Solutions of Integrable PDEs: Solving, Properties and Applications

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

Deadline for manuscript submissions: closed (30 November 2024) | Viewed by 1418

Special Issue Editors


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Guest Editor
Department of Mathematics and Statistics, University of Southern Maine, Portland, OR 9300, USA
Interests: mathematical analysis; partial differential equations; fractional calculus

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Guest Editor
Department of Electrical and Computer Engineering, University of Manitoba, 75A Chancellor’s Circle, Winnipeg, MB R3T 5V6, Canada
Interests: linear systems; signals and in computational geometry (cells, paths, cycles); proximity (closeness of fixed sets); algebraic topology (free group & homotopy theory); physics (vector fields)

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Guest Editor
Department of Mathematics, Savitribai Phule Pune University, Pune 411007, India
Interests: integral equations; differential equations; integral inequalities

Special Issue Information

Dear Colleagues,

Integrable PDEs (partial differential equations) are a special class of PDEs that can be solved exactly, meaning that an explicit expression for the solution can be obtained. This contrasts with non-integrable PDEs, where exact solutions are generally not possible, and numerical methods must be used.

Integrable PDEs have been studied extensively over the past several decades, and many powerful mathematical methods have been developed for their analysis. These methods include the inverse scattering transform, the inverse spectral method (including Riemann–Hilbert problems), the method of dressing transformations, the Painlevé test, and the Lax pair method.

Integrable PDEs have important applications in many areas of science and engineering, including fluid mechanics, quantum field theory, nonlinear optics, and soliton theory. Some examples of integrable PDEs include the Korteweg–de Vries equation, the nonlinear Schrödinger equation, and the sine-Gordon equation. The study of integrable PDEs has led to many important insights into the nature of nonlinear phenomena and has opened up new avenues for research in a wide range of fields.

Dr. Hayman Thabet
Prof. Dr. James F Peters
Prof. Dr. Subhash Kendre
Guest Editors

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Keywords

  • integrable PDEs including: Korteweg–de Vries equation
  • nonlinear Schrödinger equation
  • Sine-Gordon equation
  • Boussinesq equation
  • Kadomtsev–Petviashvili equation
  • mathematical methods for solving the integrable PDEs include: inverse scattering transform
  • Riemann–Hilbert problem
  • Lax pair method
  • inverse spectral method

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Published Papers (1 paper)

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Research

18 pages, 545 KiB  
Article
Certain Novel Dynamic Inequalities Applicable in the Theory of Retarded Dynamic Equations and Their Applications
by Sujata Bhamre, Nagesh Kale, Subhash Kendre and James Peters
Mathematics 2024, 12(3), 406; https://doi.org/10.3390/math12030406 - 26 Jan 2024
Cited by 1 | Viewed by 959
Abstract
In this article, we establish certain time-scale-retarded dynamic inequalities that contain nonlinear retarded integral equations on various time scales. These inequalities extend and generalize some significant inequalities existing in the literature to their more general forms. The qualitative and quantitative characteristics of solutions [...] Read more.
In this article, we establish certain time-scale-retarded dynamic inequalities that contain nonlinear retarded integral equations on various time scales. These inequalities extend and generalize some significant inequalities existing in the literature to their more general forms. The qualitative and quantitative characteristics of solutions to various dynamic equations on time scales involving retarded integrals can be studied using these inequalities. The results presented in this manuscript furnish a powerful tool to analyze the boundedness of nonlinear integral equations with retarded integrals on several time scales. In the end, we also include numerical illustrations to signify the applicability of these results to power nonlinear retarded integral equations on real and quantum time scales. Full article
(This article belongs to the Special Issue Solutions of Integrable PDEs: Solving, Properties and Applications)
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