Spectral transform for a two spatial dimension extension of the dispersive long wave equation

A two-dimensional nonlinear evolution equation is solved in the inverse spectral transform scheme. It coincides, when reduced to one spatial dimension, with the dispersive long wave equation. The Backlund transformation, soliton solution and superposition formula are obtained. The spectral transform is explicitly defined and the corresponding linear evolution of the spectral data is given. The inverse spectral problem is formulated as a non-local Riemann-Hilbert boundary value problem and solved.

[1]  D. J. Kaup,et al.  A Higher-Order Water-Wave Equation and the Method for Solving It , 1975 .

[2]  Ljf Lambert Broer.  Approximate equations for long water waves , 1975 .

[3]  The Dirac inverse spectral transform: Kinks and boomerons , 1980 .

[4]  S. Manakov,et al.  The inverse scattering transform for the time-dependent Schrodinger equation and Kadomtsev-Petviashvili equation , 1981 .

[5]  Athanassios S. Fokas,et al.  ON THE INVERSE SCATTERING AND DIRECT LINEARIZING TRANSFORMS FOR THE KADOMTSEV-PETVIASHVILI EQUATION , 1983 .

[6]  Athanassios S. Fokas,et al.  On the Inverse Scattering Transform for the Kadomtsev-Petviashvili Equation , 1983 .

[7]  Ronald R. Coifman,et al.  Scattering and inverse scattering for first order systems , 1984 .

[8]  Mark J. Ablowitz,et al.  A Multidimensional Inverse-Scattering Method , 1984 .

[9]  Athanassios S. Fokas,et al.  On the inverse scattering transform of multidimensional nonlinear equations related to first‐order systems in the plane , 1984 .

[10]  B. Konopelchenko,et al.  Backlund transformations via gauge transformations in 2+1 dimensions , 1985 .

[11]  B. Kupershmidt,et al.  Mathematics of dispersive water waves , 1985 .

[12]  M. Boiti,et al.  On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions , 1986 .

[13]  B. Konopelchenko.  Method of recursion operator for pseudodifferential spectral problems. I: Dispersive long wave equations , 1986 .

[14]  J. Leon,et al.  On a spectral transform of a KDV-like equation related to the Schrodinger operator in the plane , 1987 .

[15]  M. Boiti,et al.  Integrable two-dimensional generalisation of the sine- and sinh-Gordon equations , 1987 .