Infinite hierarchy of nonlinear Schrödinger equations and their solutions

Abstract

We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.

Extended criterion for the modulation instability

“…The GNLSE (14) ignores terms like * y y ¶ t ( ) 2 or y y ¶ t | | 2 , as they appear in, e.g. the Lakshmanan-Porsezian-Daniel equation [48]. Such terms are quadratic in u(z) and v(z), they have no effect on pump modulations.…”

Optical Peregrine rogue waves of self-induced transparency in a resonant erbium-doped fiber

“…In an analogous manner, our solutions can even be generalized to the infinite NLS hierarchy [17] coupled to the MB equations. This is not surprising because the whole NLS hierarchy share the same linear spectral problem constructed from a loop algebra of sl (2) [10].…”

“…To reflect the diversity and complexity of media, it requires the study of the propagation models that go beyond the scalar nonlinear Schrödinger (NLS) equation, such as the extended scalar models [15][16][17][18] and the coupled multicomponent models [19][20][21][22][23][24][25].…”

Superposition of Solitons with Arbitrary Parameters for Higher-order Equations

“…However, this is not an exact representation of reality, since both fibres and water waves exhibit deviations from NLSE pulses. In fact, an infinite number of higher-order nonlinear Schrödinger-type operators can be written down [16]. The full equation then can allow for higher orders of dispersion and nonlinearity, thus going well beyond the Kerr effect.…”

“…A general method, using a recurrence relation, to obtain each particular operator K n , is given in [16]. For convenience, we present the first few operators.…”