Brief Report - (2025) Volume 16, Issue 3
Received: 01-May-2025, Manuscript No. jpm-26-179403;
Editor assigned: 05-May-2025, Pre QC No. P-179403;
Reviewed: 19-May-2025, QC No. Q-179403;
Revised: 22-May-2025, Manuscript No. R-179403;
Published:
29-May-2025
, DOI: 10.37421/2090-0902.2025.16.532
Citation: Varga, Helena. ”Exact Solutions To Nonlinear Partial Differential Equations.” J Phys Math 16 (2025):532.
Copyright: © 2025 Varga H. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.
This article explores the development and application of novel analytical techniques for solving complex nonlinear partial differential equations that arise in fundamental areas of mathematical physics, such as fluid dynamics and nonlinear optics. The authors introduce a modified tanh-sech method, demonstrating its efficacy in deriving exact solutions for challenging models. The work highlights the power of these integrable systems in understanding emergent phenomena and predicting physical behavior [1].
The paper delves into the study of integrability and the search for exact solutions in the context of the nonlinear Schrödinger equation, a cornerstone in describing wave propagation in various physical systems. It presents a new approach to systematically generate multiple exact solutions by employing specific ansatz functions, offering a deeper insight into the behavior of nonlinear waves and their stability [2].
This research focuses on the application of advanced computational methods to find exact solutions for coupled nonlinear partial differential equations that model phenomena in plasma physics and hydrodynamics. The authors showcase the effectiveness of a symbolic computation-assisted approach, providing a streamlined pathway to discover new analytical solutions and understand the underlying physics [3].
The paper investigates the integrability properties of a specific class of nonlinear differential equations relevant to theoretical physics, particularly in the study of solitonic waves. By applying a novel transformation and algebraic methods, the authors not only confirm the integrability of the system but also derive a set of exact solutions, illuminating the dynamics of these solitary waves [4].
This article presents a comprehensive study of exact solutions for the FitzHugh-Nagumo equation, a model used in neuroscience and reaction-diffusion systems. The authors utilize the sine-Gordon expansion method to obtain a variety of traveling wave solutions, contributing to a better understanding of pattern formation and signal propagation in biological and chemical systems [5].
The research explores the application of the extended tanh-expansion method to find exact solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation, a fundamental equation in plasma physics and nonlinear wave phenomena. The paper demonstrates the method's ability to generate multiple soliton and other nonlinear wave solutions, offering valuable insights into the complex behaviors of such systems [6].
This work focuses on deriving exact solutions for the (2+1)-dimensional Riesz space fractional nonlinear diffusion equation using the improved tanh-function method. The study highlights the importance of fractional calculus in modeling anomalous diffusion processes and provides a systematic way to find analytical solutions, which are crucial for validating numerical simulations and understanding complex diffusion dynamics [7].
The authors investigate the integrability and exact solutions of a generalized (3+1)-dimensional shallow water wave equation. They employ the generalized exponential rational function method to obtain a range of novel exact solutions, including solitary and other nonlinear wave structures, offering valuable insights into the behavior of shallow water waves under various nonlinear effects [8].
This paper presents a study on the exact solutions for the (2+1)-dimensional hyperbolic tangent function equation, a significant model in nonlinear science. Using a modified approach based on the tanh method, the researchers derive multiple exact solutions, enhancing our understanding of wave propagation and related phenomena in complex nonlinear systems [9].
The research focuses on finding exact solutions for the (3+1)-dimensional Jimbo-Miwa equation, a nonlinear partial differential equation with applications in hydrodynamics and plasma physics. By applying the generalized extended tanh-expansion method, the authors successfully derive several new families of exact solutions, contributing to the understanding of nonlinear wave interactions [10].
This article explores the development and application of novel analytical techniques for solving complex nonlinear partial differential equations that arise in fundamental areas of mathematical physics, such as fluid dynamics and nonlinear optics. The authors introduce a modified tanh-sech method, demonstrating its efficacy in deriving exact solutions for challenging models. The work highlights the power of these integrable systems in understanding emergent phenomena and predicting physical behavior [1].
The paper delves into the study of integrability and the search for exact solutions in the context of the nonlinear Schrödinger equation, a cornerstone in describing wave propagation in various physical systems. It presents a new approach to systematically generate multiple exact solutions by employing specific ansatz functions, offering a deeper insight into the behavior of nonlinear waves and their stability [2].
This research focuses on the application of advanced computational methods to find exact solutions for coupled nonlinear partial differential equations that model phenomena in plasma physics and hydrodynamics. The authors showcase the effectiveness of a symbolic computation-assisted approach, providing a streamlined pathway to discover new analytical solutions and understand the underlying physics [3].
The paper investigates the integrability properties of a specific class of nonlinear differential equations relevant to theoretical physics, particularly in the study of solitonic waves. By applying a novel transformation and algebraic methods, the authors not only confirm the integrability of the system but also derive a set of exact solutions, illuminating the dynamics of these solitary waves [4].
This article presents a comprehensive study of exact solutions for the FitzHugh-Nagumo equation, a model used in neuroscience and reaction-diffusion systems. The authors utilize the sine-Gordon expansion method to obtain a variety of traveling wave solutions, contributing to a better understanding of pattern formation and signal propagation in biological and chemical systems [5].
The research explores the application of the extended tanh-expansion method to find exact solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation, a fundamental equation in plasma physics and nonlinear wave phenomena. The paper demonstrates the method's ability to generate multiple soliton and other nonlinear wave solutions, offering valuable insights into the complex behaviors of such systems [6].
This work focuses on deriving exact solutions for the (2+1)-dimensional Riesz space fractional nonlinear diffusion equation using the improved tanh-function method. The study highlights the importance of fractional calculus in modeling anomalous diffusion processes and provides a systematic way to find analytical solutions, which are crucial for validating numerical simulations and understanding complex diffusion dynamics [7].
The authors investigate the integrability and exact solutions of a generalized (3+1)-dimensional shallow water wave equation. They employ the generalized exponential rational function method to obtain a range of novel exact solutions, including solitary and other nonlinear wave structures, offering valuable insights into the behavior of shallow water waves under various nonlinear effects [8].
This paper presents a study on the exact solutions for the (2+1)-dimensional hyperbolic tangent function equation, a significant model in nonlinear science. Using a modified approach based on the tanh method, the researchers derive multiple exact solutions, enhancing our understanding of wave propagation and related phenomena in complex nonlinear systems [9].
The research focuses on finding exact solutions for the (3+1)-dimensional Jimbo-Miwa equation, a nonlinear partial differential equation with applications in hydrodynamics and plasma physics. By applying the generalized extended tanh-expansion method, the authors successfully derive several new families of exact solutions, contributing to the understanding of nonlinear wave interactions [10].
This collection of research papers focuses on the development and application of various analytical and computational methods for finding exact solutions to complex nonlinear partial differential equations. Techniques such as the modified tanh-sech method, sine-Gordon expansion, generalized exponential rational function method, and extended tanh-expansion are employed. These methods are applied to diverse models, including the KdV equation, nonlinear Schrödinger equation, FitzHugh-Nagumo equation, Kadomtsev-Petviashvili equation, Riesz space fractional nonlinear diffusion equation, shallow water wave equation, hyperbolic tangent function equation, and Jimbo-Miwa equation. The research spans areas like mathematical physics, fluid dynamics, nonlinear optics, plasma physics, hydrodynamics, and neuroscience, contributing to a deeper understanding of emergent phenomena, wave propagation, and complex system behaviors.
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