A singularly perturbed initial-boundary value problem for a parabolic equation known in applications as a Burgers-type or reaction-diffusion-advection equation is considered. An asymptotic approximation of solutions with a moving front is constructed in the case of modular and quadratic nonlinearity and nonlinear amplification. The influence exerted by nonlinear amplification on front propagation and blowing- up is determined. The front localization and the blowing-up time are estimated.
Exact solutions of a nonlinear integro-differential equation with quadratically cubic nonlinear term are found. The equation governs, in particular, stationary shock wave propagation in relaxing media. For the exponential kernel the shapes of both compression and rarefaction shocks having a finite width of the front are calculated. For media with limited "memorizing time" the difference relation permitting the construction of wave profile by the mapping method is derived. The initial equation is rather general. It governs the evolution of nonlinear waves in real distributed systems, for example, in biological tissues, structurally inhomogeneous media and in some meta-materials.
Solutions to an inhomogeneous partial differential equation of the second-order like Burgers equation are derived. Instead of the common quadratically nonlinear term, this equation contains the term with modular nonlinearity. This model describes the excitation of elastic waves in dissipative media differently reacting to tensile and compressive stresses. The equation is linear for the functions, preserving the sign. Nonlinear effects are manifested only to alternating functions. The solution for the periodic signal is found. The processes of generation of fundamental and higher harmonics are studied. The stationary wave profile is constructed. For one special kind of right-hand-side of the "modular" equation the solution in the form of S-wave is pointed out which is a bipolar single pulse.
Two models of an anharmonic oscillator that have exact solutions are considered. The equationsdescribe motion in a “modulus” potential well with a singularity at the minimum and in a double symmetricwell with a singularity at the vertex of the potential barrier. The forms and spectra of the oscillations are computed. Forced oscillations caused by a random force are analyzed on the basis of equations with Langevinsources. Nonstationary solutions of the corresponding Fokker–Planck equations are constructed. Thesesolutions describe
Solutions of the equation describing the high-intensity wave profile within the focal region are derived. This equation is similar to the previously studied models with quadratic and modular nonlinearities, but it is adapted for cubic and quadratically-cubic (QC) nonlinear media, where other physical processes are realized. This simplified one-dimensional equation can be regarded as a "projection" of a three-dimensional equation of Khokhlov-Zabolotskaya type (KZ) onto the axis of the wave beam. Stationary profiles at high intensities of focused waves turn out to be periodic sequences of half-periods of triangular shape with singularities of the derivative at extremum points. Such profiles are typical for nonlinear systems with low-frequency dispersion. There is shown to exist a saturation effect-the amplitude of the wave in the focus cannot exceed a certain maximum value, which does not depend on the initial amplitude.
Solutions of a forced (inhomogeneous) partial differential equation of the second order with two types of nonlinearity: power (quadratic) and nonanalytic (modular) are found. Equations containing each of these nonlinearities separately were studied earlier. A natural continuation of these studies is the development of the theory of wave phenomena in a medium with a double nonlinearity, which have recently been observed in experiments. Here solutions describing the profiles of intense waves are derived. Shapes of freely propagating stationary perturbations in the form of shock waves with a finite front width are found. The profiles of forced waves excited by external sources are calculated.
A one-dimensional equation is presented that generalizes the Burgers equation known in the theory of waves and in turbulence models. It describes the nonlinear evolution of waves in pipes of variable cross section filled with a dissipative medium, as well as in ray tubes, if the approximation of geometric acoustics of an inhomogeneous medium is used. The generalized equation is reduced to the common Burgers equation with a dissipative parameter-the "Reynolds-Goldberg number," depending on the coordinate. The method for solving statistical problems corresponding to specified characteristics of a noise signal at the input of the system is described. Integral expressions for exact solutions are given for the correlation function and the noise intensity spectrum experiencing nonlinear distortions during propagation in a waveguide. For waves in a dissipative medium, an approximate method of calculating statistical characteristics is given, consisting in finding an auxiliary correlation function and the subsequent nonlinear functional transformation. Solutions have a complicated form, so physical analysis of phenomena requires the numerical methods. For some correlation functions of stationary noise with initial Gaussian statistics and some waveguide systems, it is possible to obtain simple results.
Self-similar solutions are found for a quadratically cubic second-order partial differential equation governing the behavior of nonlinear waves in various distributed systems, for example, in some metamaterials. They are compared with self-similar solutions of the Burgers equation. One of them describing a single unipolar pulse is shown to satisfy both equations. The other self-similar solutions of the quadratically cubic equation behave differently from the solutions of the Burgers equation. They are constructed by matching the positive and negative branches of the solution, so that the function itself and its first derivative are continuous. One of these solutions corresponds to an asymmetric solitary N-wave of the sonic shock type. Self-similar solutions of a quadratically cubic equation describing the propagation of cylindrically symmetric waves are also found.
An equation is obtained that describes the nonlinear diffraction of a focused wave in a half-space starting from the wave source, then through the focus region up to the far zone, where the wave becomes spherically divergent. In contrast to the Khokhlov-Zabolotskaya equation (KZ), which contains two spatial variables, the calculation of the field on the beam axis is reduced to a simpler one-dimensional problem. Integral relations that are useful for numerical calculation are indicated. The profiles of a periodic wave harmonic at the input to the medium are constructed. A comparison with the results of a numerical solution of problems based on KZ is made, which revealed a good accuracy of the approximate model. The passage of a wave through the focus region, accompanied by the formation of shock fronts, diffraction phase shifts and asymmetric distortion of regions of different polarity, is traced.