Zhen Wang

This paper presents a systematic study of lump solutions and lump chain solutions in the modified Kadomtsev–Petviashvili‐I equation. Using the long‐wave limit method, we derive both standard lump solutions and plane lump solutions from line‐soliton solutions. Through detailed asymptotic analysis, it is shown that standard lump solutions exhibit normal scattering. We further investigate lump chain solutions and identify elastic interactions with explicit phase shifts, as well as distinctive resonant patterns, including a characteristic “Y”‐shaped resonance and a unique parallel resonance phenomenon. An improved long ‐ wave limit approach is introduced to construct higher‐order lump solutions from lump chain solutions, yielding second‐order lump solutions, second‐order lump chain solutions, and semi‐rational soliton solutions. Notably, the second‐order lump solutions exhibit anomalous scattering, in which individual lumps propagate along curved trajectories despite sharing the same asymptotic velocity. Our analysis also establishes a connection between lump solutions and the root structure of Yablonskii–Vorob'ev polynomials. These results deepen the understanding of nonlinear wave interactions in (2 + 1)‐dimensional integrable systems.

A nonlinear large deformation cable equation system based on arc length is employed to determine the configuration of the cable under current and buoyancy loads. Different from a traditional initial method and a shooting method, we have solved this nonlinear problem as a boundary value problem with nonlinear and global boundary conditions directly. A finite difference scheme is proposed to solve the large deformation cable equation, and the Newton–Raphson iteration is used to search for numerical approximate solution. We demonstrate that this system degenerates into the catenary equation in the case of vanishing bending stiffness for the first time. The solution of the catenary equation serves as the initial guess for the three-section cable problem. This method overcomes the disadvantages of the initial value method and step method, avoiding the need to adjust the boundary location. The spatial shapes of the large deformation cable and the effects of the length and position of the buoyancy section are discussed. The impact of ocean currents is also analyzed using Morison's formula.

The cavitation bubble dynamics inside liquid drops governed by a Rayleigh–Plesset-like equation is investigated theoretically. A strict qualitative analysis is made to determine the bubble dynamic behaviors. Analytical expressions of the collapse times and analytical solutions of the governing equation are derived for different initial conditions. The validity of these analytical solutions is studied by testing numerical algorithms and/or experimental data. As applications of the analytical solutions, analytical expressions in parametric forms for the evolutions of bubble oscillation velocity, oscillation acceleration, kinetic energy, and potential energy are also obtained. Furthermore, the relevant nonlinear bubble dynamic characteristics and motion laws are also revealed based on the obtained results.