The variational theory has a major role in physics, but it is difficult to develop the variational principle (VP) for the partial differential equations (PDEs). This article focusses on the vector nonlinear Schrödinger equation, namely the Manakov system, which can be used to describe the nonlinear interaction between two orthogonal polarized light waves in a birefringent fiber. Aided by the semi-inverse method, a trial-Lagrange function (TLF) is introduced and then updated step by step to develop the VP. Finally, the VP is proved through the Euler–Lagrange equations (ELEs) that attained via calculating the functional stationary conditions. The obtained VP can give us deeper insight into the intrinsic connections of the various physical quantities, and off new ideas on study of the numerical and variational methods.
We utilized the cosmological observations of supernovae Type Ia, expansion rate, baryonic acoustic oscillations, and cosmic background radiation to test the cosmic evolution in dark energy models having anisotropic background. For this purpose, we incorporate the parametrized EoS parameters of dark energy to explore cosmic evolution with matter and radiation. The capabilities of these generalized models to fit the above-mentioned dataset have been analyzed, and a comprehensive comparison based on the fitting results, derived cosmological parameters, and model selection criterion has been presented. The varying dark energy in these models yields the mechanism for anisotropy dissipation. These models have predicted a higher value of the anisotropy parameter as compared to the models having constant dark energy EoS.
This study presents a comprehensive investigation of two-beam coupling for controlling the polarization of light waves in nonlinear optical media (including the evolution of phase, energy, amplitude, and polarization). An analytical solution is employed to solve the four coupled differential equations governing the interaction of two beams in the perpendicular configuration. The effects of key physical parameters, including the refraction coefficient, relaxation time, and frequency ratio, are systematically examined. The results demonstrate that both the polarization and the intensity of the incident light wave can be effectively controlled by the appropriate choice of these parameters. Plasma has been selected as the nonlinear medium due to its superior suitability for high-intensity applications, its fast response compared to other materials, and its inherent flexibility in tuning physical parameters, which enables straightforward control of polarization and phase. The findings provide valuable insights into wave coupling and high-power laser physics and hold practical significance for advanced research areas such as laser fusion, particle acceleration, plasma diagnostics and high-harmonic generation.
This study investigates the geophysical Boussinesq (geo-Bouss) equation through two powerful analytical approaches: Lie symmetry analysis and Painlevé analysis. The integrability of the equation is investigated using the Weiss–Tabor–Carnevale (WTC) Painlevé test, which confirms that the geo-Bouss equation possesses the Painlevé property. This result facilitates the construction of a Bäcklund transformation, which is further utilized to derive the bilinear form of the equation. By applying the Hirota bilinear method, we construct explicit soliton solutions of the geo-Bouss equation. Additionally, by using the concept of nonlinear self-adjointness, we systematically derive associated conservation laws.
In this paper, anomalous scattering lump solutions for the generalized Korteweg-de Vries (gKdV) equation are constructed via the Hirota bilinear form combined with an improved limiting technique. The evolutionary dynamics of the two-lump anomalous scattering are investigated, and the corresponding asymptotic behaviors are explicitly derived. Furthermore, we analyze novel weak interaction structures comprising three lumps, which form triangular patterns and possess anomalous scattering characteristics. The collision phenomena between a fundamental lump and a two-lump anomalous scattering state are also investigated. These results provide new insights into the structural and dynamical features of lump solutions.
