We develop the Riemann-Hilbert (RH) method for the generalized inhomogeneous Hirota equation with zero boundary condition. The RH problem is related to two kinds of scattering data: N simple poles and one N-order pole. Here we consider that when the scattering data have one or more higher-order poles, the formulas of bound-state (BS) solitons and multiple BS solitons are obtained, and the interaction between solitons and BS solitons are shown.

Linear growth of spherically symmetric solutions to Navier-Stokes equations in R N with degenerate viscosity and free boundary

This paper focuses on the three-dimensional(3D) incompressible anisotropic Boussinesq system while the velocity of fluid only involves horizontal dissipation and the temperature has a damping term. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space H k ( R 3 ) ( k ≥ 3 ) of perturbations near the hydrostatic equilibrium. Moreover, we take an effective approach to obtain the optimal decay rates for the global solution itself as well as its derivatives. In this paper, we aim to reveal the mechanism of how the temperature helps stabilize the fluid. Additionally, exploring the stability of perturbations near hydrostatic equilibrium may provide valuable insights into specific severe weather phenomena.

In this paper, we investigate the long-term stability of classical solution to 3D one-fluid Euler-Poisson system of electrons with non-zero vorticity. It is shown that the classical solution is well-posed for a lifesapn Tδ at least εδ, where ε>0 is the size of the initial perturbed density and velocity, δ>0 is the size of the initial vorticity. This implies that the lifespan Tδ only essentially depends on the size δ of the initial vorticity.