University of Tabriz Computational Methods for Differential Equations 2345-3982 13 4 2025 10 01 Computational applications by ITEM and the variational method for solving the Hamiltonian amplitude equation 1336 1356 18514 10.22034/cmde.2024.62097.2714 EN Elvir Akhmetshin Department of Economics and Management of Elabuga Institute, Kazan Federal University, Kazan, Russia Moscow Aviation Institute (National Research University), Moscow, Russia. Ilyos Abdullayev Department of Management and Marketing, Urgench State University, Urgench, Uzbekistan. Nalbiy Tuguz Department of Higher Mathematics, Kuban State Agrarian University named after I.T. Trubilin, Krasnodar, Russia. Dmitry Fugarov Department of Automation and Mathematical Modeling in the Oil and Gas Industry, Don State Technical University, Rostov-on-Don, Russia. Diana Stepanova Higher School of Finance, Plekhanov Russian University of Economics, Moscow, Russia. Journal Article 2024 06 13 The paper presents a significant improvement to the implementation of the improved $\tan(\phi(\xi)/2)$-expansion method (ITEM) for solving the Hamiltonian amplitude equation (HAE). We seek to improve the exact solutions by applying the ITEM. Computed solutions are compared with previously published results obtained using the simplest equation method \cite{Eslami} and the $(G'/G,1/G)$-expansion method \cite{Demiray}. There is clear evidence that the new approach produces results that are as good as, if not better than, published results determined using the other methods. The main advantage of the method is that it offers further solutions.  By using this method, exact solutions, including the hyperbolic function solution, traveling wave solution, soliton solution, rational function solution,  and periodic wave solution of this equation, have been obtained. Moreover, variational principles for the HAE are formulated. The invariance identities of the HAE involving the Lagrangian $L$ and the generators of the infinitesimal Lie group of transformations have been utilized for writing down their first integrals via Noether's theorem, Logan. We demonstrate the simplest example of the application of this technique, taking the box-shaped initial pulse and an ansatz based on linear Jost functions. We consider a combination of two boxes of opposite signs, the total area of the initial pulse being thus zero. Therewith, we develop a variational approximation for finding the eigenvalues of this pulse, by a piece-wise linear ansatz and tanh functions of the piece-wise linear function.  Moreover, by using MATLAB, some graphical simulations were done to see the behavior of these solutions. https://cmde.tabrizu.ac.ir/article_18514_412a1dc0b18278ef4e76e9573c9af50c.pdf