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Nonlinear Wave Dynamics: Theory and Application
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Nonlinear Wave Dynamics: Theory and Application

A Special Issue of Mathematics (ISSN 2227-7390) belonging to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: 31 October 2026 | Viewed by 3223

Editors

Prof. Dr. Zhen Wang

E-Mail Website
Guest Editor
School of Mathematical Sciences, Beihang University, Beijing 100191, China
Interests: integrable system; application of mathematical methods; ocean engineering
Prof. Dr. Dengshan Wang

E-Mail Website
Guest Editor
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Interests: integrable systems; asymptotic analysis
Special Issues, Collections and Topics in MDPI journals
Dr. Xiangyu Yang

E-Mail Website
Guest Editor Assistant
School of Mathematics and Information Science, Zhongyuan University of Technology, Zhengzhou 451191, China
Interests: integrable system; soliton theory; wave interaction

Special Issue Information

Dear Colleagues,

Since the groundbreaking work on nonlinear wave dynamics, particularly the discovery of solitons by John Scott Russell in 1834 and the subsequent mathematical formalization by Korteweg and de Vries in 1895, there has been continuous expansion in both the theoretical advancements and practical applications of nonlinear wave theory. Nonlinear waves, often governed by nonlinear partial differential equations, play a crucial role in describing a wide range of physical phenomena, such as fluid dynamics, nonlinear optics, plasma physics, and condensed matter systems. They are not only mathematically fascinating but also have significant implications for understanding complex natural systems and engineering applications.

Nonlinear waves are a complex and fascinating aspect of fluid mechanics, with profound implications for the study of ocean waves. The interplay between nonlinearity, dispersion, and external forces leads to a rich variety of wave phenomena. The study of integrable systems has spurred the development of powerful analytical methods, including the inverse scattering transform, Riemann–Hilbert problem, Hirota bilinear method, Darboux transformation, algebraic geometry techniques, and Lie symmetry analysis. These methods enable researchers to obtain exact solutions and explore the intricate behavior of nonlinear waves.

This Special Issue provides a platform for researchers from academia and industry to present their novel and unpublished works in the domain of nonlinear wave dynamics, focusing on both theoretical developments and practical applications. By highlighting advancements in the understanding of integrable systems, ocean waves, fluid mechanics, and the physical mechanisms of nonlinear waves, this Issue aims to foster future research in this interdisciplinary field, bridging the gap between mathematical theory and real-world applications.

We look forward to receiving your contributions.

Prof. Dr. Zhen Wang
Prof. Dr. Dengshan Wang
Guest Editors

Dr. Xiangyu Yang
Guest Editor Assistant

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • integrable system
  • nonlinear wave
  • soliton
  • rogue wave
  • Riemann–Hilbert problem
  • inverse scattering method
  • Darboux transformation
  • Hirota bilinear method
  • ocean wave
  • fluid mechanics

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Published Papers (3 papers)

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Research

37 pages, 1282 KB  
Article
A Structure-Preserving Covering Method for the KdV-Burgers Equation with Exact Conservation and High-Order Compact Discretization
by Faiza Afzal and Seham S. Alzahrani
Mathematics 2026, 14(10), 1714; https://doi.org/10.3390/math14101714 - 16 May 2026
Viewed by 428
Abstract
Structure-preserving numerical methods are well-established for purely conservative or purely dissipative systems but remain underdeveloped for mixed-type equations coupling dispersion, dissipation, and nonlinearity. We investigate the Korteweg–de Vries–Burgers equation as a canonical model of this class. We develop a geometric covering method based [...] Read more.
Structure-preserving numerical methods are well-established for purely conservative or purely dissipative systems but remain underdeveloped for mixed-type equations coupling dispersion, dissipation, and nonlinearity. We investigate the Korteweg–de Vries–Burgers equation as a canonical model of this class. We develop a geometric covering method based on nonlocal symmetries that lifts the equation to an extended manifold, enabling exact conservation law preservation. As a pedagogical counterexample, we also analyze a naive recursive approximation. Both methods are implemented using sixth-order compact finite differences and fourth-order Runge–Kutta (RK4) time integration. Numerical experiments on sinusoidal waves, two-soliton collisions, and perturbed traveling waves show that the covering method reduces numerical dissipation by 50% and phase error by 90% relative to a standard second-order scheme, achieving one to two orders of magnitude higher accuracy. Mass and momentum are conserved to machine precision (below 10−14), and soliton amplitudes are preserved to within 0.3% after collision, with only 15% computational overhead. The framework offers a generalizable template for embedding nonlocal symmetries into high-order numerical methods for nonlinear wave equations. Full article
(This article belongs to the Special Issue Nonlinear Wave Dynamics: Theory and Application)
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24 pages, 1768 KB  
Article
Analytical Solutions and Analyses for the Deflection of Nonlinear Waves on Kirchhoff Plates Underlying a Pasternak-like Nonlinear Elastic Foundation
by Asma AlThemairi, Rahmatullah I. Nuruddeen and Roger Bertin Djob
Mathematics 2026, 14(1), 74; https://doi.org/10.3390/math14010074 - 25 Dec 2025
Cited by 3 | Viewed by 1286
Abstract
The present study models the deflection of nonlinear waves over a Kirchhoff plate underlying a Pasternak-like elastic foundation. A promising version of the tanh expansion analytical method has been deployed for the construction of regular exact solutions for the model, including the application [...] Read more.
The present study models the deflection of nonlinear waves over a Kirchhoff plate underlying a Pasternak-like elastic foundation. A promising version of the tanh expansion analytical method has been deployed for the construction of regular exact solutions for the model, including the application of certain ansatz functions for validations and yet construction of more solutions. The resulting frequency equation and the modulation instability spectrum have been obtained for the linearized model, including the expressions for the related phase and group velocities. In addition, the study examines the equilibrium status of the resulting dynamical system with the help of the bifurcation analysis. Numerically, nonlinear deflection and dispersion of waves have been simulated through the acquired expressions and equations. Notably, the study notes that increasing both the Pasternak-like nonlinear parameter η and time variation (for x>0) decreases the nonlinear deflection in the plate, while increasing the stiffness of the Winkler foundation increases deflection in the medium. In addition, the study establishes, concerning the determined frequency equation, that increasing the Winkler foundation stiffness increases the dispersion of nonlinear waves in the medium, while an opposite trend has been noted concerning the imposed Pasternak-like nonlinear foundation. In addition, both phase and group velocities, the gain function for modulation instability, and the resulting dynamical system have been noted to be greatly affected by the variation of the imposed foundational parameters. Lastly, this study has potential applications in various engineering fields while modeling and analysis of mechanical structures supported by additional structures. Full article
(This article belongs to the Special Issue Nonlinear Wave Dynamics: Theory and Application)
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16 pages, 1837 KB  
Article
Interactions and Soliton Dynamics for a (2+1)-Dimensional Nonlinear Integrable Model Arising in Shallow Water
by Ali Danladi, Aljethi Reem Abdullah, Ejaz Hussain and Zhao Li
Mathematics 2025, 13(21), 3474; https://doi.org/10.3390/math13213474 - 31 Oct 2025
Cited by 1 | Viewed by 738
Abstract
In this study, we consider a (2+1)-dimensional integrable Boussinesq equation, where the Hirota method of positive logarithmic transformation is used to convert it into a bilinear form. We proceeded by employing different test functions, through which we obtained breather solutions, two-wave solutions, lump-periodic [...] Read more.
In this study, we consider a (2+1)-dimensional integrable Boussinesq equation, where the Hirota method of positive logarithmic transformation is used to convert it into a bilinear form. We proceeded by employing different test functions, through which we obtained breather solutions, two-wave solutions, lump-periodic solutions, and new interaction solutions. The resulting soliton dynamics for the governing model are also derived using the enhanced modified extended tanh function method, where varieties of solutions, such as trigonometric, hyperbolic, and rational forms, were obtained. The derived solutions may hold significant potential for explaining real-world physical phenomena in fields like mathematical physics, plasma physics, and nonlinear optics. The accuracy and reliability of the solutions were tested by substituting them back into the original equation using Python, highlighting the method’s robustness, precision, and reliability. By choosing appropriate physical parameters, we showcased the rich diversity and dynamic behavior of the obtained soliton structures. In other words, the graphical representations in 3D, contour, and 2D were provided for some of the obtained results. The modulation instability analysis and gain spectrum of the model are also provided. The importance of the obtained results in the area of (2+1)-dimensional integrable equation application was also highlighted. Full article
(This article belongs to the Special Issue Nonlinear Wave Dynamics: Theory and Application)
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