Nonlinear distortion of symmetry in solutions to the convection-diffusion equation of Burgers type

Abstract

In this paper, the initial value problem of the convection-diffusion equation of Burgers type is treated. In the asymptotic profile of solutions, the nonlinearity of the equation is reflected. Regarding the solutions to this model, the Spanish school in the 1990s performed asymptotic expansions based on the linear diffusion. Those profiles exhibit symmetries characteristic of linear phenomena. In this paper, the distortion of symmetry arising from the nonlinear effects is described explicitly. Furthermore, it is demonstrated that the extent of this distortion differs significantly depending on the parity of the spatial dimension. This contradicts the conventional expectation that the manifestation of nonlinearity depends on the scale of the equation. This interpretation is supported by comparison with similar Navier--Stokes equations. The Burgers type is applicable as an indicator for considering several bilinear problems.