Compatible symmetry reductions and integrable decompositions for (2+1)-dimensional KdV-HD equation

Abstract This paper is dedicated to exploring the high-dimensional extensions of low-dimensional integrable systems. Although high-dimensional integrable systems have been constructed via deformation techniques established through conservation laws, solving such high-dimensional equations remains highly challenging. To address this, we introduce a compatible symmetry reduction method. Taking the (2+1)-dimensional KdV-HD equation as an example, and based on Lie point symmetries, we derive four distinct types of symmetry reductions, including one traveling wave reduction and three Painlevé-type reductions. These reductions directly transform the (2+1)-dimensional partial differential equations into ordinary differential equations, thereby yielding elliptic periodic wave solutions and soliton solutions. Furthermore, we extend the compatible symmetry reduction to a consistent integrable decomposition method to construct exact solutions containing arbitrary functions. Our results indicate that the (2+1)-dimensional KdV-HD system can be decomposed into lower-dimensional integrable subsystems. In particular, by selecting appropriate arbitrary functions, we reveal the existence of non-traveling twisted solitons. These novel solutions exhibit complex structural deformations and rotational behaviors, going beyond the descriptive scope of traditional planar solitons, and provide new perspectives for understanding the propagation dynamics of high-dimensional nonlinear waves.