [34].…”

Multi-rogue wave solutions for a generalized integrable discrete nonlinear Schrödinger equation with higher-order excitations

“…A mass of nonlinear evolution equations including the NLS equation, Kundu-Eckhaus equation, Hirota equation, Sasa-Satuma equaion and so on, can describe the RW phenomena [4][5][6][7][8][9][10]. In discrete integrable system, the RW solutions of the AL equation, coupled discrete NLS equation and discrete Hirota equation are also discussed based on generalized Darboux transformation(DT) and Hirota bilinear method [11][12][13]. There are great differences on RWs between the continuous integrable system and discrete integrable system.…”

Soliton, breather, lump and their interaction solutions of the ($2+1$)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

“…A set of systematic methods have been used in the literature to obtain reliable treatments of nonlinear evolution equations. So far, researchers have established several methods to find the exact solutions, including the inverse scattering transform [1], the Bäcklund transformation [2][3][4][5], the Darboux transformation [6][7][8][9][10][11][12][13][14], the Riemann-Hilbert approach [15][16][17] and Hirota's bilinear method [18][19][20][21][22][23][24][25][26][27][28], Jacobian elliptic function method and modified tanh-function method [29][30][31][32][33]. Each of these approaches has its features, Hirota's bilinear method is widely popular due to its simplicity and directness.…”