Abstract
We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1 + 1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf–Cole substitution (the Bürgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together with the scalar and three-dimensional reductions. Variants of the Kadomtsev–Petviashvili-type and Korteweg–de Vries-type equations are represented among them. Our algorithm is based on the combination of two Frobenius-type reductions and special differential reduction imposed on the matrix fields of integrable PDEs. It is shown that the derived four-dimensional nonlinear PDEs admit arbitrary functions of two variables in their solution spaces which clarifies the integrability degree of these PDEs.
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