We have investigated a rock–scissors–paper model with long-range-directed interactions in two dimensions where every site has four outgoing links but a fraction q of the outgoing links to the nearest neighbour sites are rewired to other long-distance sites chosen randomly and the lattice structure is replaced again after a Monte Carlo step. It is found that, with q increasing, the system changes from a three species coexistence self-organizing state to a global oscillation state and then to one of the homogeneous states. However when q exceeds a third threshold value, the system returns to a self-organizing state. When we restrict the maximum number of ingoing links of a site to four, the last self-organizing state disappears, the system stays in the homogeneous state forever after q exceeds the second threshold value. And when we restrict the maximum number of ingoing links of a site to five or six, the system exhibits a transition from the homogeneous state to a global oscillation state again and then to the last self-organizing state with q increasing. The comparison of results on different networks suggests that the sites with zero ingoing links should play a significant role in the emergences of the later self-organizing state and the subsequent global oscillation.

We investigate the phase transition of the pair contact process (PCP) model in a fragmented network. The network is formed by rewiring the link between two nearest neighbors to another randomly selected site in one dimension with a probability 𝑞. When the average degree ⟨𝑘⟩ = 2, the system exhibits a structure transition and the lattice is fragmented into several isolated subgraphs, it is shown that a giant cluster appears and its node fraction does not change nearly for 𝑞 > 0. Furthermore, it is found that the critical behavior of the continuous phase transition for the PCP model is different from the directed percolation (DP) class and the estimated values of the critical exponents are independent of the rewiring probability for 𝑞 > 0. We conjecture that the structure transition for ⟨𝑘⟩ = 2 takes an important role in the change of the critical behavior of the continuous phase transition.

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Using the standard truncated Painlevé analysis, we have obtained some special types of exact explicit solitary wave solutions of the (3+1)-dimensional Kadomtsev-Petviashvilli equations. Usually, one obtains the single solitary wave solution from the Bäcklund transformation related to the truncated Painlevé analysis starting from the trivial vacuum solution. In this letter, we find some special types of the multi-solitary wave solution from the truncated Painlevé analysis and the trivial vacuum solution.