(1+1)-dimensional turbulence systems reduced from (2+1)-dimensional Lax integrable dispersive long wave equation

After extending the Clarkson-Kruskal’s direct similarity reduction ansatz to a more general form, one may obtain various new types of reduction equations. Especially, some lower dimensional turbulence systems or chaotic systems may be obtained from the general type of similarity reductions of a higher dimensional Lax integrable model with nonintegrable boundary and/or the initial conditions. In this letter, Taking the (2+1)-dimensional dispersive long wave equation system as a simple example, we derive a general Korteweg de-Vries Burgers type equation and a Kuramoto-Sivashinsky equation as two special (1+1)-dimensional similarity reductions. To reduce a higher dimensional nonlinear physics model to some lower dimensional ones is one of the most important method in the study of nonlinear science. Usually one use the standard Lie group approach to reduce a higher dimensional partial differential equation (PDE) to lower dimensional ones[1]. Lately, the so-called nonclassical Lie group approach is used to find lower dimensional similarity reductions[2]. To find some lower dimensional reductions by using the classical and nonclassical Lie group approaches, one has to use some tedious algebraic procedures. In the past decade, to avoid the tedious algebraic calculation in the finding of the similarity reductions, a simple direct method is established[3, 4]. Using the direct method, various new similarity reductions of many physical models are found though these reductions are also obtained lately from the nonclassical Lie group approached[5, 6, 7]. In [8], the direct method is developed to find some types of conditional similarity reductions which have not yet ∗Email: [email protected]