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Mathematics
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New Trends in Nonlinear Waves
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New Trends in Nonlinear Waves

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 31 August 2025 | Viewed by 1949

Special Issue Editor

Dr. Mirko Tarulli

E-Mail Website
Guest Editor
1. Mathematics and Science Department, American University in Bulgaria, 2700 Blagoevgrad, Bulgaria
2. Institute of Mathematics and Informatics, Bulgarian Academy of Science, 1113 Sofia, Bulgaria
3. Department of Mathematics, University of Pisa, 56127 Pisa, Italy
Interests: nonlinear dispersive equations; Maxwell and Schrödinger equations; wave equations; oscillatory integrals and micro-local analysis; harmonic analysis

Special Issue Information

Dear Colleagues,

The Special Issue of this scientific journal delves into the forefront of mathematical research, concentrating on nonlinear waves and evolution partial differential equations. Embracing cutting-edge concepts, it specifically spotlights six critical aspects: well-posedness, blow-up phenomena, scattering phenomena, stability/instability, and variational and geometrical approaches to studying PDEs. The intricate behavior of nonlinear waves and their interactions constitutes a fundamental topic worthy of exploration within this Special Issue. Articles may analyze the long-time behavior phenomena, investigating how waves evolve and disperse over time. The quest for stability within evolving systems takes center stage. The Special Issue probes deep into the concepts of stability/instability and scattering, inspecting how solutions of nonlinear wave equations behave over extended periods and different geometric frameworks. This examination not only refines theoretical frameworks but also unveils the stability constraints crucial for real-world applications. Addressing the foundational aspect of well-posedness, the Special Issue explores new formulation and solution behaviors of nonlinear evolution PDEs. Articles aim to establish robust frameworks that ensure the existence, uniqueness, and stability of solutions, providing a critical foundation for advancing mathematical models in diverse domains. Furthermore, variational and geometric approaches to solving nonlinear PDEs have emerged as highly significant, and this Special Issue aims to showcase its efficacy in providing elegant solutions and exploring the behavior of nonlinear systems settled on general ambient spaces.

Dr. Mirko Tarulli
Guest Editor

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 100 words) can be sent to the Editorial Office for announcement on this website.

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-blind 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

  • Scrödinger-type equations
  • wave-type equations
  • well-posedness
  • blow-up
  • scattering
  • stability
  • concentration–compactness
  • Hamiltonian systems
  • nonlinear PDEs
  • geometry

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (4 papers)

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Research

27 pages, 1190 KiB  
Article
Analysis of Mild Extremal Solutions in Nonlinear Caputo-Type Fractional Delay Difference Equations
by Ravi P. Agarwal and Ekaterina Madamlieva
Mathematics 2025, 13(8), 1321; https://doi.org/10.3390/math13081321 - 17 Apr 2025
Viewed by 120
Abstract
This study investigates extremal solutions for fractional-order delayed difference equations, utilizing the Caputo nabla operator to establish mild lower and upper approximations via discrete fractional calculus. A new approach is employed to demonstrate the uniform convergence of the sequences of lower and upper [...] Read more.
This study investigates extremal solutions for fractional-order delayed difference equations, utilizing the Caputo nabla operator to establish mild lower and upper approximations via discrete fractional calculus. A new approach is employed to demonstrate the uniform convergence of the sequences of lower and upper approximations within the monotone iterative scheme using the summation representation of the solutions, which serves as a discrete analogue to Volterra integral equations. This research highlights practical applications through numerical simulations in discrete bidirectional associative memory neural networks. Full article
(This article belongs to the Special Issue New Trends in Nonlinear Waves)
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11 pages, 224 KiB  
Article
Comparison Principle for Weakly Coupled Cooperative Parabolic Systems with Delays
by Georgi Boyadzhiev
Mathematics 2025, 13(8), 1230; https://doi.org/10.3390/math13081230 - 9 Apr 2025
Viewed by 145
Abstract
In this article, the validity of the comparison principle (CP) for weakly coupled quasi-linear cooperative systems with delays is proven. This is a powerful tool for studying the qualitative properties of the solutions. The CP is crucial in the proofs of the existence [...] Read more.
In this article, the validity of the comparison principle (CP) for weakly coupled quasi-linear cooperative systems with delays is proven. This is a powerful tool for studying the qualitative properties of the solutions. The CP is crucial in the proofs of the existence and uniqueness of weak solutions to cooperative reaction–diffusion systems presented here. Other direct consequences of the CP are the stability of the solution, the attenuation of long time periods, etc. An example model is given by spatial SEIR models with delays. They are suitable for modeling disease spread in space and time and can be described using a weakly coupled cooperative reaction–diffusion system. In this paper, spatial SEIR models with delays are considered in a continuous space. The emphasis is on the qualitative properties of the solutions, which are important for providing a mathematical basis for the model. Full article
(This article belongs to the Special Issue New Trends in Nonlinear Waves)
10 pages, 296 KiB  
Article
New Insights on Keller–Osserman Conditions for Semilinear Systems
by Dragos-Patru Covei
Mathematics 2025, 13(1), 83; https://doi.org/10.3390/math13010083 - 28 Dec 2024
Viewed by 565
Abstract
In this article, we consider a semilinear elliptic system involving gradient terms of the form [...] Read more.
In this article, we consider a semilinear elliptic system involving gradient terms of the form Δyx+λ1∇yx=pxfyx,zxifx∈Ω,Δzx+λ2∇zx=qxgyxifx∈Ω, where λ1, λ2∈0,∞, Ω is either a ball of radius R>0 or the entire space RN. Based on certain standard assumptions regarding the potential functions p and q, we introduce new conditions on the nonlinearities f and g to investigate the existence of entire large solutions for the given system. The method employed is successive approximation. Additionally, for specific cases of p, q, f and g, we employ Python code to plot the graph of both the numerical solution and the exact solution. Full article
(This article belongs to the Special Issue New Trends in Nonlinear Waves)
15 pages, 304 KiB  
Article
On Small Energy Solutions of the Nonlinear Schrödinger Equation in 1D with a Generic Trapping Potential with a Single Eigenvalue
by Scipio Cuccagna and Masaya Maeda
Mathematics 2024, 12(24), 3876; https://doi.org/10.3390/math12243876 - 10 Dec 2024
Viewed by 645
Abstract
We prove in dimension d=1 a result similar to a classical paper by Soffer and Weinstein, Jour. Diff. Eq. 98 (1992), improving it by encompassing for pure power nonlinearities the whole range of exponents p>1. The proof is [...] Read more.
We prove in dimension d=1 a result similar to a classical paper by Soffer and Weinstein, Jour. Diff. Eq. 98 (1992), improving it by encompassing for pure power nonlinearities the whole range of exponents p>1. The proof is based on the virial inequality of Kowalczyk et al., J. Eur. Math. Soc. (JEMS) 24 (2022), with smoothing estimates as shown in Mizumachi J. Math. Kyoto Univ. 48 (2008). Full article
(This article belongs to the Special Issue New Trends in Nonlinear Waves)
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