Stable-Range Approach to the Equation of Nonstationary Transonic Gas Flows

Using certain finite-dimensional stable range of the nonlinear terms, we obtain large families of exact solutions parameterized by functions for the equation of nonstationary transonic gas flows discovered by Lin, Reisner and Tsien, and its three-dimensional generalization.

💡 Research Summary

The paper addresses the long‑standing difficulty of finding explicit solutions for the non‑stationary transonic gas‑flow equation introduced by Lin, Reisner, and Tsien (the LRT equation) and its three‑dimensional extension. The authors introduce the concept of a “stable range” – a finite‑dimensional subspace of functions in which the nonlinear terms of the equation remain closed under differentiation and multiplication. By restricting the nonlinear term ((\Phi_x)^2) (and its analogues in higher dimensions) to lie within such a subspace, the original highly nonlinear partial differential equation (PDE) is transformed into a system that is effectively linear in the dependent variable but nonlinear only in a set of auxiliary coefficient functions.

The analysis begins with the two‑dimensional LRT equation
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