Smooth positon solutions of the focusing modified Korteweg–de Vries equation

Abstract

The n-fold Darboux transformation T n of the focusing real modified Korteweg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the n-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues λ j and the corresponding eigenfunctions of the associated Lax equation. The nonsingular n-positon solutions of the focusing mKdV equation are obtained in the special limit λ j → λ 1 , from the corresponding n-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the n-positon solution into n single-soliton solutions, the trajectories, and the corresponding "phase shifts" of the multipositons are also investigated.

Generating mechanism and dynamic of the smooth positons for the derivative nonlinear Schrödinger equation

“…As for focusing mKdV equation, the distance of two peaks in two-positon of the focusing mKdV equation is 2c 11 (≈ ln(64t 2 ) 2 ). While the distance of two peaks in two-positon of the DNLS equation is 2c 11…”

“…It is crucial to get the smooth positons because the positon solutions above-mentioned are singular. Recently, the smooth positons of focusing mKdV equation [11], complex mKdV equation [12] and the second-type derivative nonlinear Schrödinger (DNLSII) equation [13] also have been constructed. Postion is a slowly decreasing analogue of soliton, which is closed related to Wigner-von Neumann phenomenon [14].…”

Breather Positons and Rogue Waves for the Nonlocal Fokas-Lenells Equation

“…It is always a difficult problem to find the trajectories of breather positons [10,16,17]. Not to mention, let us study the dynamic properties of breather positons like decomposing smooth positons [7][8][9].…”

“…Similarly, their team applied this mechanism to other integrable equations, such as the modified KdV equation (5) and Fokas-Lenells (FL) equation (7), to obtain higher-order rogue waves. Since Matveev [5,6] proposed singular positon solutions for the Korteweg-de Vries equation in 1992, many experts [7][8][9][10][11] have been working in this field. Among the numerous studies on positons, Wang et al [10] elaborate the connection between breather positons and rogue waves for a local integrable system.…”

Families of exact solutions of a new extended $$\varvec{(2+1)}$$(2+1)-dimensional Boussinesq equation

“…In particular, the exact solutions and their dynamics in the case of integrable (2 + 1)d equations have been studied in detail, such as the Davey-Stewartson [DS] equations [22,23], the Kadomtsev-Petviashvili-I [KPI] equation [24,25], the (3 + 1)d KP equation [26], and other physically-relevant NLEEs, see, for example, Refs. [27][28][29][30][31][32][33][34][35][36][37][38][39][40].…”