Analysis on an extended Majda--Biello system

In this paper, we begin with extended Majda–Biello system (BSAB equations): $$\{\begin{array}{l}0=A_t-D{A}_3+\mu A_1+{\mathrm{\Gamma }}_S{B}_1^S+{\mathrm{\Gamma }}_A{B}_1^A+{\left(A{B}^S\right)}_x\\ 0=B_t^S-B_3^S+{\mathrm{\Gamma }}_S{A}_1+\lambda B_1^S+\sigma B_1^A+A{A}_1\\ 0=B_t^A-B_3^A+{\mathrm{\Gamma }}_A{A}_1+\sigma B_1^S-\lambda B_1^A\end{array}$$ We conclude global well-posedness in $L^2(\mathbb{R})\times L^2(\mathbb{R})\times L^2(\mathbb{R})$ by Brougain’s method and the stability of solitary wave solutions by putting it in a framework of generalised KdV type system with three components, where Hamiltonian structure plays an important role. Both of them are bases for numerical tests.\par Last but not least, we explore the effect of interaction of two solitary waves in Majda–Biello system in a novel way : \par $\mathit{\text{While}}\mathit{\text{fixing}}\mathit{\text{initial}}\mathit{\text{data}}\mathit{\text{for}}\mathit{\text{one}}\mathit{\text{soliton}}U\mathit{\text{,}}\mathit{\text{we}}\mathit{\text{point}}\mathit{\text{out}}\mathit{\text{the}}\mathit{\text{effect}}\mathit{\text{on}}U\mathit{\text{decays,}}\mathit{\text{to}}\mathit{\text{some}}\mathit{\text{extent}}\mathit{\text{and}}\mathit{\text{in}}\mathit{\text{certain}}\mathit{\text{range,}}\mathit{\text{in}}\mathit{\text{a}}\mathit{\text{polynomial}}\mathit{\text{way.}}$ \par Since effect of interaction of two solitary waves are practically interesting, such kind of analysis, as we have explained, is likely be fundamental for generalised KdV type systems.