3,768 research outputs found

    Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy

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    We consider two-component integrable generalizations of the dispersionless 2DTL hierarchy connected with non-Hamiltonian vector fields, similar to the Manakov-Santini hierarchy generalizing the dKP hierarchy. They form a one-parametric family connected by hodograph type transformations. Generating equations and Lax-Sato equations are introduced, a dressing scheme based on the vector nonlinear Riemann problem is formulated. The simplest two-component generalization of the dispersionless 2DTL equation is derived, its differential reduction analogous to the Dunajski interpolating system is presented. A symmetric two-component generalization of the dispersionless elliptic 2DTL equation is also constructed.Comment: 10 pages, the text of the talk at NEEDS 09. Notations clarified, references adde

    `Interpolating' differential reductions of multidimensional integrable hierarchies

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    We transfer the scheme of constructing differential reductions, developed recently for the case of the Manakov-Santini hierarchy, to the general multidimensional case. We consider in more detail the four-dimensional case, connected with the second heavenly equation and its generalization proposed by Dunajski. We give a characterization of differential reductions in terms of the Lax-Sato equations as well as in the framework of the dressing method based on nonlinear Riemann-Hilbert problem.Comment: Based on the talk at NLPVI, Gallipoli, 15 page

    Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems

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    Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms ΩN−1\Omega_{N-1} of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deformations of (N-1)-dimensional linear subspaces. For the class of solutions which are Laurent polynomials in one variable these systems coincide with N-dimensional integrable systems such as Liouville equation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3), dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4) and others. Gauge invariant part of the forms ΩN−1\Omega_{N-1} provides us with the compact form of the corresponding hierarchies. Dual quasi-linear systems associated with the projectively dual Grassmannians Gr(2,N+1) are defined via the requirement of the closedness of the dual forms ΩN−1⋆\Omega_{N-1}^{\star}. It is shown that at N=3 the self-dual quasi-linear system, which is associated with the harmonic (closed and co-closed) form Ω2\Omega_{2}, coincides with the Maxwell equations for orthogonal electric and magnetic fields.Comment: 26 pages, references adde

    On the heavenly equation hierarchy and its reductions

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    Second heavenly equation hierarchy is considered using the framework of hyper-K\"ahler hierarchy developed by Takasaki. Generating equations for the hierarchy are introduced, they are used to construct generating equations for reduced hierarchies. General NN-reductions, logarithmic reduction and rational reduction for one of the Lax-Sato functions are discussed. It is demonstrated that rational reduction is equivalent to the symmetry constraint.Comment: 13 pages, LaTeX, minor misprints corrected, references adde

    On a class of reductions of Manakov-Santini hierarchy connected with the interpolating system

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    Using Lax-Sato formulation of Manakov-Santini hierarchy, we introduce a class of reductions, such that zero order reduction of this class corresponds to dKP hierarchy, and the first order reduction gives the hierarchy associated with the interpolating system introduced by Dunajski. We present Lax-Sato form of reduced hierarchy for the interpolating system and also for the reduction of arbitrary order. Similar to dKP hierarchy, Lax-Sato equations for LL (Lax fuction) due to the reduction split from Lax-Sato equations for MM (Orlov function), and the reduced hierarchy for arbitrary order of reduction is defined by Lax-Sato equations for LL only. Characterization of the class of reductions in terms of the dressing data is given. We also consider a waterbag reduction of the interpolating system hierarchy, which defines (1+1)-dimensional systems of hydrodynamic type.Comment: 15 pages, revised and extended, characterization of the class of reductions in terms of the dressing data is give
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