Bilinear Transformation Method Research Articles

Transformation formulae of the solutions of the two-dimensional Toda lattice

Two transformation formulae of the solutions of the two-dimensional Toda lattice are presented. The first formula, which can be obtained by the Zakharov-Shabat dressing method, is proved by applying the Hirota bilinear transformation method (1988) and gives the basis for the second new formula.

Two-Dimensional Multisoliton Solutions: Periodic Soliton Solutions to the Kadomtsev-Petviashvili Equation with Positive Dispersion

N -periodic soliton solutions that consist of N sequences of the algebraic localized solitons in arbitrary directions are presented to the Kadomtsev-Petviashvili equation with positive dispersion using the bilinear transformation method. The existence conditions and some properties of them are also given.

New integrable nonlinear integrodifferential equations and related solvable finite-dimensional dynamical systems

A new integrable nonlinear integrodifferential equation (NIDE) is proposed. This equation may be interpreted as a model equation for deep-water waves. The N-periodic and N-soliton solutions for the equation are constructed by means of the bilinear transformation method. These solutions have the same structure as that for the Benjamin–Ono equation which describes internal waves in stratified fluids of great depth. Furthermore, it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite-dimensional dynamical systems described by first-order nonlinear ordinary differential equations. The discussion is also made on a more general NIDE that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth.

Bilinear Transformation Method (Yoshimasa Matsuno)

Previous article Next article Bilinear Transformation Method (Yoshimasa Matsuno)Mark J. AblowitzMark J. Ablowitzhttps://doi.org/10.1137/1028013PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Bilinear Transformation Method (Yoshimasa Matsuno)." SIAM Review, 28(1), pp. 91–92 Previous article Next article FiguresRelatedReferencesCited byDetails Improved Matched Pole Zero Mapping method for design of discrete controllers Cross Ref Volume 28, Issue 1| 1986SIAM Review History Published online:18 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1028013Article page range:pp. 91-92ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Bilinearization of Nonlinear Evolution Equations. III. Bäcklund Transformations of Higher-Order Korteweg-de Vries Equations

The Backlund transformations of the higher-order Korteweg-de Vries (KdV) equations are derived on the basis of the bilinear transformation method. It is shown that these transformations are closely related to the quantities obtained from the functional derivative of conserved quantities of the modified KdV equation. The inverse scattering transforms of the higher-order KdV equations are also presented.

Bilinearization of Nonlinear Evolution Equations

A new method for transforming the Lax hierarchy of nonlinear evolution equations u t =-(∂/∂ x )(δ I n /δ u ) into bilinear forms is developed on the basis of the bilinear transformation method, and the structure of these equations is clarified, where I n is the n -th conserved quantity of some nonlinear evolution equation and δ/δ u denotes the functional derivative. The explicit calculations are carried out in the case of the higher-order Korteweg-de Vries equations.

Solutions of the Higher Order Benjamin-Ono Equation

Exact N-soliton and N-periodic wave solutions of the higher order Benjamin-Ono (B-O) equation u t =-(∂/∂ x )(δ I n /δ u ) are presented for n =5 by using bilinear transformation method, where I n is the n -th conserved quantity of the B-O equation and δ/δ u denotes the functional derivative. For n ≧6, a method for transforming the higher order equations into the bilinear forms is proposed and explicit N-soliton solutions are speculated.

N-Soliton Solution of the Higher Order Wave Equation for a Fluid of Finite Depth

Exact N -soliton solution of the higher order wave equation for a stratified fluid layer of finite depth is obtained by employing Hirota's bilinear transformation method. The solution reduces to the N -soliton solution of the higher order Korteweg-de Vries equation in the shallow water limit and to that of the higher order Benjamin-One equation in the deep water limit. The asymptotic form of the solution for large time is derived.

N-Periodic Wave and N-Soliton Solutions of the Modified Benjamin-Ono Equation

Exact N -periodic wave and N -soliton solutions of the modified Benjamin-Ono equation that is generated from the Bäcklund transform of the Benjamin-Ono equation is obtained by the bilinear transformation method.

N-Soliton and N-Periodic Wave Solutions of the Higher Order Benjamin-Ono Equation

Exact N -soliton and N -periodic wave solutions of the higher order Benjamin-Ono equation are presented by applying Hirota's bilinear transformation method.

Digital notch filters

The letter deals with the special class of digital filters that possess a prescribed notch frequency. The present techniques for synthetising such filters rest on the bilinear-transformation method, where the designer is required to synthetise a continuous s plane notch filter, prewarp it, and then apply the transformation. It is shown here that notch filters can be synthetised directly into the z−1 plane.