1,906 research outputs found

    Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation

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    We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the "energy" parameter E E . We show that as ∣E∣→∞ |E| \to \infty , NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when ∣E∣ | E | is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied

    A new extended matrix KP hierarchy and its solutions

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    With the square eigenfunctions symmetry constraint, we introduce a new extended matrix KP hierarchy and its Lax representation from the matrix KP hierarchy by adding a new Ï„B\tau_B flow. The extended KP hierarchy contains two time series tA{t_A} and Ï„B{\tau_B} and eigenfunctions and adjoint eigenfunctions as components. The extended matrix KP hierarchy and its tAt_A-reduction and Ï„B\tau_B reduction include two types of matrix KP hierarchy with self-consistent sources and two types of (1+1)-dimensional reduced matrix KP hierarchy with self-consistent sources. In particular, the first type and second type of the 2+1 AKNS equation and the Davey-Stewartson equation with self-consistent sources are deduced from the extended matrix KP hierarchy. The generalized dressing approach for solving the extended matrix KP hierarchy is proposed and some solutions are presented. The soliton solutions of two types of 2+1-dimensional AKNS equation with self-consistent sources and two types of Davey-Stewartson equation with self-consistent sources are studied.Comment: 17 page

    Generalized Darboux transformations for the KP equation with self-consistent sources

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    The KP equation with self-consistent sources (KPESCS) is treated in the framework of the constrained KP equation. This offers a natural way to obtain the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation with arbitrary functions in time tt for the KPESCS which, in contrast with the binary Darboux transformation of the KP equation, provides a non-auto-B\"{a}cklund transformation between two KPESCSs with different degrees. The formula for N-times repeated generalized binary Darboux transformation is proposed and enables us to find the N-soliton solution and lump solution as well as some other solutions of the KPESCS.Comment: 20 pages, no figure
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