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The Direct Method in Soliton Theory
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  • Cited by 895
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    This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Liu, Q P Hu, Xing-Biao and Zhang, Meng-Xia 2005. Supersymmetric modified Korteweg–de Vries equation: bilinear approach. Nonlinearity, Vol. 18, Issue. 4, p. 1597.

    Kajiwara, Kenji and Mukaihira, Atsushi 2005. Soliton solutions for the non-autonomous discrete-time Toda lattice equation. Journal of Physics A: Mathematical and General, Vol. 38, Issue. 28, p. 6363.

    Liu, Q P and Hu, Xing-Biao 2005. Bilinearization ofN= 1 supersymmetric Korteweg–de Vries equation revisited. Journal of Physics A: Mathematical and General, Vol. 38, Issue. 28, p. 6371.

    Gehlen, G von Pakuliak, S and Sergeev, S 2005. The Bazhanov–Stroganov model from 3D approach. Journal of Physics A: Mathematical and General, Vol. 38, Issue. 33, p. 7269.

    Zabusky, Norman J. 2005. Fermi–Pasta–Ulam, solitons and the fabric of nonlinear and computational science: History, synergetics, and visiometrics. Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 15, Issue. 1, p. 015102.

    Gegenhasi Zhao, Jun-Xiao Hu, Xing-Biao and Tam, Hon-Wah 2005. Pfaffianization of the discrete three-dimensional three wave interaction equation. Linear Algebra and its Applications, Vol. 407, Issue. , p. 277.

    Hu, Xing-Biao Li, Chun-Xia Nimmo, Jonathan J C and Yu, Guo-Fu 2005. An integrable symmetric (2+1)-dimensional Lotka–Volterra equation and a family of its solutions. Journal of Physics A: Mathematical and General, Vol. 38, Issue. 1, p. 195.

    Parker, Allen 2005. On the Camassa–Holm equation and a direct method of solution. III. N -soliton solutions . Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 461, Issue. 2064, p. 3893.

    Бухштабер, Виктор Матвеевич Buchstaber, Victor Matveevich Бухштабер, Виктор Матвеевич Buchstaber, Victor Matveevich Кричевер, Игорь Моисеевич Krichever, Igor Moiseevich Кричевер, Игорь Моисеевич and Krichever, Igor Moiseevich 2006. Интегрируемые уравнения, теоремы сложения и проблема Римана - Шоттки. Успехи математических наук, Vol. 61, Issue. 1, p. 25.

    Zhang, Da-jun Bi, Jin-bo and Hao, Hong-hai 2006. A modified KdV equation with self-consistent sources in non-uniform media and soliton dynamics. Journal of Physics A: Mathematical and General, Vol. 39, Issue. 47, p. 14627.

    Dimakis, Aristophanes and Müller-Hoissen, Folkert 2006. Functional representations of integrable hierarchies. Journal of Physics A: Mathematical and General, Vol. 39, Issue. 29, p. 9169.

    Gegenhasi Hu, Xing-Biao and Levi, Decio 2006. On a discrete Davey–Stewartson system. Inverse Problems, Vol. 22, Issue. 5, p. 1677.

    Aktosun, Tuncay and Mee, Cornelis van der 2006. Explicit solutions to the Korteweg–de Vries equation on the half line. Inverse Problems, Vol. 22, Issue. 6, p. 2165.

    Gegenhasi and Hu, Xing-Biao 2006. On an integrable differential-difference equation with a source. Journal of Nonlinear Mathematical Physics, Vol. 13, Issue. 2, p. 183.

    Ogawa, Minoru 2006. Integrability of One-Degree-of-Freedom Symplectic Maps with Polar Singularities. Journal of the Physical Society of Japan, Vol. 75, Issue. 6, p. 064006.

    WU, RANCHAO and SUN, JIANHUA 2006. A BRIEF SURVEY ON CONSTRUCTING HOMOCLINIC STRUCTURES OF SOLITON EQUATIONS. International Journal of Bifurcation and Chaos, Vol. 16, Issue. 10, p. 2799.

    Da-Jun, Zhang 2006. Grammian Solutions to a Non-Isospectral Kadomtsev–Petviashvili Equation. Chinese Physics Letters, Vol. 23, Issue. 9, p. 2349.

    Lü, Shu-Qiang Hu, Xing-Biao and P. Liu, Q. 2006. A Supersymmetric Ito's Equation and its Soliton Solutions. Journal of the Physical Society of Japan, Vol. 75, Issue. 6, p. 064004.

    Jin-Bing, Chen and Xian-Guo, Geng 2006. On the linearization of the coupled Harry–Dym soliton hierarchy. Chinese Physics, Vol. 15, Issue. 7, p. 1407.

    Nakamura, Akira 2006. Derivations of identities by symbolic computation. Japan Journal of Industrial and Applied Mathematics, Vol. 23, Issue. 3, p. 315.


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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