|
|
| Line 1: |
Line 1: |
| '''Dissipative solitons (DSs)''' are stable solitary localized | | The author's name is Christy. My wife and I live in Mississippi and I love every working day living right here. Credit authorising is how he tends to make money. To play lacross is 1 of the things she enjoys most.<br><br>my web site [http://www.weddingwall.com.au/groups/easy-advice-for-successful-personal-development-today/ accurate psychic predictions] |
| structures that arise in nonlinear spatially extended
| |
| [[dissipative system]]s due to mechanisms of [[self-organization]].
| |
| They can be considered as an extension of the classical
| |
| [[soliton]] concept in conservative systems. An alternative
| |
| terminology includes autosolitons, spots and pulses.
| |
| | |
| Apart from
| |
| aspects similar to the behavior of classical particles like the
| |
| formation of bound states, DSs exhibit entirely
| |
| ''nonclassical'' behavior – e.g. scattering, generation and
| |
| annihilation – all without the constraints of energy or momentum
| |
| conservation. The excitation of internal
| |
| [[degrees of freedom (physics and chemistry)|degrees of freedom]] may result in a dynamically stabilized intrinsic
| |
| speed, or periodic oscillations of the shape.
| |
| | |
| == Historical development ==
| |
| | |
| === Origin of the soliton concept ===
| |
| | |
| DSs have been experimentally observed for a long time.
| |
| [[Helmholtz]]<ref>H. Helmholtz, Arch. Anat. Physiol. 57 (1850):
| |
| 276</ref> measured the propagation velocity of nerve pulses in
| |
| 1850. In 1902, [[Otto Lehmann (physicist)|Lehmann]]<ref>O. Lehmann, Ann.
| |
| Phys. 4 (1902): 1</ref> found the formation of localized anode
| |
| spots in long gas-discharge tubes. Nevertheless, the term
| |
| "soliton" was originally developed in a different context. The
| |
| starting point was the experimental detection of "solitary
| |
| water waves" by [[John Scott Russell|Russell]] in 1834.<ref>J.
| |
| S. Russell, Report of the fourteenth meeting of the British
| |
| Association for the Advancement of Science (1845): 311</ref>
| |
| These observations initiated the theoretical work of
| |
| [[John Strutt, 3rd Baron Rayleigh|Rayleigh]]<ref>J. W. Rayleigh, Phil. Mag. 1 (1876): 257
| |
| </ref> and [[Joseph Valentin Boussinesq|Boussinesq]]<ref>J.
| |
| Boussinesq, Compt. Rend. Acad. Sc. 72 (1871): 755</ref> around
| |
| 1870, which finally led to the approximate description of such
| |
| waves by Korteweg and de Vries in 1895; that description is known today as the (conservative)
| |
| [[Korteweg-de Vries Equation|KdV]] equation.<ref>D. J. Korteweg and H. de Vries, Phil. Mag. 39 (1895):
| |
| 422</ref>
| |
| | |
| On this background the term "[[soliton]]" was
| |
| coined by [[Norman Zabusky|Zabusky]] and [[Martin David Kruskal|Kruskal]]<ref>N. J. Zabusky and M. D.
| |
| Kruskal, Phys. Rev. Lett. 15 (1965): 240</ref> in 1965. These
| |
| authors investigated certain well localised solitary solutions
| |
| of the KdV equation and named these objects solitons. Among
| |
| other things they demonstrated that in 1-dimensional space
| |
| solitons exist, e.g. in the form of two unidirectionally
| |
| propagating pulses with different size and speed and exhibiting the
| |
| remarkable property that number, shape and size are the same
| |
| before and after collision.
| |
| | |
| Gardner at al.<ref>C. S. Gardner et al., Phys. Rev. Lett. 19
| |
| (1967): 1095</ref> introduced the [[Inverse scattering transform|
| |
| inverse scattering technique]]
| |
| for solving the KdV equation and proved that this equation is
| |
| completely [[Integrable system|integrable]]. In 1972 [[Vladimir E. Zakharov|Zakharov]] and
| |
| Shabat<ref>V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl.
| |
| 8 (1974): 226</ref> found another integrable equation and
| |
| finally it turned out that the inverse scattering technique can
| |
| be applied successfully to a whole class of equations (e.g. the
| |
| [[Nonlinear Schrödinger equation|nonlinear Schrödinger]] and
| |
| [[Sine-Gordon equation|Sine-Gordon]] equations). From 1965
| |
| up to about 1975, a common agreement was reached: to reserve the term ''soliton'' to
| |
| pulse-like solitary solutions of conservative nonlinear partial
| |
| differential equations that can be solved by using the inverse
| |
| scattering technique.
| |
| | |
| === Weakly and strongly dissipative systems ===
| |
| | |
| With increasing knowledge of classical solitons, possible
| |
| technical applicability came into perspective, with the most
| |
| promising one at present being the transmission of optical
| |
| solitons via [[Fiberglass|glass fibers]] for the purpose of
| |
| [[data transmission]]. In contrast to systems with purely
| |
| classical behavior, solitons in fibers dissipate energy and
| |
| this cannot be neglected on an intermediate and long time
| |
| scale. Nevertheless the concept of a classical soliton can
| |
| still be used in the sense that on a short time scale
| |
| dissipation of energy can be neglected. On an intermediate time
| |
| scale one has to take small energy losses into account as a
| |
| perturbation, and on a long scale the amplitude of the soliton
| |
| will decay and finally vanish.<ref>Y. S. Kivshar and G. P.
| |
| Agrawal, ''Optical Solitons. From Fibers to Photonic Crystals'',
| |
| Academic press (2003)</ref>
| |
| | |
| There are however various types of systems which are capable of
| |
| producing solitary structures and in which dissipation plays an
| |
| essential role for their formation and stabilization. Although
| |
| research on certain types of these DSs has been carried out for
| |
| a long time (for example, see the research on nerve pulses culminating
| |
| in the work of [[Hodgkin and Huxley]]<ref>A. L. Hodgkin and A.
| |
| F. Huxley, J. Physiol. 117 (1952): 500</ref> in 1952), since
| |
| 1990 the amount of research has significantly increased (see e.g.
| |
| <ref>B. S. Kerner and V. V. Osipov,
| |
| ''Autosolitons. A New Approach to Problems of Self-Organization and Turbulence'',
| |
| Kluwer Academic Publishers (1995): 53</ref><ref>M. Bode and H.-G. Purwins, Physica D 86 (1995): 53</ref><ref>C.I. Christov and M.G. Velarde, Physica D 86 (1995): 323</ref><ref>N. Akhmediev and A. Ankiewicz, ''Dissipative Solitons'', Lecture Notes in Physics, Springer, Berlin (2005)</ref>)
| |
| Possible reasons are improved experimental devices and
| |
| analytical techniques, as well as the availability of more
| |
| powerful computers for numerical computations. Nowadays, it is
| |
| common to use the term ''dissipative solitons'' for solitary structures in
| |
| strongly dissipative systems.
| |
| | |
| == Experimental observations of DSs ==
| |
| | |
| Today, DSs can be found in many different
| |
| experimental set-ups. Examples include
| |
| | |
| * [[Gas-discharge lamp|Gas-discharge systems]]: [[Plasma (physics)|plasmas]] confined in a discharge space which often has a lateral extension large compared to the main discharge length. DSs arise as current filaments between the electrodes and were found in DC systems with a high-ohmic barrier,<ref>C. Radehaus et al., Phys. Lett. A 125 (1987): 92</ref> AC systems with a dielectric barrier,<ref>I. Brauer et al., Phys. Rev. Lett. 84 (2000): 4104</ref> and as anode spots,<ref>S. M. Rubens and J. E. Henderson, Phys. Rev. 58 (1940): 446</ref> as well as in an obstructed discharge with metallic electrodes.<ref>S. Nasuno, Chaos 13 (2003): 1010</ref>
| |
| | |
| <gallery caption="DSs experimentally observed in planar dc gas-discharge systems with high-ohmic barrier" widths="320" heights="269" perrow="2">
| |
| Image:Isoldissol1_en.gif| Averaged current density distribution without oscillatory tails.
| |
| Image:Isoldissol2_en.gif| Averaged current density distribution with oscillatory tails.
| |
| </gallery>
| |
| | |
| * [[Semiconductor]] systems: these are similar to gas-discharges; however, instead of a gas, semiconductor material is sandwiched between two planar or spherical electrodes. Set-ups include Si and GaAs [[PIN diode|pin diodes]],<ref>D. Jäger et al., Phys. Lett. A 117 (1986): 141</ref> n-GaAs,<ref>K. M. Mayer et al., Z. Phys. B 71
| |
| (1988): 171</ref> and Si p<sup>+</sup>-n<sup>+</sup>-p-n<sup>-</sup>,<ref>F.-J. Niedernostheide et al., Phys. stat. sol. (b) 172 (1992): 249</ref> and ZnS:Mn structures.<ref>M. Beale, Phil. Mag. B 68 (1993): 573</ref>
| |
| | |
| * [[Nonlinear optics|Nonlinear optical systems]]: a light beam of high intensity interacts with a nonlinear medium. Typically the medium reacts on rather slow time scales compared to the beam propagation time. Often, the output is [[feedback|fed back]] into the input system via single-mirror feedback or a feedback loop. DSs may arise as bright spots in a two-dimensional plane orthogonal to the beam propagation direction; one may, however, also exploit other effects like [[Polarization (waves)|polarization]]. DSs have been observed for [[saturable absorption|saturable absorbers]],<ref>V. B. Taranenko et al., Phys. Rev. A 56 (1997): 1582</ref> degenerate [[optical parametric oscillator]]s (DOPOs),<ref>V. B. Taranenko et al., Phys. Rev. Lett. 81 (1998): 2236</ref> liquid crystal [[light valve]]s (LCLVs),<ref>A. Schreiber et al., Opt. Comm. 136 (1997): 415</ref> alkali vapor systems,<ref>B. Schäpers et al., Phys. Rev. Lett. 85 (2000): 748</ref> [[photorefractive effect|photorefractive media]],<ref>C. Denz et al., ''Transverse-Pattern Formation in Photorefractive Optics'', Springer Tracts in Modern Physics (2003): 188</ref> and semiconductor microresonators.<ref>S. Barland et al., Nature 419 (2002): 699</ref>
| |
| | |
| * If the vectorial properties of DSs are considered,''[[vector dissipative soliton]]'' could also be observed in a fiber laser passively mode locked through saturable absorber,<ref>H. Zhang et al., Optics Express 17 (2009): 455.</ref>
| |
| | |
| * In addition, multiwavelength dissipative soliton in an all normal dispersion fiber laser passively mode-locked with a SESAM has been obtained. It is confirmed that depending on the cavity birefringence, stable single-, dual- and triple-wavelength dissipative soliton can be formed in the laser. Its generation mechanism can be traced back to the nature of dissipative soliton.<ref>H. Zhang et al, Optics Express 17 (2009): 12692</ref>
| |
| | |
| * Chemical systems: realized either as one- and two-dimensional reactors or via catalytic surfaces, DSs appear as pulses (often as propagating pulses) of increased concentration or temperature. Typical reactions are the [[Belousov-Zhabotinsky reaction]],<ref>C. T. Hamik et al., J. Phys. Chem. A 105 (2001): 6144</ref> the ferrocyanide-iodate-sulphite reaction as well as the oxidation of hydrogen,<ref>S. L. Lane and D. Luss, Phys. Rev. Lett. 70 (1993): 830</ref> CO,<ref>H. H. Rotermund et al., Phys. Rev. Lett. 66 (1991): 3083</ref> or iron.<ref>R. Suzuki, Adv. Biophys. 9 (1976): 115</ref> [[Action potential|Nerve pulses]]<ref>A. L. Hodgkin and A. F. Huxley, J. Physiol. 117 (1952): 500</ref> or migraine aura waves <ref>Dahlem MA, Hadjikhani N (2009) Migraine Aura: Retracting Particle-Like Waves in Weakly Susceptible Cortex. PLoS ONE 4(4): e5007</ref> also belong to this class of systems.
| |
| | |
| * Vibrated media: vertically shaken granular media,<ref>P. B. Umbanhowar et al., Nature 382 (1996): 793</ref> [[colloid|colloidal suspensions]],<ref>O. Lioubashevski et al., Phys. Rev. Lett. 83 (1999): 3190</ref> and [[Newtonian fluid]]s<ref>O. Lioubashevski et al., Phys. Rev. Lett. 76 (1996): 3959</ref> produce harmonically or sub-harmonically oscillating heaps of material, which are usually called [[oscillon]]s.
| |
| | |
| * [[Hydrodynamics|Hydrodynamic systems]]: the most prominent realization of DSs are domains of [[horizontal convective rolls|convection]] rolls on a conducting background state in binary liquids.<ref>G. Ahlers, Physica D 51 (1991): 421</ref> Another example is a film dragging in a rotating cylindric pipe filled with oil.<ref>F. Melo and S. Douady, Phys. Rev. Lett. 71 (1993): 3283</ref>
| |
| | |
| * Electrical networks: large one- or two-dimensional arrays of coupled cells with a nonlinear [[current-voltage characteristic]].<ref>J. Nagumo et al., Proc. Inst. Radio Engin. Electr. 50 (1962): 2061</ref> DSs are characterized by a locally increased current through the cells.
| |
| | |
| Remarkably enough, phenomenologically the dynamics of the DSs in many of the above systems are similar in spite of the microscopic differences. Typical observations are (intrinsic) propagation, [[scattering]], formation of [[bound state]]s and clusters, drift in gradients, interpenetration, generation, and annihilation, as well as higher instabilities.
| |
| | |
| == Theoretical description of DSs ==
| |
| | |
| Most systems showing DSs are described by nonlinear
| |
| [[partial differential equations]]. Discrete difference equations and
| |
| [[Cellular automaton|cellular automata]] are also used. Up to now,
| |
| modeling from first principles followed by a quantitative
| |
| comparison of experiment and theory has been performed only
| |
| rarely and sometimes also poses severe problems because of large
| |
| discrepancies between microscopic and macroscopic time and
| |
| space scales. Often simplified prototype models are
| |
| investigated which reflect the essential physical processes in
| |
| a larger class of experimental systems. Among these are
| |
| | |
| * [[Reaction–diffusion system]]s, used for chemical systems, gas-discharges and semiconductors.<ref>H.-G. Purwins et al., Dissipative Solitons in Reaction-Diffusion Systems, in ''Dissipative Solitons'', Lecture Notes in Physics, Springer (2005)</ref> The evolution of the state vector '''''q'''''('''''x''''', ''t'') describing the concentration of the different reagents is determined by diffusion as well as local reactions:
| |
| | |
| ::<math>\partial_t \boldsymbol{q} = \underline{\boldsymbol{D}}
| |
| \Delta \boldsymbol{q} + \boldsymbol{R}(\boldsymbol{q}).</math>
| |
| | |
| :A frequently encountered example is the two-component Fitzhugh-Nagumo-type activator-inhibitor system
| |
| | |
| ::<math> \left( \begin{array}{c} \tau_u \partial_t u\\\tau_v
| |
| \partial_t v
| |
| \end{array} \right) =
| |
| \left(\begin{array}{cc} d_u^2 &0\\0&d_v^2
| |
| \end{array}\right)
| |
| \left( \begin{array}{c} \Delta u\\ \Delta v
| |
| \end{array} \right) + \left(\begin{array}{c} \lambda u -u^3 - \kappa_3 v +\kappa_1\\u-v
| |
| \end{array}\right)
| |
| .</math>
| |
| | |
| :Stationary DSs are generated by production of material in the center of the DSs, diffusive transport into the tails and depletion of material in the tails. A propagating pulse arises from production in the leading and depletion in the trailing end.<ref>E. Meron, Phys. Rep. 218 (1992):1</ref> Among other effects, one finds periodic oscillations of DSs ("breathing"),<ref>F.-J. Niedernostheide et al. in Nonlinearity with Disorder, Springer Proc. Phys. 67 (1992): 282</ref><ref name="gurevich">S. V. Gurevich et al., Phys. Rev. E 74 (2006), 066201</ref> bound states,<ref>M. Or-Guil et al., Physica D 135 (2000): 154</ref> and collisions, merging, generation and annihilation.<ref>C. P. Schenk et al., Phys. Rev. Lett. 78 (1997): 3781</ref>
| |
| | |
| * [[Ginzburg-Landau theory|Ginzburg-Landau type systems]] for a complex scalar ''q''('''''x''''', ''t'') used to describe nonlinear optical systems, plasmas, Bose-Einstein condensation, liquid crystals and granular media.<ref>I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74 (2002): 99</ref> A frequently found example is the cubic-quintic subcritical Ginzburg-Landau equation
| |
| | |
| ::<math> \partial_t q = (d_r+ i d_i) \Delta q + l_r q + (c_r + i
| |
| c_i)|q|^2 q + (q_r + i q_i) |q|^4 q.</math>
| |
| | |
| :To understand the mechanisms leading to the formation of DSs, one may consider the energy ρ = |''q''|<sup>2</sup> for which one may derive the continuity equation
| |
| | |
| ::<math> \partial_t \rho + \nabla \cdot \boldsymbol{m} = S = d_r
| |
| (q \Delta q^{\ast} + q^{\ast} \Delta q) + 2 l_r \rho + 2 c_r
| |
| \rho^2 + 2 q_r \rho^3 \quad\text{with} \quad\boldsymbol{m} = 2
| |
| d_i \text{Im}(q^{\ast}\nabla q).</math>
| |
| | |
| :One can thereby show that energy is generally produced in the flanks of the DSs and transported to the center and potentially to the tails where it is depleted. Dynamical phenomena include propagating DSs in 1d,<ref>V. V. Afanasjev et al., Phys. Rev. E 53 (1996): 1931</ref> propagating clusters in 2d,<ref>N. N. Rosanov et al., J. Exp. Theor. Phys. 102 (2006): 547</ref> bound states and vortex solitons,<ref>L.-C. Crasovan et al., Phys. Rev. E 63 (2000): 016605</ref> as well as "exploding DSs".<ref>J. M. Soto-Crespo et al., Phys. Rev. Lett. 85 (2000), 2937</ref>
| |
| | |
| * The [[Swift-Hohenberg equation]] is used in nonlinear optics and in the granular media dynamics of flames or electroconvection. Swift-Hohenberg can be considered as an extension of the Ginzburg-Landau equation. It can be written as
| |
| | |
| ::<math>\partial_t q = (s_r+ i s_i) \Delta^2 q + (d_r+ i d_i)
| |
| \Delta q + l_r q + (c_r + i c_i)|q|^2 q + (q_r + i q_i) |q|^4
| |
| q.</math>
| |
| | |
| :For ''d<sub>r</sub>'' > 0 one essentially has the same mechanisms as in the Ginzburg-Landau equation.<ref>J. M.
| |
| Soto-Crespo and N. Akhmediev, Phys. Rev. E 66 (2002): 066610</ref> For ''d<sub>r</sub>'' < 0, in the real Swift-Hohenberg equation one finds bistability between homogeneous states and Turing patterns. DSs are stationary localized Turing domains on the homogeneous background.<ref>H. Sakaguchi and H. R. Brand,Physica D 97 (1996): 274</ref> This also holds for the complex Swift-Hohenberg equations; however, propagating DSs as well as interaction phenomena are also possible, and observations include merging and interpenetration.<ref>H. Sakaguchi and H. R. Brand, Physica D 117 (1998): 95</ref>
| |
| | |
| <gallery caption="Space-time plots showing the dynamics and interaction of DSs as numerical solutions of the above model equations in one spatial dimension" widths="215" heights="206" perrow="3">
| |
| Image:breathing_DS_reaction_diffusion.gif| Single "breathing" DS as solution of the two-component reaction-diffusion system with activator u (left half) and inhibitor v (right half).
| |
| Image:DS_collision_ginzburg_landau.gif| Collision and merging of two DSs with a mutual phase difference of π/4 in the cubic-quintic Ginzburg-Landau equation, the plot shows the amplitude |''q''|.
| |
| Image:DS_interpenetration_swift_hohenberg.gif| "Interpenetration" of two DSs with a mutual phase difference of 0 in the Swift-Hohenberg equation with ''d<sub>r</sub>'' < 0, the plot shows the amplitude |''q''|.
| |
| </gallery>
| |
| | |
| == Particle properties and universality ==
| |
| | |
| DSs in many different systems show universal particle-like
| |
| properties. To understand and describe the latter, one may try
| |
| to derive "particle equations" for slowly varying order
| |
| parameters like position, velocity or amplitude of the DSs by
| |
| adiabatically eliminating all fast variables in the field
| |
| description. This technique is known from linear systems,
| |
| however mathematical problems arise from the nonlinear models
| |
| due to a coupling of fast and slow modes.<ref>R. Friedrich,
| |
| Group Theoretic Methods in the Theory of Pattern Formation, in
| |
| ''Collective dynamics of nonlinear and disordered systems'',
| |
| Springer (2004)</ref>
| |
| | |
| Similar to low-dimensional dynamic systems, for supercritical
| |
| bifurcations of stationary DSs one finds characteristic normal
| |
| forms essentially depending on the symmetries of the system.
| |
| E.g., for a transition from a symmetric stationary to an
| |
| intrinsically propagating DS one finds the Pitchfork normal
| |
| form
| |
| | |
| :<math> \dot{\boldsymbol{v}} = (\sigma - \sigma_0)
| |
| \boldsymbol{v} - |\boldsymbol{v}|^2 \boldsymbol{v}</math>
| |
| | |
| for the velocity ''v'' of the DS,<ref>M.
| |
| Bode, Physica D 106 (1997): 270</ref> here σ
| |
| represents the bifurcation parameter and σ<sub>0</sub>
| |
| the bifurcation point. For a bifurcation to a "breathing" DS,
| |
| one finds the Hopf normal form
| |
| | |
| :<math> \dot{A} = (\sigma - \sigma_0) A - |A|^2
| |
| A</math>
| |
| | |
| for the amplitude ''A'' of the oscillation.<ref
| |
| name="gurevich"/> It is also possible to treat "weak interaction"
| |
| as long as the overlap of the DSs is not too large.<ref>M. Bode
| |
| et al., Physica D 161 (2002): 45</ref> In this way, a
| |
| comparison between experiment and theory is facilitated.,<ref>H.
| |
| U. Bödeker et al., Phys. Rev. E 67 (2003): 056220</ref>
| |
| <ref>H. U. Bödeker et al., New J. Phys. 6 (2004): 62</ref>
| |
| Note that the above problems do not arise for classical
| |
| solitons as inverse scattering theory yields complete
| |
| analytical solutions.
| |
| | |
| ==See also==
| |
| * [[soliton]]
| |
| * [[vector soliton]]
| |
| * [[fiber laser]]
| |
| * [[Nonlinear system]]
| |
| * [[compacton]], a soliton with compact support
| |
| * [[Clapotis]]
| |
| * [[Freak wave]]s may be a related phenomenon.
| |
| * [[Oscillon]]s
| |
| * [[peakon]], a soliton with a non-differentiable peak.
| |
| * [[Q-ball]] a non-topological soliton
| |
| * [[Soliton (topological)]].
| |
| * [[Soliton (optics)]]
| |
| * [[Soliton model]] of nerve impulse propagation
| |
| * [[Spatial soliton]]
| |
| * [[Solitary wave]]s in discrete media [http://www.livescience.com/technology/050614_baby_waves.html]
| |
| * [[Topological quantum number]]
| |
| * [[Sine-Gordon equation]]
| |
| * [[graphene]]
| |
| | |
| {{Commons category|Solitons}}
| |
| * [[nonlinear Schrödinger equation]]
| |
| | |
| == References ==
| |
| | |
| ===Inline===
| |
| | |
| {{reflist|2}}
| |
| | |
| === Books and overview articles ===
| |
| | |
| * N. Akhmediev and A. Ankiewicz, ''Dissipative Solitons'', Lecture Notes in Physics, Springer, Berlin (2005)
| |
| * N. Akhmediev and A. Ankiewicz, ''Dissipative Solitons: From Optics to Biology and Medicine'', Lecture Notes in Physics, Springer, Berlin (2008)
| |
| * H.-G. Purwins et al., Advances in Physics 59 (2010): 485
| |
| * A. W. Liehr: ''Dissipative Solitons in Reaction Diffusion Systems. Mechanism, Dynamics, Interaction.'' Volume 70 of Springer Series in Synergetics, Springer, Berlin Heidelberg 2013, [http://www.springer.com/physics/complexity/book/978-3-642-31250-2 ISBN 978-3-642-31250-2]
| |
| | |
| {{DEFAULTSORT:Dissipative Soliton}}
| |
| [[Category:Solitons]]
| |
| [[Category:Self-organization]]
| |
| [[Category:Systems theory]]
| |