Superposition behaviour between lump solutions and different forms of N -solitons $$(N\rightarrow \infty )$$
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Superposition behaviour between lump solutions and different forms of N-solitons (N → ∞) for the fifth-order Korteweg–de Vries equation WEI TAN1,2
,∗
and JUN LIU3
1 Department
of Mathematics, Sun Yat-sen University, Guangzhou 510275, China of Mathematics and Statistics, Jishou University, Jishou 416000, China 3 Institute of Applied Mathematics, Qujing Normal University, Qujing 655011, China ∗ Corresponding author. E-mail: [email protected] 2 College
MS received 5 June 2019; revised 27 October 2019; accepted 10 November 2019 Abstract. A lump-type solution of the (2 + 1)-dimensional generalised fifth-order Korteweg–de Vries (KdV) equation is obtained from the two-soliton solution by applying the parametric limit method. Some theorems and corollaries about the superposition behaviour between lump solutions and different forms of N -soliton (N → ∞) solutions are constructed, and detailed proofs are given. Besides,we give a large number of examples and spatial evolution graphics to illustrate the effectiveness of the described theorems and corollaries. Some new nonlinear phenomena and superposition behaviour, such as rational-exponential type, rational-cosh-cos type, rational-sin type, rational-logarithmic type etc., are simulated and shown for the first time. Finally, we also illustrate the superposition between high-order lump-type solutions and N -soliton solutions. Keywords. Fifth-order KdV equation; lump solutions; superposition behaviour; soliton solutions; Hirota’s bilinear method. PACS Nos 04.20.Jb; 02.30.Jr; 05.45.Yv
1. Introduction
on exponential-trigonometric functions to study rogue waves and lump-type solutions and so on. Very recently, In recent years, the study of lump solution has become the study of solitons related to lump solutions, such as a hot topic in the field of nonlinear partial differen- lump-type solutions, rogue waves, interaction solutions, tial equations (NLPDEs). Lump solution, as solitary hybrid solutions, semirational solutions and interacwave solution in the form of rational functions, shows tion solitons, has attracted much attention because of polynomial decay in all directions of space variables. their application value and existence theory in the fields The first mathematical description of lump solution was of hydrodynamics, nonlinear optics, electromagnetics, reported in 1977 by studying the analytical expression optical fibre communications, etc. [11–22]. However, of two-dimensional solitons in KPI equation [1]. Up we find that most of the papers contain studies on the to now, many effective methods have been proposed interaction solutions between lump solution and N to study lump solutions of NLPDEs. Ablowitz and solitons, where the soliton number N is often less than 3. Satsuma [2,3] first proposed the long wave limit method It is well known that the superposition of solutions based on exponential function in 1978 to study the often exists in linear systems, but the superposition of N -solitons and M-lump solutions of NLPDEs. Imai solutions
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