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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

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

Deadline for manuscript submissions: closed (31 August 2022) | Viewed by 45845

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Nikolai A. Kudryashov

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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
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Special Issue Information

Dear Colleagues,

Nonlinear partial differential equations are encountered in various fields of mathematics, physics, chemistry, and biology, and 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 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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Published Papers (19 papers)

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10 pages, 1936 KiB  
Article
Modeling of Mechanisms of Wave Formation for COVID-19 Epidemic
by Alexander Leonov, Oleg Nagornov and Sergey Tyuflin
Mathematics 2023, 11(1), 167; https://doi.org/10.3390/math11010167 - 29 Dec 2022
Cited by 8 | Viewed by 1870
Abstract
Two modifications with variable coefficients of the well-known SEIR model for epidemic development in the application to the modeling of the infection curves of COVID-19 are considered. The data for these models are information on the number of infections each day obtained from [...] Read more.
Two modifications with variable coefficients of the well-known SEIR model for epidemic development in the application to the modeling of the infection curves of COVID-19 are considered. The data for these models are information on the number of infections each day obtained from the Johns Hopkins Coronavirus Resource Center database. In our paper, we propose special methods based on Tikhonov regularization for models’ identification on the class of piecewise constant coefficients. In contrast to the model with constant coefficients, which cannot always accurately describe some of infection curves, the first model is able to approximate them for different countries with an accuracy of 2–8%. The second model considered in the article takes into account external sources of infection in the form of an inhomogeneous term in one of the model equations and is able to approximate the data with a slightly better accuracy of 2–4%. For the second model, we also consider the possibility of using other input data, namely the number of infected people per day. Such data are used to model infection curves for several waves of the COVID-19 epidemic, including part of the Omicron wave. Numerical experiments carried out for a number of countries show that the waves of external sources of infection found are ahead of the wave of infection by 10 or more days. At the same time, other piecewise constant coefficients of the model change relatively slowly. These models can be applied fairly reliably to approximate many waves of infection curves with high precision and can be used to identify external and hidden sources of infection. This is the advantage of our models. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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18 pages, 481 KiB  
Article
On Convergence of Support Operator Method Schemes for Differential Rotational Operations on Tetrahedral Meshes Applied to Magnetohydrodynamic Problems
by Yury Poveshchenko, Viktoriia Podryga and Parvin Rahimly
Mathematics 2022, 10(20), 3904; https://doi.org/10.3390/math10203904 - 20 Oct 2022
Viewed by 1134
Abstract
The problem of constructing and justifying the discrete algorithms of the support operator method for numerical modeling of differential repeated rotational operations of vector analysis (curlcurl) in application to problems of magnetohydrodynamics is considered. [...] Read more.
The problem of constructing and justifying the discrete algorithms of the support operator method for numerical modeling of differential repeated rotational operations of vector analysis (curlcurl) in application to problems of magnetohydrodynamics is considered. Difference schemes of the support operator method on the unstructured meshes do not approximate equations in the local sense. Therefore, it is necessary to prove the convergence of these schemes to the exact solution, which is possible after analyzing the error structure of their approximation. For this analysis, a decomposition of the space of mesh vector functions into an orthogonal direct sum of subspaces of potential and vortex fields is introduced. Generalized centroid-tensor metric representations of repeated operations of tensor analysis (div, grad, and curl) are constructed. Representations have flux-circulation properties that are integrally consistent on spatial meshes of irregular structure. On smooth solutions of the model magnetostatic problem on a tetrahedral mesh with the first order of accuracy in the rms sense, the convergence of the constructed difference schemes is proved. The algorithms constructed in this work can be used to solve physical problems with discontinuous magnetic viscosity, dielectric permittivity, or thermal resistance of the medium. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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19 pages, 2719 KiB  
Article
Mathematical Modeling of Gas Hydrates Dissociation in Porous Media with Water-Ice Phase Transformations Using Differential Constrains
by Natalia Alekseeva, Viktoriia Podryga, Parvin Rahimly, Richard Coffin and Ingo Pecher
Mathematics 2022, 10(19), 3470; https://doi.org/10.3390/math10193470 - 23 Sep 2022
Cited by 1 | Viewed by 1369
Abstract
2D numerical modeling algorithms of multi-component, multi-phase filtration processes of mass transfer in frost-susceptible rocks using nonlinear partial differential equations are a valuable tool for problems of subsurface hydrodynamics considering the presence of free gas, free water, gas hydrates, ice formation and phase [...] Read more.
2D numerical modeling algorithms of multi-component, multi-phase filtration processes of mass transfer in frost-susceptible rocks using nonlinear partial differential equations are a valuable tool for problems of subsurface hydrodynamics considering the presence of free gas, free water, gas hydrates, ice formation and phase transitions. In this work, a previously developed one-dimensional numerical modeling approach is modified and 2D algorithms are formulated through means of the support-operators method (SOM) and presented for the entire area of the process extension. The SOM is used to generalize the method of finite difference for spatially irregular grids case. The approach is useful for objects where a lithological heterogeneity of rocks has a big influence on formation and accumulation of gas hydrates and therefore it allows to achieve a sufficiently good spatial approximation for numerical modeling of objects related to gas hydrates dissociation in porous media. The modeling approach presented here consistently applies the method of physical process splitting which allows to split the system into dissipative equation and hyperbolic unit. The governing variables were determined in flow areas of the hydrate equilibrium zone by applying the Gibbs phase rule. The problem of interaction of a vertical fault and horizontal formation containing gas hydrates was investigated and test calculations were done for understanding of influence of thermal effect of the fault on the formation fluid dynamic. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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9 pages, 260 KiB  
Article
Optical Solitons of the Generalized Nonlinear Schrödinger Equation with Kerr Nonlinearity and Dispersion of Unrestricted Order
by Nikolay A. Kudryashov
Mathematics 2022, 10(18), 3409; https://doi.org/10.3390/math10183409 - 19 Sep 2022
Cited by 17 | Viewed by 1810
Abstract
The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé test is applied for finding arbitrary constants in the expansion of the general solution into [...] Read more.
The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé test is applied for finding arbitrary constants in the expansion of the general solution into the Laurent series. It is shown that the equation does not pass the Painlevé test but has two arbitrary constants in local expansion. This fact allows us to look for solitary wave solutions for equations of unrestricted order. The main result of this paper is the theorem of existence of optical solitons for equations of unrestricted order that is proved by direct calculation. The optical solitons for partial differential equations of the twelfth order are given in detail. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
16 pages, 479 KiB  
Article
On the Short Wave Instability of the Liquid/Gas Contact Surface in Porous Media
by Vladimir A. Shargatov, George G. Tsypkin, Sergey V. Gorkunov, Polina I. Kozhurina and Yulia A. Bogdanova
Mathematics 2022, 10(17), 3177; https://doi.org/10.3390/math10173177 - 3 Sep 2022
Cited by 1 | Viewed by 1836
Abstract
We consider a problem of hydrodynamic stability of the liquid displacement by gas in a porous medium in the case when a light gas is located above the liquid. The onset of instability and the evolution of the small shortwave perturbations are investigated. [...] Read more.
We consider a problem of hydrodynamic stability of the liquid displacement by gas in a porous medium in the case when a light gas is located above the liquid. The onset of instability and the evolution of the small shortwave perturbations are investigated. We show that when using the Darcy filtration law, the onset of instability may take place at an infinitely large wavenumber when the normal modes method is inapplicable. The results of numerical simulation of the nonlinear problem indicate that the anomalous growth of the amplitude of shortwave small perturbations persists, but the growth rate of amplitude decreases significantly compared to the results of linear analysis. An analysis of the stability of the gas/liquid interface is also carried out using a network model of a porous medium. It is shown that the results of surface evolution calculations obtained using the network model are in qualitative agreement with the results of the continual approach, but the continual model predicts a higher velocity of the interfacial surfaces in the capillaries. The growth rate of perturbations in the network model also increases with decreasing perturbation wavelength at a constant amplitude. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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11 pages, 266 KiB  
Article
Highly Dispersive Optical Solitons in Fiber Bragg Gratings with Kerr Law of Nonlinear Refractive Index
by Elsayed M. E. Zayed, Mohamed E. M. Alngar, Reham M. A. Shohib, Anjan Biswas, Yakup Yıldırım, Salam Khan, Luminita Moraru, Simona Moldovanu and Catalina Iticescu
Mathematics 2022, 10(16), 2968; https://doi.org/10.3390/math10162968 - 17 Aug 2022
Cited by 3 | Viewed by 1360
Abstract
This paper obtains highly dispersive optical solitons in fiber Bragg gratings with the Kerr law of a nonlinear refractive index. The generalized Kudryashov’s approach as well as its newer version makes this retrieval possible. A full spectrum of solitons is thus recovered. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
46 pages, 4783 KiB  
Article
Bifurcation Theory, Lie Group-Invariant Solutions of Subalgebras and Conservation Laws of a Generalized (2+1)-Dimensional BK Equation Type II in Plasma Physics and Fluid Mechanics
by Oke Davies Adeyemo, Lijun Zhang and Chaudry Masood Khalique
Mathematics 2022, 10(14), 2391; https://doi.org/10.3390/math10142391 - 7 Jul 2022
Cited by 15 | Viewed by 1674
Abstract
The nonlinear phenomena in numbers are modelled in a wide range of fields such as chemical physics, ocean physics, optical fibres, plasma physics, fluid dynamics, solid-state physics, biological physics and marine engineering. This research article systematically investigates a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. We [...] Read more.
The nonlinear phenomena in numbers are modelled in a wide range of fields such as chemical physics, ocean physics, optical fibres, plasma physics, fluid dynamics, solid-state physics, biological physics and marine engineering. This research article systematically investigates a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. We achieve a five-dimensional Lie algebra of the equation through Lie group analysis. This, in turn, affords us the opportunity to compute an optimal system of fourteen-dimensional Lie subalgebras related to the underlying equation. As a consequence, the various subalgebras are engaged in performing symmetry reductions of the equation leading to many solvable nonlinear ordinary differential equations. Thus, we secure different types of solitary wave solutions including periodic (Weierstrass and elliptic integral), topological kink and anti-kink, complex, trigonometry and hyperbolic functions. Moreover, we utilize the bifurcation theory of dynamical systems to obtain diverse nontrivial travelling wave solutions consisting of both bounded as well as unbounded solution-types to the equation under consideration. Consequently, we generate solutions that are algebraic, periodic, constant and trigonometric in nature. The various results gained in the study are further analyzed through numerical simulation. Finally, we achieve conservation laws of the equation under study by engaging the standard multiplier method with the inclusion of the homotopy integral formula related to the obtained multipliers. In addition, more conserved currents of the equation are secured through Noether’s theorem. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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22 pages, 2781 KiB  
Article
Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering
by Aly R. Seadawy, Hanadi Zahed and Syed T. R. Rizvi
Mathematics 2022, 10(11), 1818; https://doi.org/10.3390/math10111818 - 25 May 2022
Cited by 6 | Viewed by 1963
Abstract
This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations [...] Read more.
This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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12 pages, 1201 KiB  
Article
Highly Dispersive Optical Solitons in Birefringent Fibers with Polynomial Law of Nonlinear Refractive Index by Laplace–Adomian Decomposition
by Oswaldo González-Gaxiola, Anjan Biswas, Yakup Yıldırım and Luminita Moraru
Mathematics 2022, 10(9), 1589; https://doi.org/10.3390/math10091589 - 7 May 2022
Cited by 5 | Viewed by 5757
Abstract
This paper is a numerical simulation of highly dispersive optical solitons in birefringent fibers with polynomial nonlinear form, which is achieved for the first time. The algorithmic approach is applied with the usage of the Laplace–Adomian decomposition scheme. Dark and bright soliton simulations [...] Read more.
This paper is a numerical simulation of highly dispersive optical solitons in birefringent fibers with polynomial nonlinear form, which is achieved for the first time. The algorithmic approach is applied with the usage of the Laplace–Adomian decomposition scheme. Dark and bright soliton simulations are presented. The error measure has a very low count, and thus, the simulations are almost an exact replica of such solitons that analytically arise from the governing system. The suggested iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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11 pages, 1242 KiB  
Article
Highly Dispersive Optical Soliton Perturbation, with Maximum Intensity, for the Complex Ginzburg–Landau Equation by Semi-Inverse Variation
by Anjan Biswas, Trevor Berkemeyer, Salam Khan, Luminita Moraru, Yakup Yıldırım and Hashim M. Alshehri
Mathematics 2022, 10(6), 987; https://doi.org/10.3390/math10060987 - 18 Mar 2022
Cited by 9 | Viewed by 2812
Abstract
This work analytically recovers the highly dispersive bright 1–soliton solution using for the perturbed complex Ginzburg–Landau equation, which is studied with three forms of nonlinear refractive index structures. They are Kerr law, parabolic law, and polynomial law. The perturbation terms appear with maximum [...] Read more.
This work analytically recovers the highly dispersive bright 1–soliton solution using for the perturbed complex Ginzburg–Landau equation, which is studied with three forms of nonlinear refractive index structures. They are Kerr law, parabolic law, and polynomial law. The perturbation terms appear with maximum allowable intensity, also known as full nonlinearity. The semi-inverse variational principle makes this retrieval possible. The amplitude–width relation is obtained by solving a cubic polynomial equation using Cardano’s approach. The parameter constraints for the existence of such solitons are also enumerated. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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18 pages, 1044 KiB  
Article
Exact Traveling Waves of a Generalized Scale-Invariant Analogue of the Korteweg–de Vries Equation
by Lewa’ Alzaleq, Valipuram Manoranjan and Baha Alzalg
Mathematics 2022, 10(3), 414; https://doi.org/10.3390/math10030414 - 28 Jan 2022
Cited by 8 | Viewed by 2047
Abstract
In this paper, we study a generalized scale-invariant analogue of the well-known Korteweg–de Vries (KdV) equation. This generalized equation can be thought of as a bridge between the KdV equation and the SIdV equation that was discovered recently, and shares the same one-soliton [...] Read more.
In this paper, we study a generalized scale-invariant analogue of the well-known Korteweg–de Vries (KdV) equation. This generalized equation can be thought of as a bridge between the KdV equation and the SIdV equation that was discovered recently, and shares the same one-soliton solution as the KdV equation. By employing the auxiliary equation method, we are able to obtain a wide variety of traveling wave solutions, both bounded and singular, which are kink and bell types, periodic waves, exponential waves, and peaked (peakon) waves. As far as we know, these solutions are new and their explicit closed-form expressions have not been reported elsewhere in the literature. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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21 pages, 1229 KiB  
Article
Diverse Multiple Lump Analytical Solutions for Ion Sound and Langmuir Waves
by Abdulmohsen D. Alruwaili, Aly R. Seadawy, Syed T. R. Rizvi and Sid Ahmed O. Beinane
Mathematics 2022, 10(2), 200; https://doi.org/10.3390/math10020200 - 10 Jan 2022
Cited by 20 | Viewed by 1596
Abstract
In this work, we study a time-fractional ion sound and Langmuir waves system (FISLWS) with Atangana–Baleanu derivative (ABD). We use a fractional ABD operator to transform our system into an ODE. We investigate multiwaves, periodic cross-kink, rational, and interaction solutions by the combination [...] Read more.
In this work, we study a time-fractional ion sound and Langmuir waves system (FISLWS) with Atangana–Baleanu derivative (ABD). We use a fractional ABD operator to transform our system into an ODE. We investigate multiwaves, periodic cross-kink, rational, and interaction solutions by the combination of rational, trigonometric, and various bilinear functions. Furthermore, 3D, 2D, and relevant contour plots are presented for the natural evolution of the gained solutions under the selection of proper parameters. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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11 pages, 1365 KiB  
Article
Numerical Simulation of Cubic-Quartic Optical Solitons with Perturbed Fokas–Lenells Equation Using Improved Adomian Decomposition Algorithm
by Alyaa A. Al-Qarni, Huda O. Bakodah, Aisha A. Alshaery, Anjan Biswas, Yakup Yıldırım, Luminita Moraru and Simona Moldovanu
Mathematics 2022, 10(1), 138; https://doi.org/10.3390/math10010138 - 4 Jan 2022
Cited by 8 | Viewed by 1805
Abstract
The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known [...] Read more.
The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known results in the literature. As proven, the proposed scheme is efficient and possess a high level of accuracy. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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17 pages, 777 KiB  
Article
Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering
by Chaudry Masood Khalique and Karabo Plaatjie
Mathematics 2022, 10(1), 24; https://doi.org/10.3390/math10010024 - 22 Dec 2021
Cited by 5 | Viewed by 2577
Abstract
In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact [...] Read more.
In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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19 pages, 291 KiB  
Article
Highly Dispersive Optical Solitons with Complex Ginzburg–Landau Equation Having Six Nonlinear Forms
by Elsayed M. E. Zayed, Khaled A. Gepreel, Mahmoud El-Horbaty, Anjan Biswas, Yakup Yıldırım and Hashim M. Alshehri
Mathematics 2021, 9(24), 3270; https://doi.org/10.3390/math9243270 - 16 Dec 2021
Cited by 23 | Viewed by 2273
Abstract
This paper retrieves highly dispersive optical solitons to complex Ginzburg–Landau equation having six forms of nonlinear refractive index structures for the very first time. The enhanced version of the Kudryashov approach is the adopted integration tool. Thus, bright and singular soliton solutions emerge [...] Read more.
This paper retrieves highly dispersive optical solitons to complex Ginzburg–Landau equation having six forms of nonlinear refractive index structures for the very first time. The enhanced version of the Kudryashov approach is the adopted integration tool. Thus, bright and singular soliton solutions emerge from the scheme that are exhibited with their respective parameter constraints. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
10 pages, 333 KiB  
Article
Modeling the Dynamics of Spiking Networks with Memristor-Based STDP to Solve Classification Tasks
by Alexander Sboev, Danila Vlasov, Roman Rybka, Yury Davydov, Alexey Serenko and Vyacheslav Demin
Mathematics 2021, 9(24), 3237; https://doi.org/10.3390/math9243237 - 14 Dec 2021
Cited by 12 | Viewed by 3033
Abstract
The problem with training spiking neural networks (SNNs) is relevant due to the ultra-low power consumption these networks could exhibit when implemented in neuromorphic hardware. The ongoing progress in the fabrication of memristors, a prospective basis for analogue synapses, gives relevance to studying [...] Read more.
The problem with training spiking neural networks (SNNs) is relevant due to the ultra-low power consumption these networks could exhibit when implemented in neuromorphic hardware. The ongoing progress in the fabrication of memristors, a prospective basis for analogue synapses, gives relevance to studying the possibility of SNN learning on the base of synaptic plasticity models, obtained by fitting the experimental measurements of the memristor conductance change. The dynamics of memristor conductances is (necessarily) nonlinear, because conductance changes depend on the spike timings, which neurons emit in an all-or-none fashion. The ability to solve classification tasks was previously shown for spiking network models based on the bio-inspired local learning mechanism of spike-timing-dependent plasticity (STDP), as well as with the plasticity that models the conductance change of nanocomposite (NC) memristors. Input data were presented to the network encoded into the intensities of Poisson input spike sequences. This work considers another approach for encoding input data into input spike sequences presented to the network: temporal encoding, in which an input vector is transformed into relative timing of individual input spikes. Since temporal encoding uses fewer input spikes, the processing of each input vector by the network can be faster and more energy-efficient. The aim of the current work is to show the applicability of temporal encoding to training spiking networks with three synaptic plasticity models: STDP, NC memristor approximation, and PPX memristor approximation. We assess the accuracy of the proposed approach on several benchmark classification tasks: Fisher’s Iris, Wisconsin breast cancer, and the pole balancing task (CartPole). The accuracies achieved by SNN with memristor plasticity and conventional STDP are comparable and are on par with classic machine learning approaches. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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9 pages, 374 KiB  
Article
Implicit Solitary Waves for One of the Generalized Nonlinear Schrödinger Equations
by Nikolay A. Kudryashov
Mathematics 2021, 9(23), 3024; https://doi.org/10.3390/math9233024 - 25 Nov 2021
Cited by 62 | Viewed by 2908
Abstract
Application of transformations for dependent and independent variables is used for finding solitary wave solutions of the generalized Schrödinger equations. This new form of equation can be considered as the model for the description of propagation pulse in a nonlinear optics. The method [...] Read more.
Application of transformations for dependent and independent variables is used for finding solitary wave solutions of the generalized Schrödinger equations. This new form of equation can be considered as the model for the description of propagation pulse in a nonlinear optics. The method for finding solutions of equation is given in the general case. Solitary waves of equation are obtained as implicit function taking into account the transformation of variables. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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19 pages, 5358 KiB  
Article
Multicriteria Optimization of a Dynamic System by Methods of the Theories of Similarity and Criteria Importance
by Sergey Misyurin, German Kreynin, Andrey Nelyubin and Natalia Nosova
Mathematics 2021, 9(22), 2854; https://doi.org/10.3390/math9222854 - 11 Nov 2021
Cited by 5 | Viewed by 1927
Abstract
The problem of multicriteria optimization of a dynamic model is solved using the methods of the similarity theory and the criteria importance theory. The authors propose the original model of a positional system with two hydraulic actuators, synchronously moving a heavy object with [...] Read more.
The problem of multicriteria optimization of a dynamic model is solved using the methods of the similarity theory and the criteria importance theory. The authors propose the original model of a positional system with two hydraulic actuators, synchronously moving a heavy object with a given accuracy. In order to reduce the number of optimizing parameters, the mathematical model of the system is presented in a dimensionless form. Three dimensionless optimization criteria that characterize the accuracy, size, and quality of the dynamic positioning process are considered. It is shown that the application of the criteria importance method significantly reduces the Pareto set (the set of the best solutions). This opens up the possibility of reducing many optimal solutions to one solution, which greatly facilitates the choice of parameters when designing a mechanical object. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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Article
The Exact Solutions of Stochastic Fractional-Space Kuramoto-Sivashinsky Equation by Using (G′G)-Expansion Method
by Wael W. Mohammed, Meshari Alesemi, Sahar Albosaily, Naveed Iqbal and M. El-Morshedy
Mathematics 2021, 9(21), 2712; https://doi.org/10.3390/math9212712 - 26 Oct 2021
Cited by 33 | Viewed by 2333
Abstract
In this paper, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation forced by multiplicative noise. To obtain the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation, we apply the G′G-expansion method. Furthermore, we generalize some previous results that did not use [...] Read more.
In this paper, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation forced by multiplicative noise. To obtain the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation, we apply the G′G-expansion method. Furthermore, we generalize some previous results that did not use this equation with multiplicative noise and fractional space. Additionally, we show the influence of the stochastic term on the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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