Pattern transformation in higher-order lumps of the Kadomtsev-Petviashvili I equation

Abstract

Pattern formation in higher-order lumps of the Kadomtsev-Petviashvili I equation at large time is analytically studied. For a broad class of these higher-order lumps, we show that two types of solution patterns appear at large time. The first type of patterns comprise fundamental lumps arranged in triangular shapes, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. As time evolves from large negative to large positive, this triangular pattern reverses itself along the x-direction. The second type of patterns comprise fundamental lumps arranged in non-triangular shapes in the outer region, which are described analytically by nonzeroroot structures of the Wronskian-Hermit polynomials, together with possible fundamental lumps arranged in triangular shapes in the inner region, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. When time evolves from large negative to large positive, the non-triangular pattern in the outer region switches its x and y directions, while the triangular pattern in the inner region, if it arises, reverses its direction along the x-axis. Our predicted patterns at large time are compared to true solutions, and excellent agreement is observed.

Classification of KPI Lumps

“…Note that the conclusion in part (a) above is based on an assumption that the non-zero roots of the Wronskian-Hermite polynomials are simple [16]. It is also worth noting that some of the exotic surface patterns of the KPI multi-lumps were observed earlier [20,3,19,14] and particularly, the long time asymptotics has been reported recently [48] after our investigation was complete. However, the treatment presented in this paper is new and different from the earlier works since it utilizes the special property of the Schur functions as characteristics of irreducible representation of the symmetric group S N .…”

“…So the dominant balance for |t| [3]. However, it was observed in [48] that the extreme values p = 2, 3 are the only two possibilities although no explanation was offered. In what follows, we shall first establish that is indeed the case.…”

“…Remark 2.2. (a) Several authors [15,11,48] have made different choices for the φ n in (2.1) to construct the n × n matrix M (2.7). These are essentially of the form…”

“…Figure 5 below compares the exact and the approximate peak locations given by (4.8) for the cases n = 3, 4, 5 corresponding to N = 6, 10, 15 respectively. (b) The triangular waveform patterns in KPI rational solutions were previously found in [20,3] and more recently in [19,14,48] but no previous attempt was made to classify these KPI rational solutions. According to Proposition 3.3, the triangular lumps belong to Class (II) corresponding to self-conjugate partitions.…”

“…(d) The long time asymptotics of the approximate peak locations described in this subsection was also obtained in [48] but from a different approach as explained in Section 1.…”