Bifurcation of Soliton Families from Linear Modes in Non‐‐Symmetric Complex Potentials

Abstract

Continuous families of solitons in the nonlinear Schrödinger equation with non‐scriptPT‐symmetric complex potentials and general forms of nonlinearity are studied analytically. Under a weak assumption, it is shown that stationary equations for solitons admit a constant of motion if and only if the complex potential is of a special form g2(x)+ig′(x), where g(x) is an arbitrary real function. Using this constant of motion, the second‐order complex soliton equation is reduced to a new second‐order real equation for the amplitude of the soliton. From this real soliton equation, a novel perturbation technique is employed to show that continuous families of solitons bifurcate out from linear discrete modes in these non‐scriptPT‐symmetric complex potentials. All analytical results are corroborated by numerical examples.

Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach

“…Notably, the conditions for the existence of continuous families of solitary waves are much more general than those corresponding to PT-symmetry or to the aforementioned special form of the complex potential. The latter appears as a necessary condition for the existence of a homoclinic orbit, corresponding to a solitary wave, when starting from the linear modes of the inhomogeneous system [47]. On the contrary, in our approach, the starting point consists of the homogeneous nonlinear system which is integrable and features a homoclinic orbit, and our conditions ensure its persistence under inhomogenous perturbations, so that the resulting continuous families of solitary waves bifurcate from the nonlinear modes of the homogeneous system.…”

“…Although wave localization in PT-symmetric complex potentials has been well-studied, and the existence of continuous families of solitary waves has been shown [36], the existence of such families in non-symmetric complex potentials is still under investigation. Recently, continuous families of solitary waves were found to exist [41][42][43][44][45] in non-PT-symmetric potentials of a special form (so-called Wadati-type) [46]; it has been shown that such form is a necessary condition for the existence of continuous solitary wave families bifurcating from the linear modes of the system [47].…”

Nonlinear switching and solitons in PT‐symmetric photonic systems

“…Generalized Wadati-like potentials of the form W 2 (x) + aW (x) + idW (x)/dx were considered in Refs. [105][106][107][108], and it was shown that these special potentials are all the 1D potentials which admit bifurcation of asymmeric soliton families…”

Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

“…In the present work, we revisit these considerations. In particular, we discuss the results of the important contribution of [22]. This work suggests (and indeed conjectures) that the only complex potentials that could feature continuous families of stationary solutions although non-PT -symmetric are the ones of the Wadati type.…”