About linear superpositions of special exact solutions of Nizhnik-Veselov-Novikov equation

Physics and Mathematics of Nonlinear Phenomena 2013

Dubrovsky, V.G.^1 ; Topovsky, A.V.^1

Novosibirsk State Technical University, Karl Marx prosp. 20, Novosibirsk 630092, Russia^1

关键词: Arbitrary subsets;    Asymptotic values;    Exact special solutions;    Linear superpositions;    Nonlinear superposition;    Novikov equations;    Periodic solution;    Special solutions;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/482/1/012011/pdf
DOI  :  10.1088/1742-6596/482/1/012011

来源: IOP

【 摘 要 】

General scheme for calculations via Zakharov and Manakov -dressing method of exact solutions, nonstationary and stationary, of Nizhnik-Veselov-Novikov (NVN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u(n), n = 1,..., N is presented. Simple nonlinear superpositions are given up to a constant by the sums of solutions u(n)and calculated by -dressing of the first auxiliary linear problem with nonzero asymptotic values of potential at infinity. It is remarkable that in the zero limit of asymptotic values of potential simple nonlinear superpositions convert in to linear ones in the form of the sums of special solutions u(n). It is shown that the sums u u(kl)+ ...+ u(km), 1 ≤ k12m≤ N of arbitrary subsets of these N solutions are also exact solutions of NVN equation. The obtained results are illustrated in detail by hyperbolic version of NVN equation, i. e. by NVN-II equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions.