On a class of quasilinear hyperbolic reducible systems allowing for special wave interactions
For hyperbolic \(2\times 2\)-systems \(y_ t=A(y)y_ x\), the authors show how to construct explicit solutions in the hodograph plane, under the key assumption that two second order partial differential equations involving the left and right eigenvectors and associated eigenvalues of A hold identically. This technique is then used to construct solutions of initial value problems and to show that two smooth localized wave solutions of such an equation that travel in opposite directions will always emerge unchanged (with a phase shift) from their interaction. Examples for systems with the property mentioned above include the problem \(y_{tt}=F^ 2(y_ x,y)y_{xx}\) and a system from isentropic fluid dynamics, for a five parameter class of pressure - density relations.
scientific article; zbMATH DE number 4053236