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Bilinear transformation method. (English) Zbl 0552.35001

Mathematics in Science and Engineering, Vol. 174. Orlando etc.: Academic Press, Inc. VIII, 223 p. $ 50.00; £35.00 (1984).
The bilinear operator method was introduced by Hirota in 1971 and used to construct the N-soliton solution of the KdV equation. Since that time the method has been applied to a wide variety of nonlinear evolution equations. It has been employed to construct Bäcklund transformations, multi-soliton solutions and conservation laws. Basically, the method involves the reduction of the nonlinear equation to \(<bilinear\) \(form>\) via an appropriate dependent variable transformation. In the case of the KdV equation this is essentially the Cole-Hopf transformation. Multi- soliton solutions of the bilinear equation thus obtained may then be constructed by a perturbation procedure.
The present monograph opens with an account of the bilinearization of the KdV equation: both multi-soliton and periodic wave solutions are thereby generated. The bilinear formulation is then used to derive the auto- Bäcklund transformation for the KdV equation. The generation of an infinite number of conservation laws and links with the inverse scattering procedure are briefly discussed.
The next chapter is concerned with the Benjamin-Ono equation and its bilinearization. N-soliton and N-periodic wave solutions of this nonlinear integro-differential equation are derived. Algebraic and pole expansion methods for the construction of the N-soliton solution are also described. An auto-Bäcklund transformation for the Benjamin-Ono equation is then constructed via the bilinear operator formalism and is used to derive an infinite number of conserved quantities. There follows a discussion of the initial value problem for the Benjamin-Ono equation. The asymptotic behavior of solutions is analysed in the zero dispersion limit. A short discussion of the stability of Benjamin-Ono solitons is also presented.
The following chapter is devoted to a study of the complex interaction of Benjamin-Ono solitons. The motion of poles corresponding to the two- soliton solution is depicted in the complex plane and the nature of the interaction is thereby clarified.
In Chapter 5, nonlinear evolution equations related to the Benjamin-Ono are treated systematically by the bilinearization procedure. The discussion covers higher order KdV, modified KdV and Benjamin-Ono systems along with Joseph’s finite depth equation and its associated hierarchy. Bäcklund transformations and the inverse scattering formulation for the KdV hierarchy are constructed via the bilinear operator method.
The text concludes with a review of assorted topics linked to the Benjamin-Ono equation. Thus, nonlinear evolution equations generated by chains of Bäcklund transformations applied to the Benjamin-Ono and associated equations are described. The derivative nonlinear Schrödinger equation which governs the nonlinear self-modulation of weak periodic wave solutions of the Benjamin-Ono equation is discussed. Finally, systems described by the Benjamin-Ono equation with a small dissipation term are analysed by a multiple time-scale expansion method.
This monograph provides a sound introduction to the subject of Hirota bilinear transformations. Recent work suggests that the bilinear method plays a fundamental role in the theory of integrable nonlinear equations. Accordingly, the appearance of this valuable work is particularly timely.
Reviewer: C.Rogers

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q99 Partial differential equations of mathematical physics and other areas of application
35A30 Geometric theory, characteristics, transformations in context of PDEs