Solitary Strain Waves in Coaxial Shells with Fractional and Quadratic Physical Nonlinearity and with a Fluid Contained in the Annular Intershell Gap

We study the longitudinal nonlinear solitary strain waves in coaxial shells containing a viscous incompressible fluid in the annular intershell gap. The case is considered where the shell material has fractional and quadratic nonlinearity. We define the hydroelasticity problem for the annular channel under consideration and analyze it asymptotically using the perturbation method, which allows us to obtain a system of two evolution equations that generalize the Schamel–Korteweg–de Vries equations. Without the effect of the fluid, the system decomposes into two separate equations, which have an exact soliton solution. The evolution of soliton waves in coaxial shells is studied numerically with the newly obtained finite-difference scheme similar to the Crank–Nicolson scheme for the heat equation. The difference scheme is verified with the exact particular solution found for the case where a solitary wave of the same velocity and amplitude is specified in each of the shells. It is determined that the found nonlinear addition to the linear approximation for wave velocities, i.e., to the velocity of sound, speeds up the solitary waves, and they become supersonic. Moreover, the numerical experiments show that the solitary waves excited in the shells maintain their speeds and amplitudes over time and interact elastically, i.e., these waves are solitons.