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Advance Periodic Waves and Closed-Form Invariant Solutions for (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger (HNLS) Equation via Two Efficient Mathematical Approaches
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DOI：10.1007/s10773-026-06313-5.png)
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Abstract
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This paper mainly analyzes different analytical solutions of(2+1)-dimensional hyperbolicnonlinear Schrodinger equation (HNLS). The problem is employed to model wave propa-gation, when the dispersion relation's Hessian is neither strictly positive nor negative. Thisequation provides a foundational framework for modeling a wide range of physical phe-nomena, including the propagation of electromagnetic fields, the dynamics of optical solitontransmission, and the evolution of water wave surfaces. In this study, we derive novel anddistinct exact solutions of the equation using two effective numerical techniques: the Liesymmetry analysis, and the phi 6-expansion method that have not been reported earlier. Firstly,we apply the phi 6-expansion technique, which presents the exact soliton solutions and alsotheir dynamical structures. The Lie group method reveals nine Lie algebra generators (iso-morphism groups), which are employed to investigate the underlying symmetries of theequation. Initially, we determine the associated infinitesimal transformations by applying theone-parameter Lie symmetry approach. Subsequently, we solve the infinitesimal generatorsto reduce the governing partial differential equation (PDE) into forms with fewer indepen-dent variables. These reduced PDEs yield invariant solutions to the governing equation. Theresulting exact solutions exhibit diverse dynamic behaviors, including periodic wave solitons,interaction of periodic waves, solitary waves, multisoliton formations, as well as travelingand standing wave profiles. To illustrate these solutions, we present a variety of graphicalrepresentations, including two-dimensional (2D), three-dimensional (3D), and contour plots.Furthermore, the exact analytical solutions are computed and verified using symbolic com-putational tools such as Mathematica. A broad spectrum of novel analytical solutions withdistinct dynamical characteristics is analyzed through this approach
Keywords:
Hyperbolic nonlinear Schr & ouml
dinger (HNLS) equation
phi(6)-expansion method
Lie symmetry analysis
Similarity transformation
Infinitesimal generators
Exact solutions
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