Let m ≥ 1 be an integer,1 〈 β ≤ m ＋ 1.A sequence ε1ε2ε3 … with εi ∈{0,1,…,m} is called a β-expansion of a real number x if x = Σi εi/βi.It is known that when the base β is smaller than the generalized...Let m ≥ 1 be an integer,1 〈 β ≤ m ＋ 1.A sequence ε1ε2ε3 … with εi ∈{0,1,…,m} is called a β-expansion of a real number x if x = Σi εi/βi.It is known that when the base β is smaller than the generalized golden ration,any number has uncountably many expansions,while when β is larger,there are numbers which has unique expansion.In this paper,we consider the bases such that there is some number whose unique expansion is purely periodic with the given smallest period.We prove that such bases form an open interval,moreover,any two such open intervals have inclusion relationship according to the Sharkovskii ordering between the given minimal periods.We remark that our result answers an open question posed by Baker,and the proof for the case m = 1 is due to Allouche,Clarke and Sidorov.展开更多
The consistent tanh expansion(CTE) method is employed to the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada(CDGKS) equation. The interaction solutions between solitons and the cnoidal periodic waves are explic...The consistent tanh expansion(CTE) method is employed to the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada(CDGKS) equation. The interaction solutions between solitons and the cnoidal periodic waves are explicitly obtained. Concretely, we discuss a special kind of interaction solution in the form of tanh functions and Jacobian elliptic functions in both analytical and graphical ways. The results show that the profiles of the soliton-cnoidal periodic wave interaction solutions can be designed by choosing different values of wave parameters.展开更多
文摘Let m ≥ 1 be an integer,1 〈 β ≤ m ＋ 1.A sequence ε1ε2ε3 … with εi ∈{0,1,…,m} is called a β-expansion of a real number x if x = Σi εi/βi.It is known that when the base β is smaller than the generalized golden ration,any number has uncountably many expansions,while when β is larger,there are numbers which has unique expansion.In this paper,we consider the bases such that there is some number whose unique expansion is purely periodic with the given smallest period.We prove that such bases form an open interval,moreover,any two such open intervals have inclusion relationship according to the Sharkovskii ordering between the given minimal periods.We remark that our result answers an open question posed by Baker,and the proof for the case m = 1 is due to Allouche,Clarke and Sidorov.
基金Supported by the National Natural Science Foundation of China under Grant No.11505154the Zhejiang Provincial Natural Science Foundation of China under Grant No.LQ16A010003the Scientific Research Foundation for Doctoral Program of Zhejiang Ocean University under Grant No.Q1511
文摘The consistent tanh expansion(CTE) method is employed to the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada(CDGKS) equation. The interaction solutions between solitons and the cnoidal periodic waves are explicitly obtained. Concretely, we discuss a special kind of interaction solution in the form of tanh functions and Jacobian elliptic functions in both analytical and graphical ways. The results show that the profiles of the soliton-cnoidal periodic wave interaction solutions can be designed by choosing different values of wave parameters.
