arXiv subjects

Urvashi Joshi

Publications and source records attributed to Urvashi Joshi.

3 recordsLinked to original sources

arXiv2025

Solitonic and Exact Solutions for a Viscous Traffic Flow Model Via Lie Symmetry

This work studies a macroscopic traffic flow model driven by a system of nonlinear hyperbolic partial differential equations. Using Lie symmetry analysis, we determine the infinitesimal generators and construct an optimal system of one-dimensional subalgebras, facilitating symmetry reductions for the governing system. In addition, we discussed the classical symmetry and solution of the traffic flow model with the initial conditions left invariant. By applying the method of nonlinear self-adjointness, conservation laws associated with the model are established and are utilized to obtain exact solutions. Using these exact solutions, we construct solitonic solutions, including kink-type, peakon-type, and parabolic solitons. Additionally, using the weak discontinuity $C^1$ wave illustrates nonlinear wave dynamics in traffic evolution. Moreover, we investigate how these solutions affect traffic behavior, clarifying shock wave development and flow stability. The results provide a basis for useful applications in traffic management, real-time traffic control, and intelligent transportation systems, as well as improving mathematical knowledge of traffic dynamics.

math.AP↗

arXiv2025

Soliton Dynamics and Modulation Instability in the (3+1)-dimensional ZK equation: A Lie Symmetry Approach

The core focus of this research work is to obtain invariant solutions and conservation laws of the (3+1)-dimensional ZK equation, a higher-dimensional generalization of the Korteweg--de Vries (KdV) equation, which describes the phenomenon of wave stability and soliton propagation. Lie symmetry analysis has been applied to derive infinitesimal generators and classify the optimal subalgebras. Utilizing them, we construct exact invariant solutions that reveal how waves retain their shape as they travel, how they interact in space, and the impact of magnetic fields on wave propagation. Further, by implementing the traveling wave transformation, we derive additional exact solutions, including those exhibiting kink-type solitons. It also concludes with the conservation laws and the nonlinear self-adjointness property. Our examination is broadened to cover modulation instability and gain spectrum. To contextualize our results, we compare the solutions obtained from the Lie symmetry method with those derived using the modified simple equation (MSE) method and other symbolic techniques reported in recent literature. To validate the accuracy of the analytical solutions obtained via Lie symmetry, a numerical method is implemented. A review of the ZK equation's physical background and mathematical complexity is explored, emphasizing the limitations of symbolic approaches.

math.AP↗

arXiv2024

Invariance Analysis, Symmetry Reduction and Conservation Laws for Biological Population in Porous Media

This research paper talks about using complex mathematical tools to study and figure out the behavior of biological populations in porous media. Porous media offer a unique environment where various factors, including fluid flow and nutrient diffusion, significantly influence population dynamics. The theory of Lie symmetries is used to find inherent symmetries in the governing equation of the population model, helping to find conservation laws and invariant solutions. The derivation and analysis of the optimal system provide insights into the most influential parameters affecting population growth and distribution. Furthermore, the study explores the construction of invariant solutions, which aid in characterizing long-term population behavior. The article concludes with the non-linear self-adjointness property and conservation laws for the model.

math.AP↗