A singular limit problem for the Kudryashov-Sinelshchikov equation

Abstract

Abstract. We consider the Kudryashov-Sinelshchikov equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to the entropy ones of the Burgers equation. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p setting.

New exact solutions for Kudryashov–Sinelshchikov equation

“…If q < 0, the solutions of Eq. (3.1) are 19) where ξ = xct and α, η, c, λ, k 2 and k 21 are arbitrary constants. Specially, if μ = 0, then q < 0, the solutions of Eq.…”

“…In [18], the authors solved numerically the nonlinear time-fractional Kudryashov-Sinelshchikov equation by using radial basis function (RBF) method. In [19], the authors considered the Kudryashov-Sinelshchikov equation, which contains nonlinear dispersive effects, and proved that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to the entropy ones of the Burgers equation. In [20], the authors used Hermite transform for transforming the Wick-type stochastic Kudryashov-Sinelshchikov equation to deterministic partial differential equation and obtained exact solutions of Wick-type stochastic Kudryashov-Sinelshchikov equation by using improved Sub-equation method.…”

Existence results for the Kudryashov–Sinelshchikov–Olver equation

“…Methods to find exact solutions are in [1-3, 18-20, 30, 36, 38, 42, 44, 45, 47, 48]. Moreover, following [6,7,28,39], under the assumption (1.3), in [8], the authors used the convergence of the solution of (1.1) to the unique entropy solution of the following scalar conservation law:…”