Topic
Riccati equation
About: Riccati equation is a research topic. Over the lifetime, 10428 publications have been published within this topic receiving 210015 citations. The topic is also known as: Riccati's differential equation.
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TL;DR: In this article, the authors deduce a method for building these solutions by determining only a finite number of coefficients, which is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation.
Abstract: Many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective (hence linearizable) Riccati system. From that property, the authors deduce a method for building these solutions by determining only a finite number of coefficients. This method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation. They handle several examples. For the Henon-Heiles Hamiltonian system, they obtain several exact solutions; one of them defines a new solitary wave solution for a coupled system of Boussinesq and nonlinear Schrodinger equations. For a third order dispersive equation with two monomial nonlinearities, they isolate all cases where the general solution is single valued.
270 citations
TL;DR: In this article, the special quasiperiodic solution of the (2+1)-dimensional Kadometsev-Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the confocal involutive hierarchy associated with the nonlinearized Zakharov-Shabat eigenvalue problem.
Abstract: The special quasiperiodic solution of the (2+1)-dimensional Kadometsev–Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the well-known confocal involutive hierarchy associated with the nonlinearized Zakharov–Shabat eigenvalue problem. The explicit theta function solution of the Kadometsev–Petviashvili equation is obtained with the help of this separation technique. A generating function approach is introduced to prove the involutivity and the functional independence of the conserved integrals which are essential for the Liouville integrability.
267 citations
TL;DR: In this paper, the Riccati equation approach is used to obtain the memoryless linear state feedback control of uncertain dynamic delay systems, where uncertainties are time varying and within a given compact set.
Abstract: The authors present a procedure for obtaining the memoryless linear state feedback control of uncertain dynamic delay systems. The uncertainties are time varying and within a given compact set. This method is an extension of the Riccati equation approach proposed by I.R. Petersen and C.V. Hollot (1986). The extension is straightforward. Also the uncertainties do not need to satisfy the matching conditions. >
260 citations
TL;DR: In this article, the Hyers-Ulam stability of the quadratic functional equation (1) on a restricted domain was investigated, and the result was applied to the study of an asymptotic behavior of that equation.
258 citations