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The paper presents the solutions for the two-beam reduction of the densesoliton gas equations (or Born-Infeld equation) obtained by analytical andnumerical methods. The method proposed by the authors is used. This methodallows to reduce the Cauchy problem for two hyperbolic quasilinear PDEs to theCauchy problem for ODEs. In some respect, this method is analogous to themethod of characteristics for two hyperbolic equations. The method iseffectively applicable in all cases when the explicit expression for theRiemann-Green function for some linear second order PDE, resulting from the useof the hodograph method for the original equations, is known. The numericalresults for the two-beam reduction of the dense soliton gas equations, and theshallow water equations (omitting in the previous papers) are presented. Forcomputing we use the different initial data (periodic, wave packet).

Hodograph Method and Numerical Solution of the Two Hyperbolic Quasilinear Equations. Part III. Two-Beam Reduction of the Dense Soliton Gas Equations