We investigate the nonlinear response of flow harmonics $v_2$, $v_4$ to initial-state eccentricities ${\epsilon }_2$, ${\epsilon }_4$ within the Gubser-flow framework. By extending the perturbative solutions of Gubser flow, we derive analytic nonlinear response relations connecting the eccentricities ${\epsilon }_2$, ${\epsilon }_4$ to the flow harmonics $v_2$, $v_4$. Our results reproduce the well-known result $v_4/v_2^2\to 1/2$ in large transverse momentum $p_T$ limit. We have also studied the centrality dependence and scaling behavior of these response coefficients. Furthermore, we study the effects of a mismatch between the participant and reaction planes. We find that the conventional nonlinear response coefficients acquire an additional factor determined by the participant-plane angles, which is often approximated as statistical noise driven by event-by-event fluctuations. This factor can modify both the strength and even the sign of the effective nonlinear response coefficient, making it sensitive to the initial configuration of the colliding nuclei. Our study provides new analytical insight into the origin of collective phenomena in relativistic heavy-ion collisions.

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• Collective flow
• Quark-gluon plasma
• Relativistic heavy-ion collisions