J. C. Ndogmo
Based on a recent classification of subalgebras of the symmetry algebra of the Zabolotskaya-Khokhlov equation, all similarity reductions of this equation into ordinary differential equations are obtained. Large classes of group-invariant solutions of the equation are also determined, and some properties of the reduced equations and exact solutions are discussed.
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In this paper we show local (and partially global) in time existence for the Westervelt equation with several versions of nonlinear damping. This enables us to prove well-posedness with spatially varying $L_\infty$-coefficients, which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.
Conservation laws and exact solutions of a nonlinear acoustics equation by classical symmetry reduction
本文研究了广义韦斯特维尔特方程的对称性和保量律，通过约简发现点对称性和局部保量律，推导出潜在系统和非局部保量律，并研究了行波解导致的激波现象。
Early results concerning the shape and stability of ion acoustic waves are generalized to propagation at an angle to the magnetic field lines. Each wave has a critical angle for stability. Known soliton results are recovered as special cases. A historical overview of the problem concludes the paper.
This work is concerned with the study of fundamental models from nonlinear acoustics. In Part~I, a hierarchy of nonlinear damped wave equations arising in the description of sound propagation in thermoviscous fluids is deduced. In particular, a rigorous justification of two classical models, the Kuznetsov and Westervelt equations, retained as limiting systems for consistent initial data, is given. Numerical comparisons that confirm and complement the theoretical results are provided in Part~II.
Conservation laws of nonlinear PDEs arising in elasticity and acoustics in Cartesian, cylindrical, and spherical geometries
本文通过尺度齐次性和乘子法，研究了弹性力学与声学中非线性偏微分方程的守恒律，包括剪切波传播模型和参数化本构方程组。
We consider an inverse problem arising in nonlinear ultrasound imaging. The propagation of ultrasound waves is modeled by a quasilinear wave equation. We make measurements at the boundary of the medium encoded in the Dirichlet-to-Neumann map, and we show that these measurements determine the nonlinearity.
Well-posedness of a Nonlinear Acoustics -- Structure Interaction Model
该研究针对含Westervelt方程的非线性声学-结构相互作用耦合模型，建立了小数据下的时间局部与全局适定性，还提出了线性化系统的新型变分弱形式并探讨了多种边界条件，填补了相关研究空白。
Inverse problem of recovering a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations
针对源自黏性热弛豫流体波传播的J-M-G-T三阶非线性声学方程，研究含时非线性项恢复反问题，证明小初边值下方程适定性，借高阶线性化与CGO解推导得到非线性项恢复的唯一性。
We analyze the behavior of third-order in time linear and nonlinear sound waves in thermally relaxing fluids and gases as the sound diffusivity vanishes. The nonlinear acoustic propagation is modeled by the Jordan--Moore--Gibson--Thompson equation both in its Westervelt and in its Kuznetsov-type forms, that is, including quadratic nonlinearities of the type $(u^2)_{tt}$ and $ (u_t^2 + |\nabla u|^2)_t$. As it turns out, sufficiently smooth solutions of these equations converge in the energy norm to the solutions of the corresponding inviscid models at a linear rate. Numerical experiments illust…
In this paper, we consider an inverse problem for a nonlinear wave equation with a damping term and a general nonlinear term. This problem arises in nonlinear acoustic imaging and has applications in medical imaging and other fields. The propagation of ultrasound waves can be modeled by a quasilinear wave equation with a damping term. We show the boundary measurements encoded in the Dirichlet-to-Neumann map (DN map) determine the damping term and the nonlinearity at the same time. In a more general setting, we consider a quasilinear wave equation with a one-form (a first-order term) and a gene…
Uniqueness of some space dependent coefficients in a wave equation of nonlinear acoustics
本文针对非线性声学中的JMGT方程，利用反函数定理证明了从边界声压单次时间轨迹恢复空间依赖非线性参数、声速和衰减系数的唯一性，为超声断层成像提供理论支撑。
Acoustic nonlinearity parameter tomography with the Jordan-Moore-Gibson-Thompson equation in frequency domain
该研究将JMGT方程与频域非线性参数层析成像结合，证明了边界测量下空间相关非线性参数的局部唯一性，并探讨了可基于该结果证明收敛性的正则化Newton型重构方法。
In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures. This work analyzes a class of shape optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theoretical framework, the Westervelt and Kuznetsov equations of nonlinear acoustics are obtained as particular cases. The quadratic gradient nonlinearity, specific to the Kuznetsov equation, requires special attention throughout. To prove the existence …
We relate together different models of non linear acoustic in thermo-ellastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the L 2 norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found app…
This work deals with the convergence analysis of parabolic perturbations to quasilinear wave equations on smooth bounded domains. In particular, we consider wave equations with nonlinearities of quadratic type, which cover the two classical models of nonlinear acoustics, the Westervelt and Kuznetsov equations. By employing a high-order energy analysis, we obtain convergence of their solutions to the corresponding inviscid equations' solutions in the standard energy norm with a linear rate, assuming small data and a sufficiently short time. The smallness of initial data can, however, be imposed…
We study the Blackstock equation which models the propagation of nonlinear sound waves through dissipative fluids. Global well-posedness of the model with homogeneous Dirichlet boundary conditions is shown for small initial data. To this end, we employ a fixed-point technique coupled with well-posedness results for a linearized model and appropriate energy estimates. Furthermore, we obtain exponential decay for the energy of the solution. We present additionally a finite element-based method for solving the Blackstock equation and illustrate the behavior of solutions through several numerical …
Well-posedness of a first-order-in-time model for nonlinear acoustics with nonhomogeneous boundary conditions
本文研究了非均匀边界条件下非线性声学模型的一阶时间模型在分数Sobolev空间中的适定性，通过谱分解和Galerkin方法证明了其正则性。
This article focuses on a structural acoustic interaction system consisting of a semilinear wave equation defined on a smooth bounded domain $Ω\subset\R^3$ which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of $Ω$. In particular, the source terms acting on the wave and plate equations are allowed to have arbitrary growth order. We employ a standard Galerkin approximation scheme to establish a rigorous proof of the existence of local weak solutions. In addition, under some conditions on the parameters in the system, we prove such solutions exist gl…
The Blackstock-Crighton equations describe the motion of a viscous, heat-conducting, compressible fluid. They are used as models for acoustic wave propagation in a medium in which both nonlinear and dissipative effects are taken into account. In this article, a mathematical analysis of the Blackstock-Crighton equations with a time-periodic forcing term is carried out. For arbitrary time-periodic data (sufficiently restricted in size) it is shown that a time-periodic solution of the same period always exists. This implies that the dissipative effects are sufficient to avoid resonance within the…
Nonlinear acoustic equations of fractional higher order at the singular limit
本文研究具有反常扩散介质中分数阶高阶非线性声学方程在松弛时间趋于零时的奇异极限，证明其可近似为 Westervelt、Blackstock 或 Kuznetsov 方程，并通过针对弱奇异记忆核强制性设计的检验方法获得解的一致有界性。
Inverse problems for a quasilinear strongly damped wave equation arising in nonlinear acoustics
本文研究非线性声学中Westervelt方程的反问题，利用复几何光学构造和Isakov方法证明边界测量可唯一确定含时势q与非线性系数β，并进一步证明零初始条件下Dirichlet-to-Neumann映射也能确定q和β。
A first order in time wave equation modeling nonlinear acoustics
针对非线性声学的Navier-Stokes-Fourier系统小振幅近似问题，推导得到时间一阶系统，结合Galerkin近似与不动点论证证明其适定性，得到小初值下全局适定性与指数衰减性。
Optimization of a Nonlinear Acoustics -- Structure Interaction Model
本文研究了PDEs非线性声学-结构相互作用模型的控制与形状优化问题，通过Westervelt和4阶Kirchoff方程描述，考虑了激励、机械和形状三种控制，证明了最小化问题解的存在性并通过伴随系统表征了最优状态。
On the Jordan-Moore-Gibson-Thompson equation of nonlinear acoustics
本文系统综述JMGT方程的适定性、记忆与分数阶衰减、奇异极限以及控制与反问题，旨在弥补经典非线性声学模型无限声速悖论。
The equations of motion of lossless compressible nonclassical fluids under the so-called Green--Naghdi theory are considered for two classes of barotropic fluids: (\textit{i}) perfect gases and (\textit{ii}) liquids obeying a quadratic equation of state. An exact reduction in terms of a scalar acoustic potential and the (scalar) thermal displacement is achieved. Properties and simplifications of these model nonlinear acoustic equations for unidirectional flows are noted. Specifically, the requirement that the governing system of equations for such flows remain hyperbolic is shown to lead to re…
The derivation of different models of non linear acoustic in thermo-ellastic media as the Kuznetsov equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) from an isentropic Navier-Stokes/Euler system is systematized using the Hilbert type expansion in the corresponding perturbative and (for the KZK and NPE equations) paraxial ansatz . The use of small, to compare to the constant state perturbations, correctors allows to obtain the approximation results for the solutions of these models and to estimate the time during which they keep clos…
This paper justifies the compressible Euler and acoustic limits from quantum Boltzmann equation with Fermi-Dirac statistics (briefly, BFD) rigorously. This limit was formally derived in Zakrevskiy's thesis \cite{Zakrevskiy} by moment method. We employ the Hilbert approach. The forms of the classical compressible Euler system and the one derived from BFD are different. Our proof is based on the analysis of the nonlinear implicit transformation of these two forms, and a few novel nonlinear estimates.
Accurate simulation of nonlinear acoustic waves is essential for the continued development of a wide range of (high-intensity) focused ultrasound applications. This article explores mixed finite element formulations of classical strongly damped quasilinear models of ultrasonic wave propagation; the Kuznetsov and Westervelt equations. Such formulations allow simultaneous retrieval of the acoustic particle velocity and either the pressure or acoustic velocity potential, thus characterizing the entire ultrasonic field at once. Using non-standard energy analysis and a fixed-point technique, we est…
Well-posedness of a first-order formulation for fractionally damped nonlinear acoustics
研究非线性声学中带记忆的拟线性一阶时间系统的适定性，利用记忆项的非线性强制性质证明Westervelt型系统在边界域上的局部适定性，并推广到Kuznetsov型系统。
Forward and Inverse Problems in Nonlinear Acoustics
针对超声人体组织成像需利用非线性效应和处理分数阶衰减的需求，本文阐述非线性与分数阶阻尼建模、分析相关偏微分方程的挑战及参数渐近性，并探讨非线性参数成像这一拟线性波动方程系数识别逆问题。
Well-posedness, long-time behavior, and discretization of some models of nonlinear acoustics in velocity-enthalpy formulation
本文基于速度-焓弱形式研究 Westervelt、Kuznetsov 及 Rasmussen 非线性声学模型，证明小数据下解全局存在且指数收敛到平衡态，并利用端口哈密顿结构实现保结构离散化。
We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form $p p_t$. However, this should be considered as a low order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is $f(p)\,p_t$ and $f$ is unknown and must be recovered from data measurements. C…
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