Gyroelectric cubic-quintic dissipative solitons

A. D. Boardman; Larry Velasco

REVIEW

Gyroelectric cubic-quintic dissipative solitons

A. D. Boardman, Larry Velasco

Optics · Submitted 24 Feb 2006 · 28 references

Pith's verdict: A plausible numerical demonstration of magneto-optic control of dissipative solitons, but the key scaling step needs careful scrutiny before the non-reciprocal mechanism is taken as established.

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arXiv IDphysics/0602166 (version 1)

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CitationA. D. Boardman, Larry Velasco. "Gyroelectric cubic-quintic dissipative solitons." arXiv:physics/0602166 (2006). https://pith.science/paper/C76KQSWC

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Editor's note

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Dear Colleague,

The paper adds a spatially inhomogeneous magneto-optic term Q1(x)ψ to the cubic-quintic complex Ginzburg-Landau equation and shows in simulations that reversing the sign of Q1 (the current direction) switches dissipative solitons between self-focusing and defocusing, altering breathing, explosions, and chaos. Non-reciprocity is the real point, and it is plausible: a sign-changing potential well/barrier should do exactly this. That is the contribution: combining magneto-optics with the known CGLE regimes, plus a dissipative Lagrangian treatment.

What the paper does well: it uses standard equations and published parameter sets, so the baseline is familiar. It demonstrates concrete effects, such as suppression of breathing, changing explosion rates, and opening the limit cycle. The Lagrangian results give semi-quantitative agreement for selected moving and breathing solitons, which is useful.

The soft spots, in order of importance. First, the reduction from Maxwell to Eq. (8) drops the ∇(∇·E) derivative terms using a scaling argument that assumes the transverse scale is the beam width D0≈8 μm. The Q(x) profile itself is set by the wire-to-guide distance d≈1 μm, so derivatives of Q vary on that scale, not D0. The stress-test estimate gives a correction on the order of (λ/2πd)^2 ≈ 6%, not the four orders of magnitude claimed. If those terms survive, the effective potential is not a simple well/barrier and could include velocity-dependent forces. This needs checking before the clean non-reciprocal picture is accepted. Second, no code or data is provided, so the simulations are not independently verifiable. Third, the derivation of Eq. (8) refers to an unspecified amplitude transformation, which matters if it changes scaling of the nonlinear terms. Fourth, the variational equations contain typos and some symbols are loose (A and K).

I do not see a fatal flaw. Even if the derivative corrections are not negligible, they would modify, not eliminate, the sign-dependent potential. The paper is honest: the vortex section is explicitly modest. Who is it for? Researchers studying magneto-optic solitons or dissipative soliton control. It is not a device paper, but it is a legitimate numerical study deserving serious review. A referee should ask for the derivative-term estimate to be quantified or numerically tested.

I would send it to a serious referee and recommend that the scaling caveat be addressed. If it holds, the paper is a solid contribution.

Best,

[Your name]

Referee report

Significance

If the reported non-reciprocal control is quantitatively robust, this mechanism would be an attractive candidate for all-optical routing and blocking devices based on dissipative solitons, and the dissipative Lagrangian framework would be a useful semi-analytical tool for such systems. The paper has clear strengths: it adopts standard CGLE parameter sets from the Akhmediev/Soto-Crespo literature, examines several qualitatively distinct dynamical regimes, makes the potential-well/barrier interpretation explicit, and provides direct comparisons between variational predictions and simulations for two representative cases. The constants A and K in the magnetization model are physically motivated rather than fitted to simulation output. However, the reduction from Maxwell's equations to Eq. (8) is asserted rather than derived in several load-bearing places, and the variational comparison is presented only qualitatively; these issues need to be addressed before the central claims can be accepted.

Major objections (4)

[Section II, Eqs. (1)-(3) and Eq. (8)] The neglect of the derivative terms in Eq. (3) is the key step that turns the magneto-optic interaction into the simple real potential Q1(x)ψ in Eq. (8), but the supporting scaling argument is not valid for the profile used later. The text estimates the relative size of the derivative terms using the beam width D0 = 8 μm as the only transverse scale, whereas the profile Q(x) in Eq. (26) and its derivatives vary on the scale set by the wire-to-guide distance d ≈ 1 μm. For λ ≈ 1.55 μm, the ratio of a term such as ∂_x[(dQ/dx)E_y] to the retained (ω²/c²)Q E term is of order (λ/(2πd))² ≈ 0.06, not four orders of magnitude smaller. Please evaluate the dropped terms explicitly for the profile of Eq. (26) and either justify their omission with a correct estimate or retain them in the envelope equation. This is load-bearing because the entire non-reciprocal well/barrier picture depends on Q1(x)ψ being the sole magneto-optic effect.
[Section II, after Eq. (8)] The statement that Eq. (8) 'only reaches this form after an arbitrary transformation of the amplitudes' and that the transformation details are unimportant is not acceptable as it stands. The simulation parameters are borrowed from Refs. [15]-[17], and the effective coefficients α, β, ε, μ, ν, and Q1 all depend on the normalization of ψ. Please give the explicit amplitude transformation, state the resulting mapping between the physical parameters and the dimensionless coefficients used in the simulations, and verify that the published parameter sets are mapped correctly. In addition, the modal average in Eq. (7) uses the unperturbed linear mode of the waveguide; since Q1 is of order 0.1-1 rather than a small perturbation, the correction to the mode profile should be estimated or shown to be negligible.
[Section III, Eqs. (16)-(25)] The dissipative Lagrangian formalism is a central component of the paper, but as printed the equations are difficult to verify. The trial function in Eq. (17) is garbled, the average Lagrangian in Eq. (18) contains terms whose derivation is not shown, and Eqs. (24)-(25) are effectively unreadable in the typeset version. Please provide clean, unambiguous equations and a step-by-step derivation of Eq. (16) from Eqs. (12)-(14). Furthermore, the claimed 'excellent agreement' in Figs. 6 and 7 is only qualitative; please provide quantitative error measures (e.g., RMS deviation of the peak position and FWHM over the propagation interval) and state the range of parameters for which the single-sech trial function is expected to be adequate.
[Section IV, numerical simulations] The numerical results are the main evidence for the non-reciprocal claims, but no numerical method, grid spacing, step size, boundary conditions, or convergence checks are reported. This is particularly important for the chaotic regime in Fig. 5, where numerical artifacts can be mistaken for deterministic chaos. Please add a short numerical-methods paragraph and, for the chaotic case, provide a quantitative indicator such as a Lyapunov exponent estimate, a power spectrum, or a grid-convergence study.

Minor objections (5)

[Abstract] The sentence 'This action this leads to limit cycle plots' contains a typo and should be corrected to 'This action leads to limit cycle plots'.
[Section IV(E), Fig. 9] The caption states that both panels (a) and (b) correspond to Q1 > 0, but the text describes one case as a potential well and the other as a potential hill; one of the labels is presumably Q1 < 0. Please correct this, since the sign asymmetry is the main point of the figure.
[Section IV(D), Fig. 7] The text says 'the current wire is placed at x=0' for the breathing-soliton example, while the figure caption says the wire is at x=10; please clarify which geometry was actually simulated.
[References] There are typographical errors in the reference list: '[17]' is printed as ']17]' and Ref. [21] has 'optial fibers' instead of 'optical fibers'.
[Section IV, Eq. (26)] Please define the units and dimensions of the constants A and K in Eq. (26), and state explicitly whether Q1 is intended to be a dimensionless parameter; this would help readers connect the profile to the dimensionless coefficient in Eq. (8).

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The paper

The paper claims

An inhomogeneous magnetic field term added to the cubic-quintic Ginzburg-Landau equation makes dissipative solitons strongly non-reciprocal: reversing the current switches them between self-focusing and defocusing.

The reading

This paper asks what an externally applied, spatially nonuniform magnetic field does to the known soliton families of the cubic-quintic complex Ginzburg-Landau equation. Its answer is that the magneto-optic contribution reduces, after scaling, to a single real potential term Q1(x)ψ, and that this term acts as a controllable potential well or barrier. With the current in one direction, breathing, exploding, and chaotic solitons are pushed toward self-focusing and stable trapping; with the current reversed, the same inputs spread and are blocked or deflected. The paper also extends the averaged-Lagrangian variational method to dissipative systems and shows that it reproduces the simulations well enough to track beam position, width, and breathing. A vortex in the Faraday configuration is shown to split into a pair of bright solitons whose placement is set by the sign of Q1.

What carries the argument

The load-bearing object is the real, spatially dependent potential Q1(x)ψ appended to the cubic-quintic complex Ginzburg-Landau envelope equation. Q1(x) is the scaled magneto-optic parameter, proportional to the applied current and shaped like a tanh profile between saturated magnetisation regions; its sign reverses when the current reverses. In the equation it plays the role of a refractive-index-like potential that is not reciprocal: a beam moving into a well feels attraction, and the same beam with the sign flipped sees a barrier. The second piece of machinery is a dissipative extension of the averaged-Lagrangian variational method: a trial function with amplitude, width, velocity, position, and phase parameters is inserted into the conservative Lagrangian, and the non-conservative gain and loss terms enter as a generalized force on the right-hand side. This yields ordinary differential equations for the soliton parameters that the paper compares with direct simulations for trapped and breathing solitons.

Extended reading notes

Core claim

The central claim is that the gyroelectric correction to the dissipative soliton envelope equation changes the dynamics qualitatively, not just quantitatively. Working in the Voigt geometry of an asymmetric planar waveguide, the paper derives the full magneto-optic contribution and then argues that, at the chosen scaling, only the term Q1(x)ψ survives; the derivative terms and modal corrections are smaller by four orders of magnitude. The spatially inhomogeneous magnetisation from a wire carrying current I is modelled by a tanh profile, so the surviving term is a real potential Q1(x) whose sign follows the current direction. Feeding this term into the complex Ginzburg-Landau cubic-quintic equation produces a set of exact simulations in which positive Q1 deepens self-focusing, turns periodic breathing into a stationary state, and traps moving solitons in a well, while negative Q1 spreads the beam, blocks passage, and opens the closed limit cycle into a line. The same term converts a Faraday vortex into two bright solitons located either on the potential well or beside the barrier. The conclusions state the mechanism directly: the magnetic field creates enhanced potential wells or barriers that encourage self-focusing or self-defocusing.

Load-bearing premise

The whole switching effect rests on the scaling assumption that the only magneto-optic term large enough to survive is the potential Q1(x)ψ, with all derivative and modal-average corrections negligible; if those corrections were comparable, the clean well-versus-barrier picture would collapse.

Editorial extensions

If this is right

Reversing the sign of the driving current provides a control channel that does not touch the input beam: the same waveguide, power, and input shape either traps and self-focuses or spreads and blocks the soliton.
The limit-cycle representation in the energy-peak-intensity plane gives a compact diagnostic: magneto-optic action opens closed cycles, creates limit points, and turns a chaotic region into either a broader chaotic pattern or a strong spreading state.
The dissipative Lagrangian equations reproduce the exact simulation for trapped and breathing solitons well enough to predict beam position, width, and intensity, offering a cheap design tool for current-controlled soliton routing.
In the Faraday configuration, an input vortex of topological charge m=1 evolves into two bright solitons of opposite phase, whose location is dictated by the sign of the magneto-optic potential.
Because control is exercised through a single real potential term, the same scaled equation applies to Voigt waveguides and Faraday bulk propagation, with the same order of magnitude for Q1 in both geometries.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the single-term reduction survives more realistic modal averaging, the same Q1(x)ψ design could be applied to other dissipative envelope models to create non-reciprocal switches without redesigning the gain landscape.
The sign-switch picture suggests a concrete device concept: a current-controlled soliton router in which the wire position defines a junction and the current direction chooses which output arm a soliton enters; this is an extension the paper does not explicitly build.
A direct test of the scaling assumption would be to run the simulations with the full derivative terms retained from the magneto-optic derivation, not just Q1(x)ψ; if the switching persists, the potential picture is confirmed as the mechanism, and if not, the approximation is the actual source of the non-reciprocity.
The tanh magnetisation profile idealises saturation; using the actual field of a finite-width wire would reveal whether the switching threshold is sensitive to the shape of the potential edge, a question the paper leaves open.

Original abstract

The influence of an externally applied magnetic field upon classic cubic quintic dissipative solitons is investigated using both exact simulations and a Lagrangian technique. The basic approach is to use a spatially inhomogeneous magnetic field and to consider two important geometries, namely the Voigt and the Faraday effects. A layered structure is selected for the Voigt case with the principal aim being to demonstrate non-reciprocal behaviour for various classes of spatial solitons that are known to exist as solutions of the complex Ginzburg-Landau cubic-quintic envelope equation under dissipative conditions. The system is viewed as dynamical and an opportunity is taken to display the behaviour patterns of the spatial solitons in terms of two-dimensional dynamical plots involving the total energy and the peak amplitude of the spatial solitons. This action this leads to limit cycle plots that beautifully reveal the behaviour of the solitons solutions at all points along the propagation axis. The closed contour that exists in the absence of a magnetic field is opened up and a limit point is exposed. The onset of chaos is revealed in a dramatic way and it is clear that detailed control by the external magnetic field can be exercised. The Lagrangian approach is adjusted to deal with dissipative systems and through the choice of particular trial functions, aspects of the dynamic behaviour of the spatial are predicted by this approach. Finally, some vortex dynamics in the Faraday configuration are investigated.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,998 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.47 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.47 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The influence of an externally applied magnetic field upon classic cubic quintic dissipative solitons is investigated using both exact simulations and a Lagrangian technique. The basic approach is to use a spatially inhomogeneous magnetic field and to consider two important geometries, namely the Voigt and the Faraday effects. A layered structure is selected for the Voigt case with the principal aim being to demonstrate non-reciprocal behaviour for various classes of spatial solitons that are known to exist as solutions of the complex Ginzburg-Landau cubic-quintic envelope equation under dissipative conditions. The system is viewed as dynamical and an opportunity is taken to display the behaviour patterns of the spatial solitons in terms of two-dimensional dynamical plots involving the total energy and the peak amplitude of the spatial solitons. This action this leads to limit cycle plots that beautifully reveal the behaviour of the solitons solutions at all points along the propagation axis. The closed contour that exists in the absence of a magnetic field is opened up and a limit point is exposed. The onset of chaos is revealed in a dramatic way and it is clear that detailed control by the external magnetic field can be exercised. The Lagrangian approach is adjusted to deal with dissipative systems and through the choice of particular trial functions, aspects of the dynamic behaviour of the spatial are predicted by this approach. Finally, some vortex dynamics in the Faraday configuration are investigated.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The influence of an externally applied magnetic field upon classic cubic quintic dissipative solitons is investigated using both exact simulations and a Lagrangian technique.	contribution 0.97	0.90	0.07	asserted 0.98
s1	The basic approach is to use a spatially inhomogeneous magnetic field and to consider two important geometries, namely the Voigt and the Faraday effects.	method 0.99	0.75	0.03	asserted 0.96
s2	A layered structure is selected for the Voigt case with the principal aim being to demonstrate non-reciprocal behaviour for various classes of spatial solitons that are known to exist as solutions of the complex Ginzburg-Landau cubic-quintic envelope equation under dissipative conditions.	method 0.79	0.76	0.10	asserted 0.96
s3	The system is viewed as dynamical and an opportunity is taken to display the behaviour patterns of the spatial solitons in terms of two-dimensional dynamical plots involving the total energy and the peak amplitude of the spatial solitons.	method 0.92	0.86	0.05	asserted 0.80
s4	This action this leads to limit cycle plots that beautifully reveal the behaviour of the solitons solutions at all points along the propagation axis.	result 0.59	0.83	0.08	asserted 1.00
s5	The closed contour that exists in the absence of a magnetic field is opened up and a limit point is exposed.	result 0.99	0.82	0.10	asserted 1.00
s6	The onset of chaos is revealed in a dramatic way and it is clear that detailed control by the external magnetic field can be exercised.	result 0.58	0.86	0.40	asserted 1.00
s7	The Lagrangian approach is adjusted to deal with dissipative systems and through the choice of particular trial functions, aspects of the dynamic behaviour of the spatial are predicted by this approach.	method 0.94	0.88	0.05	asserted 0.86
s8	Finally, some vortex dynamics in the Faraday configuration are investigated.	contribution 0.96	0.90	0.04	asserted 0.95

What kind of paper the classifier reports this is: simulation study 0.99 · evidence from simulation or computation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** This shows the kind of asymmetric waveguide structur

**Figure 4.** Intensity plot of an exploding soliton propagation under the influence

Showing 2 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:physics/0602166 by the authors.

References

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N. Akhemediev, V.V. Afanasjev and J.M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation”, Physical Review E , 1996. 53(1): pp. 1190-1201. ]17] J.M. Soto-Crespo, N. Akhmediev and A. Ankiewicz, “Pulsating, Creeping, and Erupting Solitons in Dissipative Systems”. Physical Review Letters, 2000. 85...

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A.D. Boardman and M.Xie, Spatial solitons , Springer Series in Optical Sciences, 2002, pp. 417-432

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A.D. Boardman and K. Xie, “Magneto-optic spatial solitons ”, J. Opt. Soc. Am. B , 14, 1997, pp. 3102-3109

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A.D. Boardman and M. Xie, “Spatial Solitons in discontinuous magneto-optic waveguides”, J .Opt. B: Quantum semi classical optics, 3, 2001, pp. 5244-5250,

View on Pith 2001
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A.D. Boardman and M. Xie, Magneto-optics: a critical review , in Introduction to Complex Mediums for Optics and Electromagnetics , W.S. Weiglhofer and A. Lakhtakia, Editors. 2004, SPIE: Bellingham, WA

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A.D. Boardman, M. Xie and K. Xie, “Spatial bright-dark solitons in transversely magnetized coupled waveguides”, Journal optical society America B, 2005. 22(1): p. 220-227. [6]A.D.Boardman, M.Xie and K.Xie, “Surface magneto-optic solitons”, J.Phys.D:Appl.Phys., 36, 2211-2217, 2002

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F. Jonsson and C. Flytzanis, “Polarizati on state controlled multistability of a nonlinear magneto-optic cavity”, Phys Rev Lett, 82, (1997), pp. 1426-1429. 14

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A.K. Zvezdin and V.A. Kotov, Modern magneto-optics and modern magneto-optic materials, Institute of Physics Publishing, Bristol, UK, 1997

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M.S. Soskin and M.V. Vasnetsov, “Singular Optics ”, Progress in Optics, 2001. 42: p. 219- 276. Figures propagation Fig 1. This shows the kind of asymmetric waveguide structur e needed to ensure that the Voigt effect is O(Q), as opposed to O(Q 2) in the bulk or for a symmetric waveguide. It cons ists of a nonlinear layer that will support spatial solitons ...

View on Pith 2001

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [15]

A.D.Boardman,L.Velasco and P.Egan Dissipative Magneto-optic Solitons in Dissipative Solitons, N.N. Akhmediev and A. Ankiewicz, Editors. 2005, Springer: Berlin. pp. 18-35

View on Pith 2005

[2] [17]

N. Akhemediev, V.V. Afanasjev and J.M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation”, Physical Review E , 1996. 53(1): pp. 1190-1201. ]17] J.M. Soto-Crespo, N. Akhmediev and A. Ankiewicz, “Pulsating, Creeping, and Erupting Solitons in Dissipative Systems”. Physical Review Letters, 2000. 85...

View on Pith 1996

[3] [1]

A.D. Boardman and M.Xie, Spatial solitons , Springer Series in Optical Sciences, 2002, pp. 417-432

View on Pith 2002

[4] [2]

A.D. Boardman and K. Xie, “Magneto-optic spatial solitons ”, J. Opt. Soc. Am. B , 14, 1997, pp. 3102-3109

View on Pith 1997

[5] [3]

A.D. Boardman and M. Xie, “Spatial Solitons in discontinuous magneto-optic waveguides”, J .Opt. B: Quantum semi classical optics, 3, 2001, pp. 5244-5250,

View on Pith 2001

[6] [4]

A.D. Boardman and M. Xie, Magneto-optics: a critical review , in Introduction to Complex Mediums for Optics and Electromagnetics , W.S. Weiglhofer and A. Lakhtakia, Editors. 2004, SPIE: Bellingham, WA

View on Pith 2004

[7] [5]

A.D. Boardman, M. Xie and K. Xie, “Spatial bright-dark solitons in transversely magnetized coupled waveguides”, Journal optical society America B, 2005. 22(1): p. 220-227. [6]A.D.Boardman, M.Xie and K.Xie, “Surface magneto-optic solitons”, J.Phys.D:Appl.Phys., 36, 2211-2217, 2002

View on Pith 2005

[8] [7]

I.V. Lindell, A.H. Sihvola, S.A. Tretyakov and A.J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media. 1st ed. 1994, London: Artech House

View on Pith 1994

[9] [8]

J. Petykiewicz, Wave optics, Kluwer, Dordrecht, The Netherlands, 1992

View on Pith 1992

[10] [9]

the following 2D equation is appropriate 4 22 122 1 02i zx y ψψ ψ ψ++ + + ⎛⎞∂∂ ∂++ +⎜⎟∂∂ ∂ ⎝⎠ Q = (4) and the x and y wave number components are equal. Thus, for this example, an arbitrarily polarised input to a Faraday system evolves, in the linear case, as uncoupled counter- rotating, circularly polarised waves. Equation (4) is sufficient to describe 2D...

View on Pith

[11] [10]

F. Jonsson and C. Flytzanis, “Polarizati on state controlled multistability of a nonlinear magneto-optic cavity”, Phys Rev Lett, 82, (1997), pp. 1426-1429. 14

View on Pith 1997

[12] [11]

A.K. Zvezdin and V.A. Kotov, Modern magneto-optics and modern magneto-optic materials, Institute of Physics Publishing, Bristol, UK, 1997

View on Pith 1997

[13] [12]

N. Akhemediev and A. Ankiewicz, Solitons nonlinear pulses and beams , Chapman & Hall, London, UK, 1997

View on Pith 1997

[14] [13]

N. Akhmediev, General Theory of Solitons, in Soliton-driven photonics, A.D. Boardman and A.P. Sukhorukov, Editors. 2001, Kluwer Academic Publishers: London. pp. 371-395

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