10 citations · 59 across the 16 of their papers we have counts for
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A new method for solving completely integrable PDEs
Andrey Melnikov
The inverse scattering theory is a basic tool to solve linear differential equations and some Partial Differential Equations (PDEs). Using this theory the Korteweg-de Vries (KdV),…
Construction of a Sturm-Liouville vessel using Gelfand-Levitan theory. On solution of the Korteweg-de Vries equation in the first quadrant
Andrey Melnikov
Using Gelfand-Levitan theory on a half line, we construct a vessel for the class of potentials, whose spectral functions satisfy a certain regularity assumption. When the singular…
On construction of solutions of evolutionary Non Linear Schrodinger equation
A. Melnikov
In this work we present an application of a theory of vessels to solution of the evolutionary Non Liner Schrodinger (NLS) equation. The classes of functions for which the initial v…
On completely integrable polynomial PDEs arising from Sturm-Liouville differential equation using evolutionary vessels. KdV Hierarchy
Andrey Melnikov
In this work we present a scheme for construction of solutions for evolutionary PDEs of some polynomial types q'_t = P(q,q'_x,...), where P is a polynomial in a finite number of va…
Solution of the Sturm-Liouville and the Korteweg-de-Vries equations with periodic and quasi-periodic parameters using theory of vessels
Andrey Melnikov
We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s^2 y(x) with periodic and quasi-periodic potential q(x) using theory of SL vessels, impl…