In this paper, we study the (3+1)-dimensional variable-coefficient Kudryashov–Sinelshchikov (vc-KS) equation, which characterizes the evolution of nonautonomous nonlinear waves in bubbly liquids. The nonautonomous lump solutions of the vc-KS equation are produced via the Hirota bilinear technique. The characteristics of trajectory and velocity of this wave are analyzed with variable dispersion coefficients. Based on the positive quadratic function assumption, we further discuss two types of interactions between the soliton and lump under the periodic and exponential modulations. Then, we give the breathing lump waves showing the periodic oscillation behavior. Finally, we obtain the second-order nonautonomous lump solution, which also shows periodic interactions if we select trigonometric functions as the dispersion coefficients.

在本文中，我们研究了 (3+1) 维可变系数 Kudryashov-Sinelshchikov (vc-KS) 方程，该方程表征了气泡液体中非自治非线性波的演化。vc-KS 方程的非自治块解是通过 Hirota 双线性技术产生的。利用变频散系数分析了该波的轨迹和速度特性。基于正二次函数假设，我们进一步讨论了周期调制和指数调制下孤子和团块之间的两种相互作用。然后，我们给出显示周期性振荡行为的呼吸团波。最后，我们得到二阶非自治块解，如果我们选择三角函数作为色散系数，它也显示出周期性的相互作用。