Recently, for high-power ultrasonic beams, it was experimentally found that as a result of self-action, their self-focusing occurs. With self-focusing, a powerful ultrasonic beam is noticeably narrowed, has a nonlinear narrowing, and it is significantly amplified at the focus. A generalization of the three-dimensional Khokhlov–Zabolotskaya–Kuznetsov model in a cubic nonlinear medium in the presence of dissipation with a special nonlinearity coefficient describes the propagation of ultrasonic beams after self-focusing. This work is devoted to the study of submodels of this model without dissipation, which are invariant with respect to four-parameter subgroups of the main group of the equation of this model. These submodels are defined by invariant solutions of rank 0 or 1, which describe one-dimensional, plane and axisymmetric ultrasonic beams. Solutions of rank 0 and some solutions of rank 1 are found explicitly. Some of them, at each fixed moment of time, contain a destroying element in the form of an ultrasonic needle or an ultrasonic knife, which at each fixed moment of time are localized in a limited domain, on the surface of which the acoustic pressure is zero. These submodels describe one-dimensional, plane, or axisymmetric ultrasonic beams. Finding other invariant solutions of rank 1 is reduced to solving nonlinear integral equations. These invariant solutions are used to study the propagation of ultrasonic beams after self-focusing, for which either the acoustic pressure and its gradient, or the acoustic pressure and the rate of its change are given at the initial moment of time at a fixed point. Under certain additional conditions, the existence and uniqueness of the solutions to boundary value problems describing these processes are established. This makes it possible to correctly carry out numerical calculations when studying these processes. The graphs of the pressure distribution obtained as a result of the numerical solution of these boundary value problems for some values of the parameters characterizing the indicated processes are presented. These graphs show that for these submodels in the ultrasonic beams a monotonic increase in acoustic pressure occurs over time.

近来，对于大功率超声波束，通过实验发现，由于自作用，它们发生了自聚焦。通过自聚焦，强大的超声波束明显变窄，具有非线性变窄，并且在焦点处被显着放大。三维非线性Khokhlov–Zabolotskaya–Kuznetsov模型在具有特殊非线性系数的耗散存在的立方非线性介质中的一般化描述了自聚焦后超声波束的传播。这项工作致力于研究该模型的子模型而没有耗散，这些子模型相对于该模型方程组的主要组的四参数子组是不变的。这些子模型由等级0或1的不变解定义，描述了一维，平面和轴对称超声束。明确找到等级0的解和等级1的一些解。它们中的一些在每个固定的时间包含超声针或超声刀形式的破坏元件，其在每个固定的时间位于有限的区域内，在该区域内的声压为零。这些子模型描述一维，平面或轴对称超声束。寻找等级1的其他不变解可简化为求解非线性积分方程。这些不变解用于研究自聚焦后超声波束的传播，为此，在固定时间的初始时刻给出声压及其梯度，或声压及其变化率。在某些附加条件下，建立了描述这些过程的边值问题解决方案的存在性和唯一性。这使得在研究这些过程时可以正确地进行数值计算。给出了作为这些边界值问题的数值解的结果的压力分布图，对于指定的过程的某些参数值，得到了这些边界值问题的数值解。这些图表明，对于超声波束中的这些子模型，声压随时间发生单调增加。给出了作为这些边界值问题的数值解的结果的压力分布图，对于指定的过程的某些参数值，得到了这些边界值问题的数值解。这些图表明，对于超声波束中的这些子模型，声压随时间发生单调增加。给出了作为这些边界值问题的数值解的结果的压力分布图，对于指定的过程的某些参数值，得到了这些边界值问题的数值解。这些图表明，对于超声波束中的这些子模型，声压随时间发生单调增加。