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The wave model generated for laser plasma was discussed, which can be expressed as the Riemann problem of Burgers equations with initial value discontinuity. The singularly perturbed asymptotic solution of the Burgers equations with discontinuous initial values was obtained with the singularly perturbed expansion method. The solution was divided into 2 parts: an outer solution and an inner layer correction term. Since the initial condition is constant, the wave will generate the characteristic boundary in the process of propagation, and the correction term will make the parabolic characteristic boundary. The external solution was corrected at the internal layer along the characteristic lines. The existence and uniqueness of the asymptotic solution was proved through the HopfCole transform, Fourier transform and the extremum principle. Then the asymptotic expansion is obtained with the uniform validity proved. [ABSTRACT FROM AUTHOR]