We present and study an extended Gardner-like family of equations, $G(m,n;k):u_t+{(c_{+}{u}^m-c_{-}{u}^n)}_x+{\left(u^k\right)}_{\mathit{xxx}}=0,c_{\pm }>0,$ n > m > 1, k ≥ 1, endowed with a non-convex convection which may be due to two opposing mechanisms that bound the range of velocities of both solitons and compactons beyond which they dissolve, and kink and/or antikink form. Close to solitons and compactons barrier, there is a narrow strip of velocities where the wave shape undergoes a structural change and rather than grow with velocity, their top flattens and they widen rapidly with minute changes in velocity. These waves, referred to as flatons, may be viewed as an approximate amalgam of a kink and anti kink placed at any distance from each other. Typical of solitons, once flatons form they are very robust with their domain of attraction being sensitive to the amplitude at which convection reverses its direction. A multi-dimensional extension of these equations unfolds a plethora of flatons which, unless m is even and n is odd, for every admissible velocity may span an entire sequence of multi-nodal radially symmetric flatons.

我们提出并研究了类似Gardner的扩展方程组， $G（米，ñ;ķ）：{ü}_{Ť}+{（C_{+}{ü}^{米}-C_{-}{ü}^{ñ}）}_X+{（{ü}^{ķ}）}_{\mathit{xxx}}=0，C_{\pm }>0，$ n  >  m  > 1，  k ≥1，具有非凸对流，这可能是由于两个相反的机制限制了孤子和压实物体的速度范围，超过该范围它们会溶解，并且扭结和/或抗扭结形式。靠近孤子和压实壁垒，有一小段速度带，其中波形发生结构变化，而不是随速度增长，其顶部变平，随着速度的微小变化而迅速扩展。这些被称为“ flatons”的波可以看作是彼此之间任意距离放置的扭结和反扭结的近似汞齐。典型的孤子，一旦形成了扁平子，它们的鲁棒性非常强，其吸引域对对流反转其方向的幅度敏感。这些方程的多维扩展展开了过多的flatons，除非m为偶数，n为奇数，因为每个允许的速度都可能跨越整个多节点径向对称平板的整个序列。