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Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications,...
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Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 16797

Special Issue Editor

Prof. Dr. Nikolai A. Kudryashov

E-Mail Website
Guest Editor
Department of Applied Mathematics at Moscow Engineering and Physics Institute (MEPhI), 31 Kashirskoe Shosse, 115409 Moscow, Russia
Interests: nonlinear mathematical model; differential equation; exact solution; dynamical system; Painlevé equation; Painlevé test; symbolic calculations; transformations; symmetry; Lie groups; dynamic chaos
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear partial differential equations are encountered in various fields of mathematics, physics, chemistry, and biology, and have numerous applications. Exact (closed form) solutions of differential equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural and engineering sciences. Exact solutions of nonlinear equations graphically demonstrate and enable the unraveling of the mechanisms of many complex nonlinear phenomena, such as the spatial localization of transfer processes, the multiplicity or absence of steady states under various conditions, the existence of peaking regimes, and the possible nonsmoothness or discontinuity of the sought quantities. It is important to note that exact solutions of the traveling-wave and self-similar solutions often represent the asymptotics of much wider classes of solutions corresponding to different initial and boundary conditions; this makes it possible to draw general conclusions and predict the dynamics of various nonlinear phenomena and processes. Even the special exact solutions that do not have a clear physical meaning can be used as “test problems” to verify the consistency and estimate errors of various numerical, asymptotic, and approximate analytical methods. Exact solutions can serve as a basis for perfecting and testing computer algebra software packages for solving partial differential equations. 

This Special Issue aims to collect original and significant contributions to both exact solutions and topics related to symmetries, reductions, analytical methods, and different applications of nonlinear PDEs. In addition, this Special Issue may serve as a platform for the exchange of ideas between scientists of different disciplines interested in nonlinear partial differential equations and partial functional differential equations.

Prof. Dr. Nikolai A. Kudryashov
Guest Editor

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Keywords

  • nonlinear partial differential equations
  • reaction–diffusion equations
  • wave type equations
  • higher-order nonlinear PDEs
  • partial differential equations with delay
  • partial functional differential equations
  • exact solutions
  • traveling-wave solutions
  • self-similar solutions
  • generalized separable solutions
  • functional separable solutions
  • classical symmetries
  • nonclassical symmetries
  • weak symmetries
  • symmetry reductions
  • differential constraints
  • Painlevé properties
  • analytical methods for PDEs
  • methods of computer algebra

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Related Special Issues

Published Papers (13 papers)

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Research

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10 pages, 264 KiB  
Article
Group Classification of the Unsteady Axisymmetric Boundary Layer Equation
by Alexander V. Aksenov and Anatoly A. Kozyrev
Mathematics 2024, 12(7), 988; https://doi.org/10.3390/math12070988 - 26 Mar 2024
Viewed by 820
Abstract
Unsteady equations of flat and axisymmetric boundary layers are considered. For the unsteady axisymmetric boundary layer equation, the problem of group classification is solved. It is shown that the kernel of symmetry operators can be extended by no more than four-dimensional Lie algebra. [...] Read more.
Unsteady equations of flat and axisymmetric boundary layers are considered. For the unsteady axisymmetric boundary layer equation, the problem of group classification is solved. It is shown that the kernel of symmetry operators can be extended by no more than four-dimensional Lie algebra. The kernel of symmetry operators of the unsteady flat boundary layer equation is found and it is shown that it can be extended by no more than a five-dimensional Lie algebra. The non-existence of the unsteady analogue of the Stepanov–Mangler transformation is proved. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
13 pages, 266 KiB  
Article
Symmetry Analysis of the Two-Dimensional Stationary Gas Dynamics Equations in Lagrangian Coordinates
by Sergey V. Meleshko and Evgeniy I. Kaptsov
Mathematics 2024, 12(6), 879; https://doi.org/10.3390/math12060879 - 16 Mar 2024
Cited by 3 | Viewed by 697
Abstract
This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates [...] Read more.
This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
9 pages, 335 KiB  
Article
Modeling of Nonlinear Sea Wave Modulation in the Presence of Ice Coverage
by A. V. Porubov and A. M. Krivtsov
Mathematics 2023, 11(23), 4805; https://doi.org/10.3390/math11234805 - 28 Nov 2023
Viewed by 871
Abstract
A model accounting for the influence of ice coverage on the propagation of surface sea waves is suggested. The model includes higher-order linear and nonlinear terms in the equation of wave motion. The asymptotic solution is obtained to account for nonlinear modulated wave [...] Read more.
A model accounting for the influence of ice coverage on the propagation of surface sea waves is suggested. The model includes higher-order linear and nonlinear terms in the equation of wave motion. The asymptotic solution is obtained to account for nonlinear modulated wave propagation and attenuation. Two kinds of attenuation are revealed. The influence of the higher-order nonlinear, dispersion, and dissipative terms on the shape and velocity of the modulated nonlinear wave is studied. Despite the presence of higher-order terms in the original equation, the modulated solitary wave solution contains free parameters, which is important for the possible generation of such waves. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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20 pages, 1268 KiB  
Article
Bifurcations of Phase Portraits, Exact Solutions and Conservation Laws of the Generalized Gerdjikov–Ivanov Model
by Nikolay A. Kudryashov, Sofia F. Lavrova and Daniil R. Nifontov
Mathematics 2023, 11(23), 4760; https://doi.org/10.3390/math11234760 - 24 Nov 2023
Cited by 5 | Viewed by 1156
Abstract
This article explores the generalized Gerdjikov–Ivanov equation describing the propagation of pulses in optical fiber. The equation studied has a variety of applications, for instance, in photonic crystal fibers. In contrast to the classical Gerdjikov–Ivanov equation, the solution of the Cauchy problem for [...] Read more.
This article explores the generalized Gerdjikov–Ivanov equation describing the propagation of pulses in optical fiber. The equation studied has a variety of applications, for instance, in photonic crystal fibers. In contrast to the classical Gerdjikov–Ivanov equation, the solution of the Cauchy problem for the studied equation cannot be found by the inverse scattering problem method. In this regard, analytical solutions for the generalized Gerdjikov–Ivanov equation are found using traveling-wave variables. Phase portraits of an ordinary differential equation corresponding to the partial differential equation under consideration are constructed. Three conservation laws for the generalized equation corresponding to power conservation, moment and energy are found by the method of direct transformations. Conservative densities corresponding to optical solitons of the generalized Gerdjikov–Ivanov equation are provided. The conservative quantities obtained have not been presented before in the literature, to the best of our knowledge. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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20 pages, 2871 KiB  
Article
Linear Stability of Filtration Flow of a Gas and Two Immiscible Liquids with Interfaces
by Vladimir Shargatov, George Tsypkin and Polina Kozhurina
Mathematics 2023, 11(21), 4476; https://doi.org/10.3390/math11214476 - 28 Oct 2023
Viewed by 893
Abstract
The stability of the vertical flow that occurs when gas displaces oil from a reservoir is investigated. It is assumed that the oil and gas areas are separated by a layer saturated with water. This method of oil displacement, called water-alternating-gas injection, improves [...] Read more.
The stability of the vertical flow that occurs when gas displaces oil from a reservoir is investigated. It is assumed that the oil and gas areas are separated by a layer saturated with water. This method of oil displacement, called water-alternating-gas injection, improves the oil recovery process. We consider the linear stability of two boundaries that are flat at the initial moment, separating, respectively, the areas of gas and water, as well as water and oil. The instability of the interfaces can result in gas and water fingers penetrating into the oil-saturated area and causing residual oil. Two cases of perturbation evolution are considered. In the first case, only the gas–water interface is perturbed at the initial moment, and in the second case, small perturbations of the same amplitude are present on both surfaces. It is shown that the interaction of perturbations at interfaces depends on the thickness of the water-saturated layer, perturbation wavelength, oil viscosity, pressure gradient and formation thickness. Calculations show that perturbations at the oil–water boundary grow much slower than perturbations at the gas–water boundary. It was found that, with other parameters fixed, there is a critical (or threshold) value of the thickness of the water-saturated layer, above which the development of perturbations at the gas–water boundary does not affect the development of perturbations at the water–oil boundary. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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11 pages, 633 KiB  
Article
Some Latest Families of Exact Solutions to Date–Jimbo–Kashiwara–Miwa Equation and Its Stability Analysis
by Arzu Akbulut, Rubayyi T. Alqahtani and Nadiyah Hussain Alharthi
Mathematics 2023, 11(19), 4176; https://doi.org/10.3390/math11194176 - 6 Oct 2023
Cited by 4 | Viewed by 1137
Abstract
The present study demonstrates the derivation of new analytical solutions for the Date–Jimbo–Kashiwara–Miwa equation utilizing two distinct methodologies, specifically the modified Kudryashov technique and the (g′)-expansion procedure. These innovative concepts employ symbolic computations to provide a dynamic and robust mathematical procedure [...] Read more.
The present study demonstrates the derivation of new analytical solutions for the Date–Jimbo–Kashiwara–Miwa equation utilizing two distinct methodologies, specifically the modified Kudryashov technique and the (g′)-expansion procedure. These innovative concepts employ symbolic computations to provide a dynamic and robust mathematical procedure for addressing a range of nonlinear wave situations. Additionally, a comprehensive stability analysis is performed, and the acquired results are visually represented through graphical representations. A comparison between the discovered solutions and those already found in the literature has also been performed. It is anticipated that the solutions will contribute to the existing literature related to mathematical physics and soliton theory. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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17 pages, 920 KiB  
Article
Painlevé Test, Phase Plane Analysis and Analytical Solutions of the Chavy–Waddy–Kolokolnikov Model for the Description of Bacterial Colonies
by Nikolay A. Kudryashov and Sofia F. Lavrova
Mathematics 2023, 11(14), 3203; https://doi.org/10.3390/math11143203 - 21 Jul 2023
Cited by 9 | Viewed by 1149
Abstract
The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equation which corresponds to the fourth-order partial differential equation. The restrictions on [...] Read more.
The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equation which corresponds to the fourth-order partial differential equation. The restrictions on the mathematical model parameters for ordinary differential equations to pass the Painlevé test are obtained. It is determined that the method of the inverse scattering transform does not solve the Cauchy problem for the original mathematical model, since the corresponding nonlinear ordinary differential equation passes the Painlevé test only when its solution is stationary. In the case of the stationary solution, the first integral of the equation is obtained, which makes it possible to represent the general solution in the quadrature form. The stability of the stationary points of the investigated mathematical model is carried out and their classification is proposed. Periodic and solitary stationary solutions of the Chavy–Waddy–Kolokolnikov model are constructed for various parameter values. To build analytical solutions, the method of the simplest equations is also used. The solutions, obtained in the form of a truncated expansion in powers of the logistic function, are represented as a closed formula using the formula for the Newton binomial. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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12 pages, 274 KiB  
Article
Hamiltonians of the Generalized Nonlinear Schrödinger Equations
by Nikolay A. Kudryashov
Mathematics 2023, 11(10), 2304; https://doi.org/10.3390/math11102304 - 15 May 2023
Cited by 11 | Viewed by 1424
Abstract
Some types of the generalized nonlinear Schrödinger equation of the second, fourth and sixth order are considered. The Cauchy problem for equations in the general case cannot be solved by the inverse scattering transform. The main objective of this paper is to find [...] Read more.
Some types of the generalized nonlinear Schrödinger equation of the second, fourth and sixth order are considered. The Cauchy problem for equations in the general case cannot be solved by the inverse scattering transform. The main objective of this paper is to find the conservation laws of the equations using their transformations. The algorithmic method for finding Hamiltonians of some equations is presented. This approach allows us to look for Hamiltonians without the derivative operator and it can be applied with the aid of programmes of symbolic calculations. The Hamiltonians of three types of the generalized nonlinear Schrödinger equation are found. Examples of Hamiltonians for some equations are presented. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
14 pages, 3821 KiB  
Article
Optical Solitons for the Concatenation Model with Differential Group Delay: Undetermined Coefficients
by Anjan Biswas, Jose Vega-Guzman, Yakup Yıldırım, Luminita Moraru, Catalina Iticescu and Abdulah A. Alghamdi
Mathematics 2023, 11(9), 2012; https://doi.org/10.3390/math11092012 - 24 Apr 2023
Cited by 28 | Viewed by 1432
Abstract
In the current study, the concatenation model of birefringent fibers is explored for the first time, and we present optical soliton solutions to the model. The integration algorithm used to achieve this retrieval is the method of undetermined coefficients, which yields a wide [...] Read more.
In the current study, the concatenation model of birefringent fibers is explored for the first time, and we present optical soliton solutions to the model. The integration algorithm used to achieve this retrieval is the method of undetermined coefficients, which yields a wide range of soliton solutions. The parameter constraints arise naturally during the derivation of the soliton solutions, which are essential for such solitons to exist. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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18 pages, 351 KiB  
Article
On Construction of Partially Dimension-Reduced Approximations for Nonstationary Nonlocal Problems of a Parabolic Type
by Raimondas Čiegis, Vadimas Starikovičius, Olga Suboč and Remigijus Čiegis
Mathematics 2023, 11(9), 1984; https://doi.org/10.3390/math11091984 - 22 Apr 2023
Cited by 1 | Viewed by 1190
Abstract
The main aim of this article is to propose an adaptive method to solve multidimensional parabolic problems with fractional power elliptic operators. The adaptivity technique is based on a very efficient method when the multidimensional problem is approximated by a partially dimension-reduced mathematical [...] Read more.
The main aim of this article is to propose an adaptive method to solve multidimensional parabolic problems with fractional power elliptic operators. The adaptivity technique is based on a very efficient method when the multidimensional problem is approximated by a partially dimension-reduced mathematical model. Then in the greater part of the domain, only one-dimensional problems are solved. For the first time such a technique is applied for problems with nonlocal diffusion operators. It is well known that, even for classical local diffusion operators, the averaged flux conjugation conditions become nonlocal. Efficient finite volume type discrete schemes are constructed and analysed. The stability and accuracy of obtained local discrete schemes is investigated. The results of computational experiments are presented and compared with theoretical results. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
11 pages, 466 KiB  
Article
New Solitary Wave Patterns of the Fokas System in Fiber Optics
by Melike Kaplan, Arzu Akbulut and Rubayyi T. Alqahtani
Mathematics 2023, 11(8), 1810; https://doi.org/10.3390/math11081810 - 11 Apr 2023
Cited by 10 | Viewed by 1397
Abstract
The Fokas system, which models wave dynamics using a single model of fiber optics, is the design under discussion in this study. Different types of solitary wave solutions are obtained by utilizing generalized Kudryashov (GKP) and modified Kudryashov procedures (MKP). These novel concepts [...] Read more.
The Fokas system, which models wave dynamics using a single model of fiber optics, is the design under discussion in this study. Different types of solitary wave solutions are obtained by utilizing generalized Kudryashov (GKP) and modified Kudryashov procedures (MKP). These novel concepts make use of symbolic computations to come up with a dynamic and powerful mathematical approach for dealing with a variety of nonlinear wave situations. The results obtained in this paper are original and have the potential to be useful in mathematical physics. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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25 pages, 568 KiB  
Article
Quiescent Optical Solitons for the Concatenation Model with Nonlinear Chromatic Dispersion
by Yakup Yıldırım, Anjan Biswas, Luminita Moraru and Abdulah A. Alghamdi
Mathematics 2023, 11(7), 1709; https://doi.org/10.3390/math11071709 - 3 Apr 2023
Cited by 35 | Viewed by 1636
Abstract
This paper recovers quiescent optical solitons that are self-sustaining, localized wave packets that maintain their shape and amplitude over long distances due to a balance between nonlinearity and dispersion. When a soliton is in a state of quiescence, it means that it is [...] Read more.
This paper recovers quiescent optical solitons that are self-sustaining, localized wave packets that maintain their shape and amplitude over long distances due to a balance between nonlinearity and dispersion. When a soliton is in a state of quiescence, it means that it is stationary in both space and time. Quiescent optical solitons are typically observed in optical fibers, where nonlinearity and dispersion can lead to the formation of solitons. The concatenation model is considered to understand the behavior of optical pulses propagating through nonlinear media. Here, we consider the familiar nonlinear Schrödinger equation, the Lakshmanan–Porsezian–Daniel equation, and the Sasa–Satsuma equation. The current paper also addresses the model with nonlinear chromatic dispersion, a phenomenon that occurs in optical fibers and other dispersive media, where the chromatic dispersion of the material is modified by nonlinear effects. In the presence of nonlinearities, such as self-phase modulation and cross-phase modulation, the chromatic dispersion coefficient becomes a function of the optical intensity, resulting in nonlinear chromatic dispersion. A full spectrum of stationary optical solitons, along with straddled stationary solitons, are obtained. There are four integration schemes that made this retrieval possible. The numerical simulations are also included for these solitons. The parameter constraints also indicate the existence criteria for these quiescent solitons. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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Review

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57 pages, 754 KiB  
Review
Roadmap of the Multiplier Method for Partial Differential Equations
by Juan Arturo Alvarez-Valdez and Guillermo Fernandez-Anaya
Mathematics 2023, 11(22), 4572; https://doi.org/10.3390/math11224572 - 7 Nov 2023
Viewed by 1794
Abstract
This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking [...] Read more.
This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)
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