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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.

    Adem, Abdullahi Rashid Yildirim, Yakup and Yaşar, Emrullah 2019. Soliton solutions to the non-local Boussinesq equation by multiple exp-function scheme and extended Kudryashov’s approach. Pramana, Vol. 92, Issue. 2,

    Liu, Wenjun Zhang, Yujia Triki, Houria Mirzazadeh, Mohammad Ekici, Mehmet Zhou, Qin Biswas, Anjan and Belic, Milivoj 2019. Interaction properties of solitonics in inhomogeneous optical fibers. Nonlinear Dynamics, Vol. 95, Issue. 1, p. 557.

    Bekki, Naoaki Ishii, Keisho and Endo, Kazushige 2019. Soliton for Nonlinear Rayleigh Surface Waves on Homogeneous Isotropic Materials. Journal of the Physical Society of Japan, Vol. 88, Issue. 1, p. 014001.

    Liu, Yaobin Qian, Chao Mihalache, Dumitru and He, Jingsong 2019. Rogue waves and hybrid solutions of the Davey–Stewartson I equation. Nonlinear Dynamics, Vol. 95, Issue. 1, p. 839.

    Paul, G C Rashedunnabi, A H M and Haque, M D 2019. Testing efficiency of the generalised $$\left( {{{G}'}/G} \right) $$G′/G-expansion method for solving nonlinear evolution equations. Pramana, Vol. 92, Issue. 2,

    Geng, Xianguo Zhai, Yunyun Xue, Bo and Wei, Jiao 2019. A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation. Journal of Nonlinear Mathematical Physics, Vol. 26, Issue. 1, p. 1.

    Stalin, S. Senthilvelan, M. and Lakshmanan, M. 2019. Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions. Nonlinear Dynamics, Vol. 95, Issue. 1, p. 343.

    Mao, Jin-Jin Tian, Shou-Fu Zou, Li Zhang, Tian-Tian and Yan, Xing-Jie 2019. Bilinear formalism, lump solution, lumpoff and instanton/rogue wave solution of a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation. Nonlinear Dynamics,

    Hafez, M. G. 2019. Face to Face Collisions of Ion Acoustic Multi-Solitons and Phase Shifts in a Dense Plasma. Brazilian Journal of Physics,

    Trejo-Garcia, David Gonzalez-Hernandez, Diana López-Aguayo, Daniel and Lopez-Aguayo, Servando 2018. Stable Hermite-Gaussian solitons in optical lattices. Journal of Optics, Vol. 20, Issue. 12, p. 125501.

    Wang, Bao Chang, Xiang-Ke Hu, Xing-Biao and Li, Shi-Hao 2018. On moving frames and Toda lattices of BKP and CKP types. Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue. 32, p. 324002.

    Liu, Wei and Li, Xiliang 2018. General soliton solutions to a $$\varvec{(2+1)}$$(2+1)-dimensional nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions. Nonlinear Dynamics, Vol. 93, Issue. 2, p. 721.

    Chen, Xiao-Min Hu, Xing-Biao and Müller-Hoissen, Folkert 2018. Non-isospectral extension of the Volterra lattice hierarchy, and Hankel determinants. Nonlinearity, Vol. 31, Issue. 9, p. 4393.

    Wazwaz, Abdul-Majid 2018. A variety of negative-order integrable KdV equations of higher orders. Waves in Random and Complex Media, p. 1.

    Abenda, Simonetta and Grinevich, Petr G. 2018. Rational Degenerations of $${{\mathtt{M}}}$$M-Curves, Totally Positive Grassmannians and KP2-Solitons. Communications in Mathematical Physics, Vol. 361, Issue. 3, p. 1029.

    Wu, Pinxia Zhang, Yufeng Muhammad, Iqbal and Yin, Qiqi 2018. Lump, periodic lump and interaction lump stripe solutions to the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation. Modern Physics Letters B, Vol. 32, Issue. 07, p. 1850106.

    Mao, Jin-Jin Tian, Shou-Fu and Zhang, Tian-Tian 2018. Rogue waves, homoclinic breather waves and soliton waves for a (3 + 1)-dimensional non-integrable KdV-type equation. International Journal of Numerical Methods for Heat & Fluid Flow,

    Yu, Jian-Ping Ma, Wen-Xiu Sun, Yong-Li and Khalique, Chaudry Masood 2018. N-fold Darboux transformation and conservation laws of the modified Volterra lattice. Modern Physics Letters B, Vol. 32, Issue. 33, p. 1850409.

    Liu, Yaqing and Wen, Xiaoyong 2018. Fission and fusion interaction phenomena of mixed lump kink solutions for a generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation. Modern Physics Letters B, Vol. 32, Issue. 15, p. 1850161.

    Guo, Ding and Tian, Shou-Fu 2018. Stability analysis, soliton waves, rogue waves and interaction phenomena for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Modern Physics Letters B, Vol. 32, Issue. 28, p. 1850345.


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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