Examples of exact solutions of the non-local wave equation with nonlinear sources

The scope of the study is a wave equation with a source in a medium with weak spatial nonlocality. The equation is distinguished by an additional term containing the function sought as a fourth order partial derivative of the spatial coordinate. The transformation of independent variables enables the construction of accurate partial solutions in the form of waves generated by a nonlinear source which depend on the desired function. The velocity regime of the wave (subsonic, sonic, supersonic) is characterized by the Mach number equal to the ratio of the velocity of the wave to the propagation velocity of small perturbations. A source function similar to the classical case for the double sine-Gordon equation is considered. A kink solution corresponds to two equilibrium states of the medium-source system. The relation between the source and the kink structure (the area of the solution, the sign of the kink obliquity, and the velocity of its movement) has been established. It is shown that in relation to the dimensionless parameter of nonlocality, the square of the Mach number is a monotonic increasing/decreasing function for the supersonic/subsonic velocity mode. In relation to one of the source parameters, the square of the Mach number is a non-monotonic function with a minimum/maximum in the supersonic/subsonic cases. The source functions corresponding to the extreme modes differ from each other by the inversion of the areas where these functions are positive and negative. For the sine-Gordon equation, the comparison of the classical and nonlocal processes are different not only in the areas of the solutions, but also in the velocity modes (subsonic/supersonic) of the motion of the kink. The cubic nonlinearity of the source gives solutions representing a weak discontinuity of the function sought or a solitary wave. A kink solution depends on the wave coordinate and is determined by a hyperbolic tangent. The paper provides a comparative analysis of the properties of the polynomial (third and fifth degree) functions of sources generating a wave in classical and nonlocal media.