In this paper, by geometric singular perturbation method, we first prove the existence of a class of viscous shock wave solutions to a scalar balance law for sufficiently small viscosity, which extends the previous results in the strictly convex nonlinearity case to nonconvex case.

By applying the geometric singular perturbation theory, it is proved that steady traveling wave fronts is maintained when the delay is sufficiently small for a type of convolution kernel.

By the geometric singular perturbation theory combining with linear chains technology and Fredholm theory, we first establish the existence of such wavefronts when the sixth and fourth order terms have sufficiently small coefficients.

By geometric singular perturbation method, for cross-diffusion rate in the second equation sufficiently large, we show that the system has traveling waves with transition layers connecting two semi-trivial equilibrium points and the waves have locally unique slow speed.

Forif b_1/b_2<a_1/a_2<c_1/c_2,by geometric singular perturbation method, for cross-diffusion rateγ2 in the second equation sufficiently large, there exists traveling waves with transition layers connecting two semi-trivial equilibrium points (0,a_2/c_2) and (a_1/b_1,0) and there exi.

For vector disease model with distributed delay,when the distributed delay kernel is the general Gamma distribution delay kernel,the existence of travelling wave solutions is obtained by using the linear chain trick and geometric singular perturbation theory.

Under the condition that the distributed delay kernel is the strong kernel,by the linear chain trick and geometric singular perturbation theory,the existence of travelling wave solutions for the two-species competition-diffusion model with nonlocal delays is obtained.