Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy

Liu, Q. P.; Mañas, Manuel

doi:10.1007/s003329900070

Cited by 24 publications

(10 citation statements)

References 34 publications

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“…In the limit γ → ∞, a KPI lump solution [7] is recovered. Multi-lump generalizations [7,14,15] can be obtained from N × N Jordan type matrices L, R. 7 , β = − 1 2 , γ = 300, a lump moves between two line solitons (left plot). For α = 2, β = − 1 2 , γ = 100, the right line soliton develops a lump, shortly after swallowed by the left one (right plot).…”

Section: Some Solutions Of the Scalar Kp Hierarchymentioning

confidence: 99%

From nonassociativity to solutions of the KP hierarchy

Dimakis

Müller‐Hoissen

2006

Czech J Phys

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A recently observed relation between 'weakly nonassociative' algebras A (for which the associator (A, A 2 , A) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus A ′ of A) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with A ′ a matrix algebra, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.

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Section: Some Solutions Of the Scalar Kp Hierarchymentioning

confidence: 99%

From nonassociativity to solutions of the KP hierarchy

Dimakis

Müller‐Hoissen

2006

Czech J Phys

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“…Bäcklund-Darboux transformations (BDTs) are well-known as a versatile tool in spectral theory as well as for integrable nonlinear equations (see, for instance, [3,7,9,13,14,15,19,25,34,42,43,44,47,49,51,85] and references therein). BDT transforms initial equation or system into another one from the same class and transforms also solutions of the initial equation into solutions of the transformed one.…”

Section: Introductionmentioning

confidence: 99%

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Sakhnovich

2010

Math. Model. Nat. Phenom.

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Abstract. A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N -wave equation are reviewed. New results on explicit construction of the wave functions for radial Dirac equation are obtained.

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“…Vectorial Darboux transformations constitiue one more approach to Darboux transformations, applied mostly in 2 + 1-dimensional case [43,44]. Although this technique needs no analogue of the Darboux matrix but the Darboux transformation is expressed by a Cauchy-like matrix and important role is played by operator identities like (7.12).…”

Section: Transfer Matrix Form Of the Darboux Matrixmentioning

confidence: 99%

Algebraic construction of the Darboux matrix revisited

Cieśliński¹

2009

J. Phys. A: Math. Theor.

105

100

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We present algebraic construction of Darboux matrices for 1+1-dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Bäcklund transformation, based on different λ-dependencies of the Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix form. We derive symmetric N -soliton formulas in the general case. The matrix spectral parameter and dressing actions in loop groups are also discussed. We describe reductions to twisted loop groups, unitary reductions, the matrix Lax pair for the KdV equation and reductions of chiral models (harmonic maps) to SU (n) and to Grassmann spaces. We show that in the KdV case the nilpotent Darboux matrix generates the binary Darboux transformation. The paper is intended as a review of known results (usually presented in a novel context) but some new results are included as well, e.g., general compact formulas for N -soliton surfaces and linear and bilinear constraints on the nonisospectral Lax pair matrices which are preserved by Darboux transformations.

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Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy

Cited by 24 publications

References 34 publications

From nonassociativity to solutions of the KP hierarchy

From nonassociativity to solutions of the KP hierarchy

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Algebraic construction of the Darboux matrix revisited

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