Analytical integrability and physical solutions of d - KdV equation

A new idea of electron inertia-induced ion sound wave excitation for transonic plasma equilibrium has already been reported. In such unstable plasma equilibrium, a linear source driven Korteweg-de Vries ( d - KdV ) equation describes the nonlinear ion sound wave propagation behavior. By numerical techniques, two distinct classes of solution (soliton and oscillatory shocklike structures) are obtained. Present contribution deals with the systematic methodological efforts to find out its ( d - KdV ) analytical solutions. As a first step, we apply the Painleve method to test whether the derived d - KdV equation is analytically integrable or not. We find that the derived d - KdV equation is indeed analytically integrable since it satisfies Painleve property. Hirota’s bilinearization method and the modified sine-Gordon method (also termed as sine-cosine method) are used to derive the analytical results. Perturbative technique is also applied to find out quasistationary solutions. A few graphical plots are provided to offer a glimpse of the structural profiles obtained by different methods applied. It is conjectured that these solutions may open a new scope of acoustic spectroscopy in plasma hydrodynamics.