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nonlinear evolution equations


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1: Peter A. Clarkson
2: 29.19 Physical Applications
Clarkson (1991) solves nonlinear evolution equations. …
3: Mark J. Ablowitz
Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. …
4: 23.21 Physical Applications
§23.21(ii) Nonlinear Evolution Equations
5: Bibliography
  • M. J. Ablowitz and P. A. Clarkson (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge.
  • 6: Bibliography C
  • P. A. Clarkson (1991) Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.
  • 7: Bibliography S
  • H. Segur and M. J. Ablowitz (1981) Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent. Phys. D 3 (1-2), pp. 165–184.
  • 8: Bibliography K
  • M. D. Kruskal (1974) The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.
  • 9: 9.16 Physical Applications
    These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …
    10: 22.19 Physical Applications
    The subsequent time evolution is always oscillatory with period 4 K ( k ) / 1 + 2 η and modulus k = 1 / 2 + η - 1 : …
    §22.19(iii) Nonlinear ODEs and PDEs
    Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. These include the time dependent, and time independent, nonlinear Schrödinger equations (NLSE) (Drazin and Johnson (1993, Chapter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the Korteweg–de Vries (KdV) equation (Kruskal (1974), Li and Olver (2000)), the sine-Gordon equation, and others; see Drazin and Johnson (1993, Chapter 2) for an overview. … …