delta -dressing and exact solutions for the (2+1)-dimensional Harry Dym equation

Dubrovsky, V. G.; Konopelchenko, B. G.

doi:10.1088/0305-4470/27/13/035

Cited by 88 publications

(37 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is easy to see that these properties are shared by the whole dKP hierarchy, associated with time operatorsL n involving higher powers of λ. An example of integrable system not associated with the commutation of vector fields, but exhibiting an inverse problem in which the solution is implicit, is the Harry Dym equation ( [29,30]), connected with the KdV equation by a hodograph transformation.…”

Section: Nonlinear Rh Inverse Problemmentioning

confidence: 99%

On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

Manakov¹,

Santini²

2008

J. Phys. A: Math. Theor.

128

View full text Add to dashboard Cite

We have recently solved the inverse scattering problem for oneparameter 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. We showed, in particular, that the associated inverse problems can be expressed in terms of nonlinear Riemann -Hilbert problems on the real axis. In this paper we make use of the nonlinear Riemann -Hilbert problem of dKP, i) to construct the longtime behaviour of the solutions of its Cauchy problem; ii) to characterize a class of implicit solutions; iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dKP, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking.

show abstract

Section: Nonlinear Rh Inverse Problemmentioning

confidence: 99%

On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

Manakov¹,

Santini²

2008

J. Phys. A: Math. Theor.

128

View full text Add to dashboard Cite

show abstract

“…As a concrete example, we consider the celebrated (2+1)-dimensional Broer-Kaup-Kupershmidt(BKK) system [6] …”

Section: New Variable Separation Solutions Of the (2+1)-dimensional Bmentioning

confidence: 99%

New Variable Separation Excitations of a (2+1)-Dimensional Broer-Kaup-Kupershmidt System Obtained by an Extended Mapping Approach

Zheng

Fang

Chen

2004

Zeitschrift Für Naturforschung A

View full text Add to dashboard Cite

show abstract

“…Remarks: When m → 1, the Jacobin doubly periodic solutions (10) and (11) all degenerate to (5), and the Jacobin doubly periodic solution (12) degenerates to (8).…”

Section: (12)mentioning

confidence: 99%

Periodic Wave Solutions of a Generalized KdV-mKdV Equation with Higher-Order Nonlinear Terms

2010

Zeitschrift Für Naturforschung A

View full text Add to dashboard Cite

The Jacobin doubly periodic wave solution, the Weierstrass elliptic function solution, the bell-type solitary wave solution, the kink-type solitary wave solution, the algebraic solitary wave solution, and the triangular solution of a generalized Korteweg-de Vries-modified Korteweg-de Vries equation (GKdV-mKdV) with higher-order nonlinear terms are obtained by a generalized subsidiary ordinary differential equation method (Gsub-ODE method for short). The key ideas of the Gsub-ODE method are that the periodic wave solutions of a complicated nonlinear wave equation can be constructed by means of the solutions of some simple and solvable ODE which are called Gsub-ODE with higherorder nonlinear terms.

show abstract

delta -dressing and exact solutions for the (2+1)-dimensional Harry Dym equation

Cited by 88 publications

References 16 publications

On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

New Variable Separation Excitations of a (2+1)-Dimensional Broer-Kaup-Kupershmidt System Obtained by an Extended Mapping Approach

Periodic Wave Solutions of a Generalized KdV-mKdV Equation with Higher-Order Nonlinear Terms

Contact Info

Product

Resources

About