### Functionals with Values in the Non-Archimedean Field of Laurent Series and Method for Numerical Calculations of Shocks and Soliton Like Solutions of Some Conservation Laws

Abstract: Functionals with values in Non-Archimedean field of Laurent series is considered. Introduced objects are natural generalization of Sobolev-Schwartz distributions. They applied to the definition of generalized solution (in the form of soliton and shock wave) of the Hopf equation and equations of elasticity theory. Calculation method for the profile of infinitely narrow soliton and shock wave is proposed. This method based on the new concept of solution of the conservation laws, functionals with values in Non-Archimedean field of Laurent series and orthogonal system of the Hermite functions. Applying this method, calculations of profiles are reduced to the nonlinear system of algebraic equations in $\mathbf{R}^{n+1}$, $n>1$. It is shown that there is a possibility to find out some of the solutions of this system using the Newton iteration method. Examples and numerical tests are considered.

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