A chain of connected genes with activation-repression links is analysed. It is shown that for various promoter activity functions (parametrised by Hill coefficient) the equations describing the concentrations of transcription factors are perturbed completely integrable differential-difference of KdV-type. In the case of large Hill coefficient the proteomic signal along the gene network is given...

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u = 2(u)x, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is uniformly globally well-posed in Hs (s ≥ 1) for all ǫ ∈ (0, 1]. Moreover, we prove that for any s ≥ 1 and T > 0, its solution converges in C([0, T ]; Hs) to that of the MKdV equation if ǫ tends to 0.

In this paper, the Homotopy Analysis Method (HAM) is presented for obtaining the approximate solution of new coupled modified Korteweg-de Vries (MKdV) system. The approximate analytical solution is obtained by using this method in the form of a convergent power series with components that are easily computable. For the problems considered, only a few terms are required to obtain an approximate ...

We study the problem of 2-soliton collision for the generalized Korteweg-de Vries equations, completing some recent works of Y. Martel and F. Merle [22, 23]. We classify the nonlinearities for which collisions are elastic or inelastic. Our main result states that in the case of small solitons, with one soliton smaller than the other one, the unique nonlinearities allowing a perfectly elastic co...