Solving the dispersionless Dym equation by hodograph transformations

Chen, Yu-Tung; Tu, Ming-Hsien

doi:10.1088/0305-4470/36/38/305

Cited by 11 publications

(14 citation statements)

References 15 publications

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“…In recent years, research in the dispersionless hierarchies has become quite active (see, for example, [1]- [11] and references therein). Dispersionless hierarchies arise as the quasiclassical limit of the original dispersionfull hierarchies [2].…”

Section: Introductionmentioning

confidence: 99%

Integrable dispersionless KdV hierarchy with sources

Yang¹,

Xiao²,

Zeng³

2006

J. Phys. A: Math. Gen.

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Section: Introductionmentioning

confidence: 99%

Integrable dispersionless KdV hierarchy with sources

Yang¹,

Xiao²,

Zeng³

2006

J. Phys. A: Math. Gen.

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“…which together with (13) yields Lax equations (5) and (11) (5), (11), and canonical relation (12). We call (f, g,f,ĝ) the twistor data of the associated solution.…”

Section: Dressing Operator Approachmentioning

confidence: 97%

Additional symmetries and Bäcklund transformations for the dispersionless Harry Dym hierarchy

Chen

Chang

2007

Theor Math Phys

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We give a dispersionless Toda-like extension to the dispersionless Harry Dym (dDym) hierarchy. The extended dDym (EdDym) hierarchy has a dressing formulation, and its underlying solution structure can be investigated through the twistor construction. We show that additional symmetries of the solution space generate Bäcklund transformations of the EdDym hierarchy. We present some examples of constructing new solutions of the (2+1)-dimensional dDym and EdDym equations via Bäcklund transformations.

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“…In 1999, the dHD hierarchy was defined by a given classical r-matrix on a Poisson algebra [16]. Some progresses have been made for the dHD hierarchy such as hodograph solutions [6], Miura map and bi-Hamiltonian formulation [7], additional symmetries and Bäcklund transformation [5] and Hydrodynamic reduction [4]. We notice that the Miura map between the dmKP hierarchy and the dHD hierarchy is triggered by the "eigenfunctions "of the dmKP hierarchy and depends on a transformation of independent variables.…”

Section: Introductionmentioning

confidence: 99%

New extension of dispersionless Harry Dym hierarchy

Wu¹,

Liu²,

Zeng³

2021

JNMP

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The symmetry constraint for dispersionless Harry Dym (dHD) hierarchy is derived for the first time by taking dispersionless limit of that for 2+1 dimensional Harry Dym hierarchy. Then, the dHD is extended by means of the symmetry constraint which we derived. From the zero-curvature equation of the new extended dHD hierarchy, two types of dHD equations with self-consistent sources (dHDESCS) together with their associated conservation equations are obtained. Moreover, the hodograph solutions to the first type of dHDESCS are given. Finally, Bäcklund transformation between the extended dispersionless mKP hierarchy and extended dHD Hierarchy are also constructed.

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Solving the dispersionless Dym equation by hodograph transformations

Cited by 11 publications

References 15 publications

Integrable dispersionless KdV hierarchy with sources

Integrable dispersionless KdV hierarchy with sources

Additional symmetries and Bäcklund transformations for the dispersionless Harry Dym hierarchy

New extension of dispersionless Harry Dym hierarchy

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