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In this paper, we investigate the fifth-order modified Korteweg-de Vries (fomKdV) equation on the half-line via the Fokas unified transformation approach. We show that the solution u(x,t) of the fomKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ. The jump matrix L(x,t,θ) has an explicit representation depended on x, t and it can be represented exactly by the two pairs of spectral functions y(θ),z(θ) (obtained from the initial value u0(x)) and Y(θ),Z(θ) (obtained from the boundary conditions v0(t), vk(t), k=1,2,3,4). Furthermore, the two pairs spectral functions y(θ),z(θ) and Y(θ),Z(θ) are not independent of each other, but are related to the compatibility condition, the so-called global relation.

In this work, we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schrödinger (DNLS) equation. By establishing a matrix Riemann–Hilbert problem and reconstructing potential function q(x, t) from eigenfunctions { G j ( x , t , η ) } 1 3 in the inverse problem, the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed. Moreover, we also obtain that the spectral functions f(η), s(η), F(η), S(η) are not independent of each other, but meet an important global relation. As applications, the generalized DNLS equation can be reduced to the Kaup–Newell equation and Chen–Lee–Liu equation on the half-line.

With the rapid development of information technology, the travel mode of " Internet + Transportation " has brought convenience to residents while causing a series of urban traffic problems. Based on GPS data from Didi Chuxing, this paper relates the residents' travel behaviour to their daily activities, and studies the regularity of the pick-up and drop-off activities of residents from the dimensions of time and space. The results are as follows: Firstly, the daily variation trend of residents' travel is unbalanced. At the same time, the travel volume of residents is at a higher level during night. Secondly, the travel of urban residents is closely related to urban areas and urban functional structures. On the administrative scale, the distribution of residents' travel is concentric, and the farther the distance from the downtown area is, the smaller the number of orders is. On the functional scale, the hot functional areas of residents' travel are related to residents' needs. At last, mastering the distribution of residents' travel hot spots and time will help to understand the residents' travel mode and the current situation of urban traffic, and provide new ideas for solving traffic problems in big cities.