The Direct Method in Soliton Theory
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.
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Structure of soliton equations
antisymmetric matrix arbitrary functions Backlund transformation formulae bi-logarithmic transformation bilinear form boundary conditions coefficient conserved quantities consider coupled KP equation D-operator defined dependent variable transformation derivatives differential operators direct method Dx(D equa equivalent exact solution exchange formula expressed in terms figj given gives grammian Hence Hirota introduce Jacobi identity KdV equation KdV-type bilinear equation KP equation Laplace expansion lattice equation Lax pair left-hand side linear differential equations Liouville equation matrix Maya diagram expression Mikio Sato molecule equation N+2 N+3 nonlinear differential equations nonlinear partial differential notation nth-order pfaffian obtain parameter partial differential equation perturbation method pfaffian entries pfaffian identities Pliicker relation polynomial prove Remark respect rewritten right-hand side satisfy the linear Sato Section sine-Gordon equation solitary wave solitary wave solution soliton equations soliton solutions Substituting tion two-dimensional Toda lattice two-dimensional Toda molecule two-soliton solution uxxx V-soliton written