Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time t evolves from large negative to large positive, the triangle lump reverses itself along the y -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the y -direction when time t evolves from large negative to large positive. The obtained results improve our understanding of time evolution mechanisms of high-order lumps.

In this work, we study the high-order rogue wave solution for the Fokas-Lenells equation using the Kadomtsev-Petviashvili (KP) reduction method. These rogue wave patterns consist of triangle, pentagon, heptagon, nonagon, which are analytically described by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy. On the other hand, we also report the other types of rogue wave patterns including heart-shaped, fan-shaped, two-arc+triangle, arc+pentagon, etc., which are analytically described by the root structures of Adler-Moser polynomials. These polynomials are the generalizations of the Yablonskii-Vorob'ev polynomial hierarchy, because of the arbitrariness of complex parameter . In addition, these rogue wave patterns are formed by the Peregrine solitons undergoing dilation, rotation, stretch, shear and translation. We also compare the prediction solutions with the corresponding true solutions and show the good consistency between them.

In this work, we extend the finite-difference contrast source inversion (FD-CSI) method to the frequency-domain elastic wave equations, where the parameters describing the subsurface structure are simultaneously reconstructed. The FD-CSI method is an iterative nonlinear inversion method, which exhibits several strengths. First, the finite-difference operator only relies on the background media and the given angular frequency, both of which are unchanged during inversion. Therefore, the matrix decomposition is performed only once at the beginning of the iteration if a direct solver is employed. This makes the inversion process relatively efficient in terms of the computational cost. In addition, the FD-CSI method automatically normalizes different parameters, which could avoid the numerical problems arising from the difference of the parameter magnitude. We exploit a parallel implementation of the FD-CSI method based on the domain decomposition method, ensuring a satisfactory scalability for large-scale problems. A simple numerical example with a homogeneous background medium is used to investigate the convergence of the elastic FD-CSI method. Moreover, the Marmousi II model proposed as a benchmark for testing seismic imaging methods is presented to demonstrate the performance of the elastic FD-CSI method in an inhomogeneous background medium.

In this work, we show several types of (semi)-rational solutions for the two dimensional Mel’nikov equation. We derive multi-breather wave solutions. They present the oblique breathers, homoclinic ...