Existence threshold for the ac-driven damped nonlinear Schrödinger solitons

doi:10.1016/S0167-2789(99)00055-X

Physica D: Nonlinear Phenomena

Volume 132, Issue 3, 1 August 1999, Pages 363-372

https://doi.org/10.1016/S0167-2789(99)00055-X Get rights and content

Abstract

It has been known for some time that solitons of the externally driven, damped nonlinear Schrödinger equation can only exist if the driver’s strength, $h$ , exceeds approximately $(2/π)γ$ , where $γ$ is the dissipation coefficient. Although this perturbative result was expected to be correct only to the leading order in $γ$ , recent studies have demonstrated that the formula $h_{thr} =(2/π)γ$ gives a remarkably accurate description of the soliton’s existence threshold prompting suggestions that it is, in fact, exact. In this note we evaluate the next order in the expansion of $h_{thr} (γ)$ showing that the actual reason for this phenomenon is simply that the next-order coefficient is anomalously small: $h_{thr} =(2/π)γ+0.002γ^{3}$ . Our approach is based on a singular perturbation expansion of the soliton near the turning point; it allows to evaluate $h_{thr} (γ)$ to all orders in $γ$ and can be easily reformulated for other perturbed soliton equations.

Introduction

The externally driven, damped nonlinear Schrödinger (NLS) equation, $iΨ_{t} + Ψ_{xx} +2∣ Ψ ∣^{2} Ψ =− i γ Ψ −h e^{iΩt},$ arises in a variety of fields including plasma and condensed matter physics, nonlinear optics and superconducting electronics. In some of these applications (e.g., in the study of the optical soliton propagation in a diffractive or dispersive ring cavity in the presence of an input forcing beam [1], [2], [3], [4]; in the description of easy-axis ferromagnets in an external rotating magnetic field perpendicular to the easy axis [5], [6], [7], [8], [9]; in the theory of rf-driven waves in plasma [10], [11]) Eq. (1) has a direct interpretation. In others – like for instance in charge-density-wave conductors with external electric field [12], [13], [14]; shear flows in nematic liquid crystals [15]; ac-driven long Josephson junctions [16], [17], [18], [19], [20], and periodically forced Frenkel–Kontorova chains [21], [22], [23], [24] – it occurs as an amplitude equation for small and slowly changing solutions of the externally driven, damped sine-Gordon equation: $q_{tt} +λq_{t} −q_{xx} + sin q= Γ cos (ωt).$

Without loss of generality $Ω$ in Eq. (1) can be normalized to unity [25], [26], [27]; hence, the driver’s strength $h$ and dissipation coefficient $γ$ are the only two essential control parameters. Given some $h$ and $γ$ , a fundamental question is what nonlinear attractors will arise at this point of the $(γ,h)$ -plane. In their pioneering paper [28] Kaup and Newell considered Eq. (1) on the infinite line under the vanishing boundary conditions at infinity. By means of the Inverse Scattering-based perturbation theory, these authors have demonstrated that for small $h$ and $γ$ Eq. (1) exhibits two soliton solutions phase-locked to the frequency of the driver. As $h$ is decreased for the fixed $γ$ , the two solitons approach each other and eventually merge in a turning point for $h=(2/π)γ$ [28]. Consequently, this value plays the role of a threshold; no solitons exist below $h=(2/π)γ$ . Later the same existence threshold was reobtained by Terrones, McLaughlin, Overman and Pearlstein [26] in a regular perturbation construction of solutions to Eq. (1) in powers of $h$ and $γ$ (see also [29]).

In Ref. [27] Eq. (1) was studied, numerically, in the full range of $h$ and $γ$ . It was found that the two soliton solutions persist for $γ$ up to approximately $0.7$ . For each $γ≲0.7$ there is a turning point at some $h=h_{thr}$ at which one branch of solitons turns into another, and which plays the role of the lower boundary of the existence region². Amazingly, Kaup and Newell’s approximate relation $h_{thr} =(2/π)γ$ was found to remain valid even for not very small $γ$ . For example, for $γ=0.48$ the ratio $h_{thr} /γ$ is different from $2/π$ by only one part in a thousand [27].

A completely different approach was put forward by Kollmann, Capel and Bountis [31] who regarded Eq. (1) as the continuous limit of a discrete NLS equation which they studied by means of the fixed point analysis and the Melnikov-function method. In particular, the lower boundary was obtained from the tangential intersection of the invariant manifolds of a hyperbolic fixed point. A remarkable accuracy of Kaup and Newell’s linear law detected in [27] as well as conclusions of their own Melnikov-function analysis prompted the authors of [31] to suggest that the relation $h_{thr} =(2/π)γ$ can be exact, at least for sufficiently small $γ$ .

The aim of the present note is to demonstrate that this relation is, in fact, not exact, and the actual reason why it appears to be so accurate for small $γ$ is simply because the coefficient of the next term in the expansion of $h_{thr} (γ)$ in powers of $γ$ is anomalously small. We do this by reconstructing the two solitons in the vicinity of the lower boundary of their existence domain by means of a singular (rather than regular) perturbation expansion. This novel expansion constitutes the main technical achievement of our work; its scope of applicability is obviously much wider than just the damped-driven NLS equation (1).

The reason why the regular expansion is not adequate near the solitons’ existence boundary is quite simple. The point is that the regular expansion is based on the assumption that solutions can be continued along rays $h= h γ$ ( $h = const$ .) But since the boundary is not exactly a straight line $h=(2/π)γ$ (as will be shown below, it slowly recedes upwards from this straight line), the ray $h= h γ$ with $h$ slightly above $2/π$ will hit the boundary at some small $γ=γ_{0}$ . Consequently, the regular expansion fails to continue the solution beyond $γ_{0}$ . In the singular expansion, on the other hand, the solution is continued along a curve whose shape is calculated simultaneously with finding perturbative corrections to the solution itself. In this way the existence boundary can be found to any desirable accuracy. (In this paper we restrict ourselves to terms $∼γ^{3}$ .) This idea should remain applicable to other perturbed soliton-bearing equations.

The outline of this paper is as follows. We start by discussing the regular asymptotic expansion as $h$ and $γ→0$ (Section 2). The procedure is similar to the one in [26]; the only difference is that since we now deal with solutions decaying at infinities ( $Ψ_{x} →0$ ) rather than periodic as in [26], we will be able to find perturbative corrections in closed form. In Section 3 we explain why the perturbation series for $Ψ$ breaks down as $h$ approaches the turning point, and replace it by a singular expansion. This allows us to find the next terms in the expansion of $h_{thr} (γ)$ . The resulting asymptotic formula is compared then with the threshold $h_{thr} (γ)$ obtained in a high-precision numerical analysis of Eq. (3) for several values of $γ$ . Next, after we have achieved an understanding of why the regular expansion fails and how it can be cured near the existence threshold, a natural step is to try to develop a unified expansion which would be equally applicable near and far from the threshold. This is done in Section 4. Finally, some concluding remarks are made in Section 5 followed by a brief summary of our results.

Section snippets

Regular perturbation expansion

By making a substitution $Ψ (x,t)=ψ(x,t) e^{it}$ Eq. (1) can be reduced to an autonomous equation $i ψ_{t} +ψ_{xx} +2∣ψ∣^{2} ψ−ψ=− i γψ−h.$ We will be interested in time-independent solutions of Eq. (2); these satisfy the stationary equation $ψ_{xx} +2∣ψ∣^{2} ψ−ψ=− i γψ−h$ with the boundary conditions $ψ_{x} (x)→0 as ∣x∣→∞.$

We start with developing a regular perturbation expansion away from the turning point. As the authors of [26], we assume that we are approaching the origin on the ( $γ,h$ )-plane along a straight line $h= h γ$ where $h$ is a

Singular perturbation expansion at the turning point

The reason for the breaking down of the expansion is that it was implicitly assumed in Eq. (4b) that $v_{1} = O (1)$ whereas in the actual fact, in the limit $h →2/π$ we have $v_{1} →0$ . Let us now explicitly take this fact into account by writing $u=γu_{1} +γ^{2} u_{2} …, v=v_{0} +γ^{2} v_{2} +⋯,$ where $v_{0} = sech x$ . We also expand $h$ : $h = h_{0} + h_{1} γ+ h_{2} γ^{2} +⋯, h_{0} = 2 π .$ (Thus, we have fixed $θ=π/2$ and $α=0$ straight away.) Substituting into Eq. (3), the first order in $γ$ yields Eq. (9) where one should only replace $h cos α→ h_{0}$ . Its solution is given by the same ,

A unified viewpoint

The singular expansion of the previous section was designed as a continuation and generalization of the regular expansion and presented in the form that allows for a straightforward comparison with the latter. A natural step now is to try to develop a unified approach which would be valid both near and far from the turning point. Such a unified formalism could provide an additional insight into the structure of solutions and have some technical advantages.

The unification is achieved if one

Concluding remarks and conclusions

In the undamped case ( $γ=0$ ) for any $h∈(0, 2/27)$ Eq. (3) has two explicit solutions [27]: $ψ^{(±)} (x)=ψ_{∞} 1+ 2 sinh^{2} β 1± cosh (A x) cosh β,$ where $ψ_{∞} = 2(1+2 cosh^{2} β)^{−1/2}$ is the asymptotic value of $ψ^{(±)} (x)$ as $∣x∣→∞$ ; the parameter $β$ is defined by inverting the relation $h= 2 cosh^{2} β (1+2 cosh^{2} β)^{3/2},$ and $A$ is given by $A =2ψ_{∞} sinh β= 2 sinh β 1+2 cosh^{2} β .$

For the given value of $h$ the $ψ^{(+)}$ branch merges with $ψ^{(−)}$ at some $γ_{thr}$ (defined, approximately, by $h=(2/π)γ_{thr} + h_{2}^{(0)} γ_{thr}^{3}$ ). If $h$ is small then this $γ_{thr}$ is also small, so that the point of

Acknowledgements

We thank Dmitry Pelinovsky for reading the manuscript and making a useful observation (which gave rise to the unified formalism of Section 4). Michael Kollmann’s and Mikhail Bogdan’s remarks and Nora Alexeeva’s computational assistance are also highly appreciated. I.B. is grateful to George Tsironis, Theo Tomaras and Nikos Flytzanis for their hospitality at the University of Crete. This research was supported by the FRD of South Africa and the URC of the University of Cape Town. I.B. was also

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¹: On sabbatical leave from Department of Mathematics, University of Cape Town, Rondebosch 7701, South Africa. E-mail: [email protected].

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Existence threshold for the ac-driven damped nonlinear Schrödinger solitons

Abstract

Introduction

Section snippets

Regular perturbation expansion

Singular perturbation expansion at the turning point

A unified viewpoint

Concluding remarks and conclusions

Acknowledgements

J. Magnetism and Magnet. Materials

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Phys. Rev. Lett.

Opt. Lett.

Opt. Commun.

Opt. Lett.

Phys. Rev. B

Phys. Rep.

Phys. Rev. Lett.

Phys. Rev. B