Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation
نویسنده
چکیده
We solve the inverse scattering problem for multidimensional vector fields and we use this result to construct the formal solution of the Cauchy problem for the second heavenly equation of Plebanski, a scalar nonlinear partial differential equation in four dimensions relevant in General Relativity, which arises from the commutation of multidimensional Hamiltonian vector fields.
منابع مشابه
Inverse Scattering Problem for Vector Fields and the Heavenly Equation
We solve the inverse scattering problem for multidimensional vector fields and we use this result to construct the formal solution of the Cauchy problem for the second heavenly equation of Plebanski, a scalar nonlinear partial differential equation in four dimensions relevant in General Relativity, which arises from the commutation of multidimensional Hamiltonian vector fields.
متن کامل8 On the solutions of the second heavenly and Pavlov equations
We have recently solved the inverse scattering 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 connected with the commutation of multidimensional vector fields, like the heavenly equation of Plebanski, the dispersionless Kadomtsev Petviashvili (dKP)...
متن کاملInverse scattering problem for the Impulsive Schrodinger equation with a polynomial spectral dependence in the potential
In the present work, under some di¤erentiability conditions on the potential functions , we rst reduce the inverse scattering problem (ISP) for the polynomial pencil of the Scroedinger equation to the corresponding ISP for the generalized matrix Scrödinger equation . Then ISP will be solved in analogy of the Marchenko method. We aim to establish an e¤ective algorithm for uniquely reconstructin...
متن کاملN ov 2 00 8 The dispersionless 2 D Toda equation : dressing , Cauchy problem , longtime behaviour , implicit solutions and wave breaking
We have recently solved the inverse spectral 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, arising as commutation ...
متن کاملN ov 2 00 7 On the solutions of the dKP equation : nonlinear Riemann Hilbert problem , longtime behaviour , implicit solutions and wave breaking
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 particu...
متن کامل