# Inverse Scattering and Soliton Solutions of Nonlocal Complex Reverse-Spacetime Modified Korteweg-de Vries Hierarchies

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## Abstract

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## 1. Introduction

## 2. Nonlocal Symmetric Reductions and Nonlocal mKdV Hierarchies

#### 2.1. Multicomponent AKNS Hierarchy

#### 2.2. Nonlocal Reverse-Spacetime mKdV Hierarchies

## 3. Inverse Scattering Transforms

#### 3.1. Property of Eigenfunctions

#### 3.2. Riemann-Hilbert Problems

#### 3.3. Time Evolution of the Scattering Data

#### 3.4. Gelfand-Levitan-Marchenko Type Integral Equations

#### 3.5. Recovery of the Potential

## 4. Soliton Solutions

#### 4.1. Basic Formulation of Solutions

#### 4.2. Nonreduced Case

#### 4.3. Nonlocal Case

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Ling, L.; Ma, W.-X.
Inverse Scattering and Soliton Solutions of Nonlocal Complex Reverse-Spacetime Modified Korteweg-de Vries Hierarchies. *Symmetry* **2021**, *13*, 512.
https://doi.org/10.3390/sym13030512

**AMA Style**

Ling L, Ma W-X.
Inverse Scattering and Soliton Solutions of Nonlocal Complex Reverse-Spacetime Modified Korteweg-de Vries Hierarchies. *Symmetry*. 2021; 13(3):512.
https://doi.org/10.3390/sym13030512

**Chicago/Turabian Style**

Ling, Liming, and Wen-Xiu Ma.
2021. "Inverse Scattering and Soliton Solutions of Nonlocal Complex Reverse-Spacetime Modified Korteweg-de Vries Hierarchies" *Symmetry* 13, no. 3: 512.
https://doi.org/10.3390/sym13030512