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Painlevé analysis and exact solutions of the Korteweg-de Vries equation with a source. (English) Zbl 1321.35203
The paper deals with the Korteweg-de Vries equation with a source in the form of a polynomial with respect to the solution, with constant coefficients. The integrability of such equations is studied using the Painlevé test applied to the traveling wave solutions. It is shown that the first step of the test is not passed for polynomials of degree \(n=3\) and \(n\geq 5\). As for \(n=2\) and \(n=4\), some explicit solutions corresponding to these cases are given by using the method of \(Q\)-functions (satisfying the Riccati equation).

35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text: DOI
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