Integrable nonlocal Hirota equations

Abstract

We construct several new integrable systems corresponding to nonlocal versions of the Hirota equation, which is a particular example of higher order nonlinear Schrödinger equations. The integrability of the new models is established by providing their explicit forms of Lax pairs or zero curvature conditions. The two compatibility equations arising in this construction are found to be related to each other either by a parity transformation P, by a time reversal T or a PT -transformation possibly combined with a conjugation. We construct explicit multi-soliton solutions for these models by employing Hirota's direct method as well as Darboux-Crum transformations. The nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behaviour. The multi-soliton solutions are of varied type, being for instance nonlocal in space, nonlocal in time of time crystal type, regular with local structures either in time/space or of rogues wave type. Integrable nonlocal Hirota equationswith constants ε, α, β, γ ∈ R. Besides the NLSE for ε = 0, four cases are known to be integrable. When the ratio of the constants are taken to be α : β : γ = 0 : 1 : 1 or α : β : γ = 0 : 1 : 0 one obtains the derivative NLSE of type I [8] and II [9], respectively, which are in fact related to each other by a dependent variable transformation [10]. For α : β : γ = 1 : 6 : 3 one obtains the Sasa-Satsuma equation [11] and for α : β : γ = 1 : 6 : 0 the Hirota equation [12]. Variations of the latter are the subject of this manuscript.We notice that the additional term in the HNLSE when compared to the NLSE, i.e. (1.1) for ε = 0, shares the same PT -symmetry with the NLSE, as it is invariant with respect to PT :Hence HNLSEs may also be viewed as PT -symmetric extensions of the NLSE. Similarly as for many other PT -symmetric nonlinear integrable systems [13], various other PT -symmetric generalizations have been proposed and investigated by adding terms to the original equation, e.g. [14,15,16]. A further option, that will be important here, was explored by Ablowitz and Musslimani [17,18] who identified a new class of nonlinear integrable systems closely related to the NLSE by exploiting a hitherto unexplored PTsymmetry present in the zero curvature condition. Exploring this option below for the Hirota equation will lead us to new integrable systems with nonlocal properties.Our manuscript is organized as follows: In section 2 we discuss the zero curvature condition or AKNS-equation for the new class of integrable systems. The solutions to these systems involve fields at different points in space or time and reduce in certain limits to the standard Hirota equation, so that we refer to them as nonlocal Hirota equations. The equations possess two types of solutions of qualitatively different behaviour and parameter dependence. We identify the origin for this novel feature within the context of Hirota's direct method as w...

Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau–Lifschitz equation

“…These equations may be viewed with q(x, t) and r(x, t) as entirely independent functions, but most commonly one imposes the relation r(x, t) = q * (x, t), such that the two equations become their mutual conjugates and are therefore essentially reduced to one equation only -the Hirota equation. Recently [28] alternative possibilities that exploit PT -symmetry have been proposed, such as taking r(…”

“…These type of novel variants of integrable systems are the main focus of this manuscript. It was shown in [28] that the different versions display quite distinct and varied behaviour and therefore deserve to be investigated in their own right.…”

“…As commented above, when specifying the (A 0 ) 21 -entry to r(x, t) = κq * (x, t) with κ = ±1, the two equations (2.9) and (2.10) reduce to standard local Hirota equation [42,28]. The first equation in (2.12) then implies that G 11 = G * 22 and G 21 = κG * 12 , reducing the four equations resulting from each matrix entry to the two equations…”

“…For the nonlocal choice r(x, t) = κq * (−x, t) the first equation in (2.12) implies that G 11 =G * 22 and G 21 = −κG * 12 . We adopt here the notation from [28] and suppress the explicit dependence on (x, t), indicating the functional dependence on (−x, t) by a tilde, i.e.q := q(−x, t),G * 12 := G * 12 (−x, t), etc. The first equation in (2.12) then reduces to the two equations a x = −κb * u, b x =ã * u, (5.13) so that in this case the components of the vector s become 14) which are solutions to the nonlocal extended Landau-Lifschitz equations (5.3), (5.4).…”

“…x f · f = −2κ |g| 2 , (ln g) xx = 0, (5.16) with q = g/f , g ∈ C, f ∈ R. We notice that the first relation in (5.16), following directly from (5.15), corresponds to one of the bilinear equations into which the Hirota equation can be converted with an additional constraint. Here D x denotes a Hirota derivative, see [28] for more details. Evidently the additional constraint is not satisfied by all solutions to the Hirota equation.…”