Dynamics of the traveling wave reduction and exact solutions of the date Jimbo Kashiwara Miwa equation for sensor modeling

The \((2+1)\) -dimensional Date Jimbo Kashiwara Miwa (DJKM) equation is an integrable model of multidimensional dispersive waves. A traveling wave transformation reduces it to the planar Hamiltonian system \(u''=v\) , \(v''=au+bu^{2}\) , where the coefficients \(a\) and \(b\) are fixed by the wave parameters and the equation coefficients. This system is a quadratic nonlinear oscillator. Periodically forced oscillators with quadratic and cubic nonlinearities are standard models of MEMS resonators and sensors, so the reduced DJKM dynamics supply a wave based source of this oscillator class. The study classifies the two equilibria as a saddle and a center that exchange type at \(a=0\) , and it links the homoclinic loop to a solitary wave profile. Under periodic forcing, finite-time Lyapunov exponents are positive mainly for trajectories that escape past the saddle, with escape times from 5.5 to 51.6. This escape corresponds to loss of bounded motion in a resonator, and the results do not establish a bounded chaotic attractor. The Khater III method gives three periodic singular solutions, and the enhanced Kudryashov method gives four kink type solutions with bell shaped derivatives. The Vakhitov Kolokolov criterion classifies two kink branches as stable at one parameter value, and one branch is marginal. He’s variational iteration method reproduces the Khater III solution with absolute errors below \(1.4\times 10^{-9}\) at the sampled points. The analysis covers representative parameters of a deterministic, conservative model without damping or stochastic effects.