We discuss the amazing interconnections between normal form theory, classical invariant theory and transvectants, modular forms and Rankin–Cohen brackets, representations of the Heisenberg algebra, differential invariants, solitons, Hirota operators, star products and Moyal brackets and coherent states. † Supported in part by NSF Grant DMS 98–03154.

We consider GL(K|M)-invariant integrable supersymmetric spin chains with twisted boundary conditions and elucidate the role of Bäcklund transformations in solving the difference Hirota equation for eigenvalues of their transfer matrices. The nested Bethe ansatz technique is shown to be equivalent to a chain of successive Bäcklund transformations " undressing " the original problem to a trivial ...

We employ the idea of Hirota’s bilinear method, to obtain some new exact soliton solutions for high nonlinear form of (2+1)-dimensional potential Kadomtsev-Petviashvili equation. Multiple singular soliton solutions were obtained by this method. Moreover, multiple singular soliton solutions were also derived. Keywords—Hirota bilinear method, potential KadomtsevPetviashvili equation, Multiple sol...

Extending the gauge-invariance principle for τ functions of the standard bilinear formalism to the supersymmetric case, we define N = 1 supersymmetric Hirota operators. Using them, we bilinearize SUSY KdV equation. The solution for multiple collisions of super-solitons is given.

We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called virtual solitons whose nonlinear interactions produce no phase shifts.

A new Darboux transformation is presented for the Hirota-Satsuma coupled KdV system. It is shown that this Darboux transformation can be constructed by means of two methods: Painlevé analysis and reduction of a binary Darboux transformation. By iteration of the Darboux transformation, the Grammian type solutions are found for the coupled KdV system.

We study the structure of the Goulden-Jackson-Vakil formula that relates Hurwitz numbers to some conjectural “intersection numbers” on a conjectural family of varieties Xg,n of dimension 4g − 3 + n. We give explicit formulas for the properly arranged generating function for these “intersection numbers”, and prove that it satisfies Hirota equations. This generalizes and substantially simplifies ...

© 2012 Hirota and Terada, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Endocytosis of Particle Formulations by Macrophages and Its Application to Clinic...

Rational solutions for the Painlevé IV equation are investigated by Hirota bilinear formalism. It is shown that the solutions in one hierarchy are expressed by 3-reduced Schur functions, and those in another two hierarchies by Casorati determinant of the Hermite polynomials, or by special case of the Schur polynomials.

* This investigation was supported by research grant #AI 03242 from the National Institutes of Health, USPHS. A brief account of it was presented at the annual meeting of the American Society for Microbiology, Washington, D. C., May 6, 1964. t Recipient of Career Development Award #5-K3-GM-6312 from the National Institutes of Health, USPHS. I Supported by training grant TI AI 80 from the Nation...