Exact Analytic Solutions of Pochammer-Chree and Boussinesq Equations by Invariant Painlevé Analysis and Generalized Hirota Techniques - Russo, Matthew; Choudhury, S. Roy (Discontinuity, Nonlinearity, and Complexity, 2026)
Combinations of truncated Painlevé expansions, invariant Painlevé analysis, and generalized Hirota series are used to solve (’partially reduce to quadrature’) the integrable Boussinesq and the cubic and quintic generalized Pochammer-Chree (GPC) equation families. Although the multisolitons of the Boussinesq equation are very well-known, the solutions obtained here for all the three NLPDEs are novel, and non-trivial. All of the solutions obtained via invariant Painlevé analysis are complicated rational functions, with arguments which themselves are trigonometric functions of various distinct traveling wave variables. This is reminiscent of doublyperiodic elliptic function solutions when nonlinear ODE systems are reduced to quadratures. The solutions obtained using recently-generalized Hirota-type expansions are closer in functional form to conventional hyperbolic secant solutions, although with non-trivial traveling-wave arguments which are distinct for the two GPC equations.