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The Direct Method in Soliton Theory
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    Abenda, Simonetta and Grinevich, Petr G. 2018. Rational Degenerations of $${{\mathtt{M}}}$$M-Curves, Totally Positive Grassmannians and KP2-Solitons. Communications in Mathematical Physics,

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    Yao, Yuqin Zhang, Juhui Lin, Runliang Liu, Xiaojun and Huang, Yehui 2018. Bilinear Identities and Hirota’s Bilinear Forms for the (γn, σk)-KP Hierarchy. Journal of Nonlinear Mathematical Physics, Vol. 25, Issue. 2, p. 309.

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    Chang, Xiangke He, Yi Hu, Xingbiao Li, Shihao Tam, Hon-wah and Zhang, Yingnan 2018. Coupled modified KdV equations, skew orthogonal polynomials, convergence acceleration algorithms and Laurent property. Science China Mathematics, Vol. 61, Issue. 6, p. 1063.

    Fang, Chun-Mei Tian, Shou-Fu Feng, Yang and Dai, Jin-Hua 2018. On the integrability and Riemann theta functions periodic wave solutions of the Benjamin Ono equation. Nonlinear Dynamics, Vol. 92, Issue. 2, p. 235.

    Chang, Jen-Hsu 2018. Soliton interaction in the modified Kadomtsev–Petviashvili-(II) equation. Applicable Analysis, p. 1.

    Xie, Xi-Yang and Meng, Gao-Qing 2018. Dark-soliton collisions for a coupled AB system in the geophysical fluids or nonlinear optics. Modern Physics Letters B, Vol. 32, Issue. 04, p. 1850039.

    Li, Li and Yu, Fajun 2018. Optical discrete rogue wave solutions and numerical simulation for a coupled Ablowitz–Ladik equation with variable coefficients. Nonlinear Dynamics, Vol. 91, Issue. 3, p. 1993.

    Zuo, Da-Wei and Jia, Hui-Xian 2018. Multi-soliton solutions of the generalized variable-coefficient Bogoyavlenskii equation. Waves in Random and Complex Media, p. 1.

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    The Direct Method in Soliton Theory
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Book description

The bilinear, or Hirota's direct, method was invented in the early 1970s as an elementary means of constructing soliton solutions that avoided the use of the heavy machinery of the inverse scattering transform and was successfully used to construct the multisoliton solutions of many new equations. In the 1980s the deeper significance of the tools used in this method - Hirota derivatives and the bilinear form - came to be understood as a key ingredient in Sato's theory and the connections with affine Lie algebras. The main part of this book concerns the more modern version of the method in which solutions are expressed in the form of determinants and pfaffians. While maintaining the original philosophy of using relatively simple mathematics, it has, nevertheless, been influenced by the deeper understanding that came out of the work of the Kyoto school. The book will be essential for all those working in soliton theory.


'Overall, the book under review is a concise and essentially self-contained book, written by one of the leading researchers associated with the development of soliton theory … provides an interesting insight into the development of a straight forward method for obtaining exact solutions for wide classes of nonlinear equations.'

Peter Clarkson - University of Kent

' … a nice example of a mathematical writing that can be read at nearly normal pace, which is extremely rare nowadays.'

Source: Zentralblatt MATH

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