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Solitons

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Nonlinear Dynamics and Chaotic Phenomena: An Introduction

Part of the book series: Fluid Mechanics and Its Applications ((FMIA,volume 103))

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Abstract

Studies of nonlinear phenomena of various systems have revealed that certain generic equations can be used to describe the nonlinear evolution of a large class of physical systems. This is due to the fact that the underlying dynamics of many diverse physical systems have a degree of commonality which can be condensed into relatively simple model evolution equations. The similarity of the asymptotic behavior of solutions to these generic equations further suggests the possibility of a unified treatment of nonlinear phenomena and hence an apparent underlying “simplicity” in their description. Some of these solutions exhibit remarkably stable structures called solitons. In this chapter, we will do a discussion centered around the Korteweg-de Vries equation which is a prime example of such nonlinear evolution equations.

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Notes

  1. 1.

    There was in fact a fourth person, Mary Tsingou, who also participated in this numerical exploration (Douxois 2008).

  2. 2.

    The Korteweg-de Vries equation was apparently first obtained (Grimshaw 2007), by Boussinesq (1877).

  3. 3.

    The truncations in equations (7.3) and (7.4) signify that the long waves described by equation (7.5) are weakly nonlinear and weakly dispersive.

  4. 4.

    Equation (7.89) turns out to be the limiting member of a three-parameter family of periodic traveling wave solutions of equation (7.2), called cnoidal waves (see Sect. 7.6.6).

  5. 5.

    The amplitude-wave length relationship for a Korteweg-de Vries solitary wave is

    $$ a = \biggl(\frac{1}{\lambda}\biggr)^2\frac{12}{\sigma}. $$
  6. 6.

    Solitary waves were first observed by J. Scott Russell on the Edinburgh-Glasgow canal as a moving “well-defined heap of water” in 1834. Russell also performed laboratory experiments on solitary waves and empirically deduced that the speed u of the solitary wave is given by

    $$ u^2=g (h_0+a) $$

    a being the amplitude of the wave and h 0 being the undisturbed depth of water. However, the great significance of Russell’s discoveries had to wait more than 130 years for proper appreciation.

  7. 7.

    \(\operatorname{cn}(\zeta,s)\), for s away from 1, is qualitatively similar to cosζ. In fact,

    $$ \operatorname{cn}(\zeta, 0)\equiv\cos\zeta. $$
  8. 8.

    It is of interest to note that (7.111) implies, in conjunction with (7.97),

    $$ u=\sigma\alpha-k^2,\quad k^2\equiv\frac{\sigma}{3}(\alpha-\gamma) $$

    which agrees with the dispersion relation given by equation (7.2) for linear waves propagating on the basic state ϕ 0=α.

  9. 9.

    Alternatively, noting further that

    $$ \operatorname{cn}(\zeta,1)=\operatorname{sech}\zeta $$

    equation (7.109) leads to

    $$ \phi(\xi)=\gamma+(\alpha-\gamma)\operatorname{sech}^2 \biggl(\sqrt{\frac{\sigma(\alpha-\gamma)}{12}}\xi\biggr) $$

    as in (7.113)!

    The phase-plane portrait (see figure above) also shows the existence of periodic solutions corresponding to closed curves about a center. The homoclinic orbit starting and ending at the saddle point x has infinite period and represents the solitary wave!

  10. 10.

    Such a “self truncation” is believed (Hietarinta 1990) to signify integrability of the nonlinear evolution equation.

  11. 11.

    Hirota transformation has some analogy with the well known Cole-Hopf transformation (Cole 1951; Hopf 1950): Consider Burgers equation,

    $$ \phi_t+\phi\phi_x=\nu\phi_{xx}. $$

    Putting,

    $$ \phi=-2\nu(\ln\psi)_x $$

    we obtain

    $$ \psi_t =\nu\psi_{xx} $$

    which is the linear diffusion equation!

  12. 12.

    Putting f 2≡0, leads to F k ≡0, ∀k>1, and substituting (7.125), (7.118) leads to the solitary wave solution (7.88) for ϕ.

  13. 13.

    There is no phase change, by contrast, for two interacting linear waves, even though they emerge unchanged by the interaction as well.

  14. 14.

    Zakharov and Shabat (1972) and Ablowitz et al. (1974) showed that this method can be adapted to solve other nonlinear evolution equations.

  15. 15.

    One may indeed view the bound states as the analytic continuation of the scattering states, defined on the real k-axis, to the upper half of the complex k-plane.

  16. 16.

    Note that,

    $$ (NM-MN)[\varPsi]=-(\phi_t-6\phi\phi_x+\phi_{xxx})\varPsi+3\phi_x M[\varPsi]. $$
  17. 17.

    Goursat problem (Garabedian 1984) has to do with finding the solution of a linear hyperbolic PDE satisfying conditions prescribed on a characteristic curve ξ=const and a monotonically increasing curve ξ=ξ(x).

  18. 18.

    One may also view the inverse-scattering problem, as represented by the GLM equation as a Riemann-Hilbert boundary-value problem in the scattering space (Ablowitz and Clarkson 1991).

  19. 19.

    Here, we have used the identity,

    $$ \varGamma(p)\varGamma(1-p)=\frac{\pi}{\sin p \pi} $$

    or

    $$ \varGamma \biggl(\frac{1}{2}+p \biggr)\varGamma \biggl(\frac{1}{2}-p \biggr)=\frac{\pi}{\cos p \pi}. $$

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Authors and Affiliations

  1. Dept. Mathematics, University of Central Florida, Orlando, FL, USA

    Bhimsen K. Shivamoggi

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Shivamoggi, B.K. (2014). Solitons. In: Nonlinear Dynamics and Chaotic Phenomena: An Introduction. Fluid Mechanics and Its Applications, vol 103. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7094-2_7

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