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Darboux and Liouvillian Integrability
2017Darboux and Liouvillian integrability is mainly concerned with algebraic aspects of the integrability of differential systems, which is related to many subjects, such as real and complex analysis, algebraic geometry, differential algebra, differential Galois theory, and so on.
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DARBOUX POLYNOMIALS AND ALGEBRAIC INTEGRABILITY OF THE CHEN SYSTEM
International Journal of Bifurcation and Chaos, 2007In this paper, we characterize all the Darboux polynomials of the Chen system, [Formula: see text], and prove that the system is not algebraic integrability. For proving the results, we use the weight homogeneous polynomials and the method of characteristics.
Tinghua Lü, Xiang Zhang
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The Darboux-type transformations of integrable lattices
Reports on Mathematical Physics, 2001zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On the Darboux integrable hyperbolic equations
Physics Letters A, 1995Abstract We prove that if the sequence of Laplace invariants for the linearization operator of a nonlinear hyperbolic equation is finite then the equation is Darboux integrable.
Sokolov, V. V., Zhiber, A. V.
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Darboux Integrability in More Than Two Dimensions
Applicable Algebra in Engineering, Communication and Computing, 2001The method of Darboux is a classical technique for solving a hyperbolic second-order PDE for one function of two variables, but is also applicable to first-order systems for two functions of two variables [see \textit{R. Bryant, P. Griffiths} and \textit{L. Hsu}, Sel. Math., New Ser. 1, 21-112 (1995; Zbl 0853.58102)].
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Darboux's problem of quadratic integrals
Journal of Physics A: Mathematical and General, 1984Consider a Hamiltonian system with two degrees of freedom, determined by \(H=1/2(p^ 2_ x+p^ 2_ y)+V(x,y)\). The problem of determining all potentials V, for which the system admits a second constant of the motion, quadratic in the momenta, was posed for the first time, but not completely resolved by Darboux.
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Darboux-integrable discrete systems
Theoretical and Mathematical Physics, 2008We extend Laplace’s cascade method to systems of discrete “hyperbolic” equations of the form ui+1,j+1 = f(ui+1,j, ui,j+1 , ui,j), where uij is a member of a sequence of unknown vectors, i, j ∊ ℤ. We introduce the notion of a generalized Laplace invariant and the associated property of the system being “Liouville.” We prove several statements on the ...
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Integrability, Darboux transformation and exact solutions for nonlocal couplings of AKNS equations
Applied Mathematics Letters, 2021Yarong Xia, Hanze Liu
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