Dissipative solitons

doi:10.1016/0167-2789(95)00111-G

Physica D: Nonlinear Phenomena

Volume 86, Issues 1–2, 1 September 1995, Pages 323-347

Dedicated to Prof. Norman J. Zabusky and to Prof. Morikazu Toda

https://doi.org/10.1016/0167-2789(95)00111-G Get rights and content

Abstract

A generalization of the Korteweg-de Vries equation (KdVE) is considered in which additional terms belonging to the Kuramoto-Sivashinsky equation (KSE) are incorporated to account for a production-dissipation (input-output) energy balance. Two different situations are thoroughly investigated.

First, the production-dissipation part of the equation is taken as a small perturbation to the KdVE, proportional to a smallness parameter ε. It is shown that within times limited by ε⁻¹ (and beyond) the KdVE seches interact similarly to the Zabusky and Kruskal's findings, save the “aging” they experience. At longer times the localized solutions adopt the terminal shape and phase velocity, and different humps can form bound states. The increase of the production-dissipation parameter exaggerates the effects through reducing the “practical infinity” for the time scale. For ε > 2 with the rest of parameters equal to unity, the solution goes chaotic. These results outline the region where the long enough transients can be approximately considered as solitons, albeit imperfect ones.

The second situation is when the KS part of the equation is predominant. This happens either when ε is not small enough or for very long times (t → ∞ or t ⪢ ε⁻¹) when strictly permanent shapes are attained which are in fact short waves and the dissipation (higher-order derivative) is dominant. It is shown that the solution to KSE of homoclinic shape (a hump) does not qualify for a wave-particle/solution since it does not persist as a permanent shape after collision and yields to a chaotic régime. The heteroclinic shapes (kinks/bores/hydraulic jumps/shocks) do behave as particles but the interactions appear to be completely inelastic. After two such wave-particles collide they stick to each other and deform to produce a single structure of the same kind which carries the total momentum of the system. This kind of (really imperfect) solitons may be called “clayons” to emphasize the fact that upon collisions they behave as clay balls.

Thus the Zabusky and Kruskal's soliton concept is extended in two directions: to “long” transients practically “permanent” and solitonic in the time scale ε⁻¹ set by production-dissipation processes and to true permanent wave-particles with, however, inelastic behaviour upon collisions.

References (84)

H.-C. Chang et al.
Laminarizing effect of dispersion in an active-dissipative nonlinear medium
Physica D
(1993)
H.-C. Chang et al.
Nonlinear evolution of waves on a vertically falling film
J. Fluid Mech.
(1993)
C.I. Christov et al.
Inelastic interaction of Boussinesq solitons
Int. J. Bifurc. Chaos
(1994)
B. Janiaud et al.
The Eckhaus instability for travelling waves
Physica D
(1992)
R.S. Johnson
Shallow water waves on a viscous fluid. The undular bore
Phys. Fluids
(1972)
S.W. Joo et al.
Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers
J. Fluid Mech.
(1991)
V.Ya. Shkadov
Wave flow regimes of a thin viscous liquid layer subjected to gravity force
Izv. Akad. Nauk SSSR Mekh. Zhid. Gaza N
(1967)
G. Benettin
Ordered and chaotic motions in dynamical systems with many degrees of freedom
J. Math. Phys.
(1966)
G. Ciccotti et al.
D.J. Benney
Long waves in liquid film
J. Math. Phys.
(1966)
T.B. Benjamin
Wave formation in laminar flow down an inclined plane
J. Fluid Mech.
(1957)

J.V. Boussinesq

Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal reactangulaire

C. R. Hebd. Seances l'Acad. Sci.

(1871)

J.V. Boussinesq

Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horisontal en communiquant au liquid contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond

J. Math. Pures Appl.

(1872)

J.V. Boussinesq

Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal

C. R. Hebd. Seances l'Acad. Sci

(1987)

F.K. Browand et al.

Large scales in the developing mixing layer

J. Fluid Mech.

(1976)

D. Campbell

An introduction to nonlinear phenomena

H.-C. Chang

Evolution of nonlinear waves on vertically falling films — a normal form analysis

Chem. Eng. Sci.

(1987)

C.I. Christov, Flows with coherent structures: application of random point functions and stochastic functional...

C.I. Christov

Localized solutions for fluid interfaces via method of variational imbedding

C.I. Christov

Numerical investigation of the long-time evolution and interaction of localized waves

C.I. Christov

Stochastic régimes for Kuramoto-Sivashinsky equation. I. Random point approximation

Ann. Univ. Sofia, Fac. Math. Inform

(1987)

C.I. Christov et al.

Application of the method of variational imbedding for calculating shapes of solitons on falling thin films

C.I. Christov et al.

Random Point Functions and Large-Scale Turbulence

(1992)

C.I. Christov et al.

On localized solutions of an equation governing Bénard-Marangoni convection

Appl. Math. Modelling

(1993)

P.G. Drazin et al.

Solitons: an Introduction

(1987)

C. Elphick et al.

Interacting localized structures with Galilean invariance

Phys. Rev. A

(1991)

E. Fermi et al.

Studies of nonlinear problems

Los Alamos Nat. Lab. Report LA-1940

(1955)

E. Fermi et al.

A.N. Garazo et al.

Dissipative Korteweg-de Vries description of Marangoni-Bénard oscillatory convection

Phys. Fluids A

(1991)

P.L. García-Ybarra et al.

Bénard-Marangoni convection with a deformable interface and poorly conducting boundaries

Phys. Fluids

(1987)

C.S. Gardner et al.

Method for solving Korteweg-de Vries equation

Phys. Rev. Lett.

(1967)

B. Gjevik

Occurrence of finite-amplitude surface waves on falling liquid films

Phys. Fluids

(1970)

G.M. Homsy

Model equations for wavy viscous film flow

Lect. Appl. Math.

(1974)

J.M. Hyman et al.

Coherence and chaos in Kuramoto-Velarde equation

J.M. Hyman et al.

The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems

Physica D

(1986)

M. Kac et al.

On the Van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model

J. Math. Phys.

(1963)

P.L. Kapitza

Wave flow in thin layer of viscous liquid

Zh. Exp. Theor. Fiz.

(1948)

P.L. Kapitza et al.

Wave flow in thin layer of a viscous liquid — iii. experimental study of undulatory flow conditions

Zh. Exp. Theor. Fiz.

(1949)

P.L. Kapitza et al.

Wave flow of thin layers of a viscous fluid

T. Kawahara et al.

Pulse interactions in an unstable dissipative-dispersion nonlinear system

Phys. Fluids

(1988)

P. Kolodner

Collisions between pulses of travelling-wave convection

Phys. Rev. A

(1991)

B.K. Kopbosynov et al.

Thermocapillary motion in thin liquid layer

D.J. Korteweg et al.

On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves

Phil. Mag., ser. 5

(1895)

M.V.G. Krishna et al.

Non-linear stability of a viscous film with respect to three-dimensional side-band disturbances

Phys. Fluids

(1977)

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The dynamical evolution of one-dimensional lattice of active units (otherwise said particles) endowed with anharmonic Morse interactions and active–dissipative van der Pol negative friction is here studied both analytically and numerically. One of the directions of this evolution is the formation and stable propagation of dissipative soliton-like perturbations (in short solitons) against the background of some stationary displacement of units (pedestal). It is shown that the evolution of units at leading-front of the soliton is determined by mutual repulsive forces, while that of units at trailing-tail of the soliton is determined by the active–dissipative forces. Approximate analytically solvable models of the leading-front and the trailing-tail are obtained, which make it possible to predict soliton parameters and admissible range of pedestal values with high accuracy. There is also a scenario of N-vertices soliton formation emerging from a single soliton and (N-1)-vertices soliton. The possibility of controlling the number of vertices of multi-vertices solitons is offered, a feature which can be used in the problems of energy accumulation and transport in periodic structures.
Electrohydrodynamic instability of viscous liquid films: Electrostatically modified second-order Benney and Kuramoto–Sivashinsky models
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Citation Excerpt :
Thus the system is capable of sustaining the soliton propagation, as defined for conservative systems [7–9], as long as the energy balance is maintained at steady conditions. References [2–6] originated in the study of oscillatory Bénard-Maragoni instability [10,11] where predictions and experimental observations were done of Boussinesq-Korteweg-de Vries interfacial solitary wave behavior in one, two and three dimensions [5,10,11]. Let us insist that the concept of dissipative soliton extends the classical theory to non-conservative systems where energy (rather than being conserved) is pumped and dissipated in an appropriate balance, thus exciting and, eventually, maintaining past an instability threshold the localized structure (or a periodic nonlinear wave train).
A few salient soliton-like wave evolutionary features of one- and two-dimensional lattices of interacting active units are provided here. In the latter case, particular attention is given to the crystal-like triangular lattice. On the one hand, the units are coupled with nearest neighbors anharmonic forces (Morse potential). On the other hand the units are endowed with the possibility of an input-output energy balance that permits evolution to a steady state and the appearance of metastable states which on occasion can be quite long lasting. The lifetimes of such metastable states depend on the lattice parameter values and the wave front width. Eventually, all metastable states evolve to steady translational modes especially under influence of noise.
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¹: On leave from National Institute of Meteorology and Hydrology, Sofia, Bulgaria.

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Dissipative solitons

Abstract

Physica D

J. Fluid Mech.

Int. J. Bifurc. Chaos

Physica D

Phys. Fluids

J. Fluid Mech.

Izv. Akad. Nauk SSSR Mekh. Zhid. Gaza N

Ordered and chaotic motions in dynamical systems with many degrees of freedom

J. Math. Phys.

Long waves in liquid film

J. Math. Phys.

Wave formation in laminar flow down an inclined plane

J. Fluid Mech.

Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal reactangulaire

C. R. Hebd. Seances l'Acad. Sci.

Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horisontal en communiquant au liquid contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond

J. Math. Pures Appl.

Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal

C. R. Hebd. Seances l'Acad. Sci

Large scales in the developing mixing layer

J. Fluid Mech.

An introduction to nonlinear phenomena

Evolution of nonlinear waves on vertically falling films — a normal form analysis

Chem. Eng. Sci.

Localized solutions for fluid interfaces via method of variational imbedding

Numerical investigation of the long-time evolution and interaction of localized waves

Stochastic régimes for Kuramoto-Sivashinsky equation. I. Random point approximation

Ann. Univ. Sofia, Fac. Math. Inform

Application of the method of variational imbedding for calculating shapes of solitons on falling thin films

Random Point Functions and Large-Scale Turbulence

On localized solutions of an equation governing Bénard-Marangoni convection

Appl. Math. Modelling

Solitons: an Introduction

Interacting localized structures with Galilean invariance

Phys. Rev. A

Studies of nonlinear problems

Los Alamos Nat. Lab. Report LA-1940

Dissipative Korteweg-de Vries description of Marangoni-Bénard oscillatory convection

Phys. Fluids A

Bénard-Marangoni convection with a deformable interface and poorly conducting boundaries

Phys. Fluids

Method for solving Korteweg-de Vries equation

Phys. Rev. Lett.

Occurrence of finite-amplitude surface waves on falling liquid films

Phys. Fluids

Model equations for wavy viscous film flow

Lect. Appl. Math.

Coherence and chaos in Kuramoto-Velarde equation

The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems

Physica D

On the Van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model

J. Math. Phys.

Wave flow in thin layer of viscous liquid

Zh. Exp. Theor. Fiz.

Wave flow in thin layer of a viscous liquid — iii. experimental study of undulatory flow conditions

Zh. Exp. Theor. Fiz.

Wave flow of thin layers of a viscous fluid

Pulse interactions in an unstable dissipative-dispersion nonlinear system

Phys. Fluids

Collisions between pulses of travelling-wave convection

Phys. Rev. A

Thermocapillary motion in thin liquid layer

On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves

Phil. Mag., ser. 5

Non-linear stability of a viscous film with respect to three-dimensional side-band disturbances

Phys. Fluids