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Lie symmetries, bifurcations, and exact soliton solutions of the Lonngren wave model in nonlinear media
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DOI：10.1007/s11082-026-09077-8.png)
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Abstract
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This paper presents a comprehensive analysis of the Lonngren wave equation, which serves as a fundamental model for nonlinear electronic signal transmission in semiconductor materials, particularly in tunnel diodes. By applying a travelling wave transformation technique, the governing nonlinear partial differential equation is reduced to a nonlinear ordinary differential equation and examined within the framework of nonlinear dynamical systems theory. A comprehensive Lie symmetry analysis is carried out to identify the admitted point symmetries, which validate the establishment of similarity reductions and invariant solutions. To derive precise analytical solutions, the $$\left( \frac{G'}{G^2}\right) $$ -expansion method and sine–cosine method are applied, yielding families of exact soliton solutions. By exploiting spatial and temporal continuous translation symmetries, the governing fourth-order mixed partial differential equation is successfully mapped into a lower-dimensional, autonomous second-order ordinary differential equation. The primary mathematical novelty and original contribution of this work lie in the simultaneous extraction of diverse closed-form travelling wave topologies. Alongside the verification of classical bright, dark, kink, and periodic wave envelopes, this dual-expansion methodology yields W-shaped dual-dip soliton solutions. To the best of our knowledge, such solutions have not been reported previously for the classical Lonngren wave equation. To investigate the structural resilience of these wave packages, I establish a conservative Hamiltonian energy framework and perform localized parameter sensitivity analyses. The phase plane contours reveal distinct homoclinic and heteroclinic separatrix topologies, confirming that the isolated soliton envelopes form stable, non-evanescent energy packets. The proposed analytical framework provides parameter conditions for stable wave propagation and may be useful for the analysis of nonlinear electronic and optical transmission systems.
Keywords:
\(\left( \frac{G'}{G^2}\right) \)-expansion method
Sine–cosine method
Lie symmetry analysis
Phase portrait
Soliton
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