Classification of qt=P(x, t, q, qx, qxx (original) (raw)
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Physica A: Statistical Mechanics and its Applications, 1989
In a previous paper (I) an extension of the direct iinearization method was developed for obtaining solutions of multicomponent generalizations of integrable nonlinear partial differential equations (PDE's). The method is based on a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of the spectral parameter. In I the general framework has been presented, with as immediate application the direct linearization of multicomponent versions of the nonlinear Schr6dinger equation and the (complex) modified Korteweg-de Vries equation. In the present paper we treat a variety of examples of other multicomponent PDE's, and we also discuss Miura transformations and gauge equivalences. The examples include the direct linearization of multicomponent generalizations of the isotropic Heisenberg spin chain equation, the complex sine-Gordon equation, the Getmanov equation, the derivative nonlinear Schr6dinger equation and the massive Thirring model equations.
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A finite-dimensional involutive system is presented, and the Wadati-Konno-Ichikawa (WKI) hierarchy of nonlinear evolution equations and their commutator representations are discussed in this article. By this finite-dimensional involutive system, it is proven that under the so-called Bargmann constraint between the potentials and the eigenfunctions, the eigenvalue problem (called the WKI eigenvalue problem) studied by Wadati, Konno, and Ichikawa [J. Phys. Sot. Jpn. 47, 1698 (1979)] is nonlinearized as a completely integrable Hamiltonian system in the Liouville sense. Moreover, the parametric representation of the solution of each equation in the WKI hierarchy is obtained by making use of the solution of two compatible systems. 0 199.5 American Institute of Physics.
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We consider an integrable generalization of the nonlinear Schrödinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.