A KdV-like advection-dispersion equation with some remarkable properties

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  author =       "Abhijit Sen and Dilip P. Ahalpara and 
                 Anantanarayanan Thyagaraja and Govind S. Krishnaswami",
  title =        "A {KdV}-like advection-dispersion equation with some
                 remarkable properties",
  journal =      "Communications in Nonlinear Science and Numerical
  volume =       "17",
  number =       "11",
  pages =        "4115--4124",
  year =         "2012",
  ISSN =         "1007-5704",
  DOI =          "doi:10.1016/j.cnsns.2012.03.001",
  URL =          "http://www.sciencedirect.com/science/article/pii/S100757041200113X",
  keywords =     "genetic algorithms, genetic programming, Advection
                 dispersion equation, Travelling waves, Recurrence",
  abstract =     "We discuss a new non-linear PDE, u t + ( 2 u xx / u )
                 u x = ?\mu u xxx , invariant under scaling of dependent
                 variable and referred to here as SIdV. It is one of the
                 simplest such translation and space-time
                 reflection-symmetric first order advection-dispersion
                 equations. This PDE (with dispersion coefficient unity)
                 was discovered in a genetic programming search for
                 equations sharing the KdV solitary wave solution. It
                 provides a bridge between non-linear advection,
                 diffusion and dispersion. Special cases include the
                 mKdV and linear dispersive equations. We identify two
                 conservation laws, though initial investigations
                 indicate that SIdV does not follow from a polynomial
                 Lagrangian of the KdV sort. Nevertheless, it possesses
                 solitary and periodic travelling waves. Moreover,
                 numerical simulations reveal recurrence properties
                 usually associated with integrable systems. KdV and
                 SIdV are the simplest in an infinite dimensional family
                 of equations sharing the KdV solitary wave. SIdV and
                 its generalisations may serve as a testing ground for
                 numerical and analytical techniques and be a rich
                 source for further explorations.",

Genetic Programming entries for Abhijit Sen Dilip P Ahalpara Anantanarayanan Thyagaraja Govind S Krishnaswami