Self-similar multishock implosions for ultrahigh compression of matter

We present a class of self-similar solutions describing ultrahigh compression of a uniform-density target by spherically converging, stacked shock waves. Extending the classical Guderley model, we derive a scaling law for the final density of the form <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>ρ</a:mi> <a:mi>r</a:mi> </a:msub> <a:mo>/</a:mo> <a:msub> <a:mi>ρ</a:mi> <a:mn>0</a:mn> </a:msub> <a:mo>∝</a:mo> <a:msup> <a:mover accent="true"> <a:mi>P</a:mi> <a:mo>̂</a:mo> </a:mover> <a:mrow> <a:mi>β</a:mi> <a:mo>(</a:mo> <a:mi>N</a:mi> <a:mo>−</a:mo> <a:mn>1</a:mn> <a:mo>)</a:mo> </a:mrow> </a:msup> </a:mrow> </a:math> , where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>N</c:mi> </c:math> is the number of shocks, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mover accent="true"> <d:mi>P</d:mi> <d:mo>̂</d:mo> </d:mover> </d:math> the stage pressure ratio, and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>β</f:mi> </f:math> a numerical exponent determined by the adiabatic index <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>γ</g:mi> </g:math> . One-dimensional hydrodynamic simulations confirm the validity of this scaling across a broad parameter range. Notably, the relation remains accurate even in the strongly nonlinear regime up to <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mover accent="true"> <h:mi>P</h:mi> <h:mo>̂</h:mo> </h:mover> <h:mo>∼</h:mo> <h:mn>70</h:mn> </h:mrow> </h:math> , well beyond the perturbative limit, highlighting the robustness and practical relevance of the model. Owing to its volumetric geometry, this compression scheme inherently avoids the Rayleigh-Taylor instability, which typically compromises shell-based implosions, and thereby establishes a theoretical benchmark for instability-free compression in inertial confinement fusion.