Latest revision as of 10:56, 18 February 2026

Abstract

We study the integrability via conservation laws and discuss the nonlinearity of the fourth-order biharmonic equations in (2+ 1) dimensions related to quantum field models based on the potential functions h(u). Lie symmetry reduction is performed, and the forms of the invariant solutions are presented, including travelling wave solutions. Variational analysis has been performed based on the various potential functions h(u). Corresponding Euler-Lagrange equations and conservation laws are investigated by Noether’s theorem and presented in the form of conserved vectors. The obtained conserved flows define energy, momentum and flow dynamics supporting the system integrability. Furthermore, detailed lump and breather solutions are presented for each potential h(u) using Bilinear forms illustrating various localized and oscillatory field characteristics.OPEN ACCESS Received: 08/11/2025 Accepted: 17/12/2025

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