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Traveling wave solutions of fourth order PDEs for image processing. (English) Zbl 1082.35080
The paper considers the problem of the existence of traveling shock-wave solutions in two equations combining the Burgers’ nonlinearity and fourth-order nonlinear diffusion. One equation is $u_t + uu_x = -[(1+u_{xx}^2)^{-1}u_{xx}]_{xx},$ and the other is $u_t + uu_x = -[(1+u_{xx}^2)^{-1}u_{xxx}]_{x}.$ These equations are used for removal of noise in image processing: while the diffusion suppresses the noise components in the image’s spectrum, the highly nonlinear character of the diffusion provides for conservation of true features of the image rich in higher harmonics, such as sharp corners. In this paper, it is rigorously proven that the first equation does not have shock-wave solutions for large values of the shock-driving jump, while the second equation always supports shock waves. The analysis is based on reduction of the problem to one for the corresponding ODE, and comparison with a result for a simplified equation. The results are visualized by means of plots in the phase plane of the ODE. The results are also compared with direct simulations of the full PDEs, that support the conclusions derived from the ODE analysis.

##### MSC:
 35K55 Nonlinear parabolic equations 68U10 Computing methodologies for image processing 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 35K25 Higher-order parabolic equations 74J30 Nonlinear waves in solid mechanics
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