Partially integrable systems in multidimensions by a variant of the dressing method. 1

Abstract

In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially integrable''. Such a construction is achieved using a suitable modification of the classical dressing scheme, consisting in assuming that the kernel of the basic integral operator of the dressing formalism be nontrivial. This new hypothesis leads to the construction of: 1) a linear system of compatible spectral problems for the solution $U(\lambda ;x)$ of the integral equation in 3 independent variables each (while the usual dressing method generates spectral problems in 1 or 2 dimensions); 2) a system of nonlinear partial differential equations in $n$ dimensions ($n>3$), possessing a manifold of analytic solutions of dimension ($n-2$), which includes one largely arbitrary relation among the fields. These nonlinear equations can also contain an arbitrary forcing.

21 pages

References in corpus (1)

• Inverse Scattering Problem for Vector Fields and the Heavenly Equation

Cited by in corpus (5)

• Combination of inverse spectral transform method and method of characteristics: deformed Pohlmeyer equation
• Dressing method based on homogeneous Fredholm equation: quasilinear PDEs in multidimensions
• On integration of multidimensional version of $n$-wave type equation
• Particular solutions to multidimensional PDEs with KdV-type nonlinearity
• New multidimensional partially integrable generalization of S-integrable N-wave equation