Multi‐breather and high‐order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

Abstract

We construct the multi-breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit t → ±∞, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi-and higher-order rogue wave solution.

Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation

“…Exact solutions for rogue waves on the travelling wave background were constructed in our previous work [15,17] by using an algebraic method with one eigenvalue. Similar exact solutions for rogue waves were constructed in [21].…”

“…The procedure of computing a new solution ψ =û to the NLS equation (1.1) from another solution ψ = u is well-known [12,15,17,21]. Let ϕ = (p 1 , q 1 ) t be any nonzero solution to the linear equations (2.1) and (2.2) for a fixed value λ = λ 1 .…”

Observation of modulation instability and rogue breathers on stationary periodic waves

“…For both cases of periodic waves, it was shown in Refs. [19,20,22] that the roots of the polynomial P(λ) for which = 0 can be used to construct the rogue breather (or rogue wave, RW) solutions ψ RW on the corresponding periodic background. Such solutions generalize the well-known Peregrine's breather (or rogue wave) on the continuous-wave background.…”

“…Note that the plane wave is just a limiting case and thus a special case of dn-periodic waves. These stationary periodic waves are highly relevant in the studies of extreme wave formation and their generalization, resulting from MI in more practical wave conditions [18][19][20] and from the development of integrable turbulence [21].…”

All-Optical Graphene-Based Modulation of Surface Plasmon Polaritons via Modulation Instability for Secure Optical Communication

“…Here, the nonlinear dynamics interfering in the modulation process may violate this demand. In fact, an absolute spatial MI regime causes the breather and rogue waves breeding thus, an irreversible instability [49][50][51][52][53][54][55]. In this connection, spatial amplitude profile is simulated in the proposed Mach-Zehnder modulator as shown in Fig.…”