Higher-order smooth positons and breather positons of Sine-Gordon equation

According to the N-soliton solution derived from Hirota's bilinear method,higher-order smooth positons and breather positons are obtained efficiently through an ingenious limit approach.This paper takes the Sine-Gordon equation as an example to introduce how to utilize this technique to generate these higher-order smooth positons and breather positons in detail.The dynamical behaviors of smooth positons and breather positons are presented by some figures.During the procedure of deduction,the approach mentioned has the strengths of concision and celerity.In terms of feasibility and practicability,this approach can be exploited widely to study higherorder smooth positons and breather positons of other integrable systems.

Smooth Positons of the Second-Type Derivative Nonlinear Schr?dinger Equation

We construct the soliton solution and smooth positon solution of the second-type derivative nonlinear Schr¨odinger(DNLSII) equation. Additionally, we present a detailed discussion about the decomposition of the positon solution, and display its approximate orbits and variable "phase shift". The second and third order breather-positon solutions are also constructed.

Positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg-de Vries equations

Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations are decomposed into multi-solitons as well as the(1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.

Soliton molecules and dynamics of the smooth positon for the Gerdjikov–Ivanov equation

Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent "phase shifts" of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.

Wronskian Determinant Solutions for the (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation

In this paper, we consider (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Based on the bilinear form, we derive exact solutions of (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation by using the Wronskian technique, which include rational solutions, soliton solutions, positons and negatons.

Nonsingular Positon Solutions of a Variable-Coefficient Modified KdV Equation

The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, the nonsingular positon solutions of the variable-coefficient modified KdV equation are firstly discovered analytically and graphically.

一类广义Boussinesq方程的Wronskian行列式解

为了构造非线性孤子方程的Wronskian行列式新解,进一步研究了Wronskian技巧.本文首先给出非线性广义Boussinesq方程的双线性形式,利用Wronskian技巧构造出该非线性方程所满足的一个线性偏微分条件方程组,然后求解该微分条件方程组,得到了广义Boussinesq方程的各种Wronskian行列式解.

Multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas–Lenells equation

In this letter,we investigate multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas–Lenells equation over a nonzero background.First,we obtain 2 n-soliton solutions with a nonzero background via n-fold Darboux transformation,and find that these soliton solutions will appear in pairs.Particularly,2 n-soliton solutions consist of n‘bright’solitons and n‘dark’solitons.This phenomenon implies a new form of integrability:even integrability.Then interactions between solitons with even numbers and breathers are studied in detail.To our best knowledge,a novel nonlinear superposition between a kink and 2 n-soliton is also generated for the first time.Finally,interactions between some different smooth positons with a nonzero background are derived.