Large-time wave patterns in higher-order lumps of the Mel’nikov equation

Abstract

In this paper, we study large-time wave patterns in higher-order lumps of the Mel’nikov equation. We show that two types of lump patterns appear when time is large. For the first type of lump patterns, it contains fundamental lumps uniformly distributed in triangular shapes, and this triangular pattern reverses its shape along the x- and y- directions when time goes from large negative to large positive. For the second type of lump patterns, it contains fundamental lumps distributed in non-triangular shapes in the outer region, together with possible fundamental lumps arranged in triangular shapes in the inner region. When time goes from large negative to large positive, the non-triangular pattern in the outer region occurs a linear transformation which involves actions such as dilation, rotation, stretch, shear and transitions, while the triangular pattern, when it arises in the inner region, reverses its shapes along the x- and y- axis. In general, these wave patterns are related to root structures of the Yablonskii-Vorob’ev polynomials and the Wronskian–Hermit polynomials. Leading-order predictions of lump locations at large time also analytically derived, and excellent agreement is observed when compared to true solutions.