Browsing by Subject "Nonlocal reductions"

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    Discrete symmetries and nonlocal reductions
    (Elsevier, 2020) Gürses, Metin; Pekcan, A.; Zheltukhin, K.
    We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
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    Nonlocal hydrodynamic type of equations
    (Elsevier, 2020-03-01) Gürses, Metin; Pekcan, A.; Zheltukhin, K.
    We show that the integrable equations of hydrodynamic type admit nonlocal reductions. We first construct such reductions for a general Lax equation and then give several examples. The reduced nonlocal equations are of hydrodynamic type and integrable. They admit Lax representations and hence possess infinitely many conserved quantities.
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    On Sawada-Kotera and Kaup-Kuperschmidt integrable systems
    (Institute of Physics Publishing Ltd., 2024-11-15) Gürses, Metin; Pekcan, A.
    To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods that are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the 2-extension of integrable such scalar equations. For illustration, we give Korteweg–de Vries, Kaup-Kupershmidt, and SawadaKotera equations as examples. By the use of such an extension of integrable scalar equations, we obtain some new integrable systems with recursion operators. We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.
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    The method of Mn-extension: the KdV equation
    (Elsevier BV, 2025-01-07) Gürses, Metin; Pekcan, Aslı
    In this work we generalize $M_2$-extension that has been introduced recently. For illustration we use the KdV equation. We present five different M3-extensions of the KdV equation and their recursion operators. We give a compact form of Mn-extension of the KdV equation and recursion operator of the coupled KdV system. The method of Mn-extension can be applied to any integrable scalar equation to obtain integrable multi-field system of equations. We also present unshifted and shifted nonlocal reductions of an example of M3-extension of KdV.