Nonlinear longitudinal spatially localized deformation waves propagating in a Bishop rod located in a magnetic field and having material damage

We study the propagation of longitudinal waves in a homogeneous, nonlinearly elastic rod located in an external nonstationary magnetic field, in the presence of damage in the rod material. The dynamic behavior of the rod is determined by Bishop’s theory. We consider the initial system of equations in two limiting cases and in the general case. In the first limiting case, we assume that under conditions of a strong magnetic field, the rod material has a high electrical resistance. In the second limiting case, we assume that the rod material has the property of ideal conductivity. In each particular case, the system reduces to a single nonlinear fifth-order equation for the longitudinal displacement of rod particles. Taking the short relaxation time into account, we obtain evolution equations for the longitudinal deformation function representing the well-known wave dynamics equation—the Kuramoto–Sivashinsky equation and its generalization containing an additional quadratically nonlinear term. We find exact solutions of the obtained evolution equations using the simplest equations method. We show that the solutions describe spatially localized deformation waves in the form of solitons and shock waves. We analyze the dependences of the characteristic parameters of stationary waves (amplitude, front width, and propagation velocity) on the system parameters. In the general case, the system reduces to a nonlinear seventh-order equation. In ordinary derivatives and under certain relations between the parameters, the equation transforms into an anharmonic oscillator equation with two types of quadratic nonlinearity. We find the first integral of the equation. The performed qualitative analysis shows the possibility of propagation of deformation waves in the system: nonlinear periodic and spatially localized soliton-type waves.