A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

Manakov, S. V.; Santini, P. M.

doi:10.1007/s11232-007-0084-2

Cited by 63 publications

(106 citation statements)

References 19 publications

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“…The Cauchy problem for the v = 0 reduction of (1), the dKP equation, was also presented in [24], while the Cauchy problem for the u = 0 reduction of (1), an integrable system introduced in [16], was given in [25]. This IST and its associated nonlinear Riemann-Hilbert (RH) dressing turn out to be efficient tools to study several properties of the solution space, such as (i) the characterization of a distinguished class of spectral data for which the associated nonlinear RH problem is linearized and solved, corresponding to a class of implicit solutions of the PDE (as it was done for the dKP equation in [26] and for the Dunajski generalization [27] of the second heavenly equation in [28]); (ii) the construction of the longtime behaviour of the solutions of the Cauchy problem [26]; (iii) the possibility to establish whether or not the lack of dispersive terms in the nonlinear PDE causes the breaking of localized initial profiles and, if yes, to investigate in a surprisingly explicit way the analytic aspects of such a wave breaking (as it was recently done for the (2+1)-dimensional dKP model in [26]).…”

Section: Introductionmentioning

confidence: 99%

Large Remanent Polarization in Sm-Substituted BiFeO₃ Thin Film Formed by Chemical Solution Deposition

Zhong

Sugiyama

Ishiwara

2010

Jpn. J. Appl. Phys.

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We have recently solved the inverse spectral problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev-Petviashvili (dKP) equation, arising as commutation of vector fields. In this paper, we make use of the above theory (i) to construct the nonlinear Riemann-Hilbert dressing for the so-called twodimensional dispersionless Toda equation (exp(ϕ)) tt = ϕ ζ 1 ζ 2 , elucidating the spectral mechanism responsible for wave breaking; (ii) we present the formal solution of the Cauchy problem for the wave form of it: (exp(ϕ)) tt = ϕ xx +ϕ yy ; (iii) we obtain the longtime behaviour of the solutions of such a Cauchy problem, showing that it is essentially described by the longtime breaking formulae of the dKP solutions, confirming the expected universal character of the dKP equation as prototype model in the description of the gradient catastrophe of two-dimensional waves; (iv) we finally characterize a class of spectral data allowing one to linearize the RH problem, corresponding to a class of implicit solutions of the PDE.

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Section: Introductionmentioning

confidence: 99%

Large Remanent Polarization in Sm-Substituted BiFeO₃ Thin Film Formed by Chemical Solution Deposition

Zhong

Sugiyama

Ishiwara

2010

Jpn. J. Appl. Phys.

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“…A simplest case of general hierarchy (4) with N = 1 is connected with the system recently introduced by Manakov and Santini [3]. For this hierarchy we have two series…”

Section: Manakov-santini Hierarchymentioning

confidence: 99%

“…The problem (19) is connected with a class of integrable equations, which can be represented as a commutation relation for vector fields containing a derivative on the spectral variable. We give a sketch of the dressing scheme corresponding to the general (N+2)-dimensional integrable hierarchy (3), (4). First, we chose γ as a unit circle.…”

Section: Dressing Schemementioning

confidence: 99%

A class of multidimensional integrable hierarchies and their reductions

Bogdanov

2009

Theor Math Phys

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A class of multidimensional integrable hierarchies connected with commutation of general (unreduced) (N+1)-dimensional vector fields containing derivative over spectral variable is considered. They are represented in the form of generating equation, as well as in the Lax-Sato form. A dressing scheme based on nonlinear vector Riemann problem is presented for this class. The hierarchies connected with ManakovSantini equation and Dunajski system are considered as illustrative examples.

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“…As we deal with the restricted Lax operators (29) and (30), the images of the Poisson tensors (24) do not have to span proper subspace of w, i.e., the images of (24) do not have to lie in the space spanned by l t ν q,k .…”

Section: Finite-field Lax Operatorsmentioning

confidence: 99%

Central extensions of cotangent universal hierarchy: (2+1)-dimensional bi-Hamiltonian systems

Sergyeyev

Szablikowski

2008

Physics Letters A

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614]. Then we construct a (2 + 1)-dimensional double central extension of the cotangent universal hierarchy and show that this extension is bi-Hamiltonian. This yields, as a byproduct, the central extension of the original universal hierarchy.The so-called universal hierarchy [1][2][3][4][5] is now a subject of intense research. This hierarchy is, in its original form, an infinite hierarchy of coupled dispersionless (1 + 1)-dimensional integrable systems. The universal hierarchy can be thought of as a model equation of soliton theory from which one can obtain many well-known and new soliton equations upon imposing suitable differential constraints. It is natural to ask whether we can construct a (2 + 1)-dimensional extension of this hierarchy which could yield, upon imposing suitable constraints, (2 + 1)-dimensional integrable systems.However, there appears to be no straightforward way to include the universal hierarchy into the standard Lie-algebraic R-matrix scheme in spirit of [6][7][8][9][10][11] which would enable one to construct the (2 + 1)-dimensional extension of the hierarchy in question and prove integrability thereof using the central extension approach.To circumvent this difficulty, we first lift, in spirit of [12][13][14], the universal hierarchy (3) to the cotangent universal hierarchy, see Eq. (14) below. This cotangent universal hierarchy is already amenable to the approach of [7-9], and integrability of central extensions of (14) follows from the general theory presented there.In fact, we go even further than that. Motivated by [13,14], we perform the double central extension of (14) using two cocycles rather than one. Commutativity of so constructed flows and integrability of the resulting hierarchy (27) still follows from the general results of the R-matrix scheme of [7][8][9]. The first of the cocycles in question, (16), introduces the second space variable y, thus yielding a (2 + 1)-dimensional rather than (1 + 1)-dimensional hierarchy, while the second cocycle, (17), introduces dispersion. A quite different way of introducing dispersion that leads to infinite-order differential equations can be found in [15].What is more, (27) contains a subhierarchy (28) which is precisely the central extension of the original universal hierarchy (3), and integrability of (28) follows from that of (27).The hierarchy (27) is bi-Hamiltonian, and, quite interestingly, the Poisson brackets (22) of (27) are not of the operand type, see e.g. [10, [16][17][18] and references therein for the latter, but rather of the same type that occurs in (1 + 1) dimensions and also for (2 + 1)-dimensional hydrodynamic-type systems [11], as is explicitly revealed by Examples 1 and 2. The corresponding recursion operators, being the ratios of

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A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

Cited by 63 publications

References 19 publications

Large Remanent Polarization in Sm-Substituted BiFeO₃ Thin Film Formed by Chemical Solution Deposition

Large Remanent Polarization in Sm-Substituted BiFeO₃ Thin Film Formed by Chemical Solution Deposition

A class of multidimensional integrable hierarchies and their reductions

Central extensions of cotangent universal hierarchy: (2+1)-dimensional bi-Hamiltonian systems

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A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

Cited by 63 publications

References 19 publications

Large Remanent Polarization in Sm-Substituted BiFeO3 Thin Film Formed by Chemical Solution Deposition

Large Remanent Polarization in Sm-Substituted BiFeO3 Thin Film Formed by Chemical Solution Deposition

A class of multidimensional integrable hierarchies and their reductions

Central extensions of cotangent universal hierarchy: (2+1)-dimensional bi-Hamiltonian systems

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Product

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Large Remanent Polarization in Sm-Substituted BiFeO₃ Thin Film Formed by Chemical Solution Deposition

Large Remanent Polarization in Sm-Substituted BiFeO₃ Thin Film Formed by Chemical Solution Deposition