Wave Motion最新文献

In this paper we determine the functional form of those planar velocity fields for which the associated system of two ordinary differential equations are automatically invariant under a Lorentz transformation. For planar motion we determine first order partial differential equations for the velocity components $u(x,y,t)$ and $w(x,y,t)$ in the $x-$ and $y-$directions respectively and their general solutions in terms of two arbitrary functions. These partial differential equations and the associated partial differential relations connecting energy and momentum are fully compatible with the Lorentz invariant energy–momentum relations and appear not to have been given previously in the literature. For a particular special relativistic model, one example is given involving similarity solutions of the wave equation. An interesting special case gives rise to a family of particle paths which are characterized by a single arbitrary function, and for which the magnitude of their velocities is the speed of light. This is indicative of the abundant possibilities existing in the “fast-lane”.

The principal algebra $L_p$ of coupled nonlinear Schrödinger equations describing the propagation of polarized optical pulses and involving four-wave mixing terms is obtained in terms of the three arbitrary labelling parameters of the system of equations. This algebra is found to be five-dimensional, indicating the richness in symmetries of the system. The most general symmetry group transformation by generators of $L_p$ is found and it is shown that this transformation preserves the boundedness of solutions, and an example of transformation of a soliton solution into a completely new soliton of a different type is presented. Moreover, it is explained how an infinite sequence of bounded solutions can thus be generated, yielding new solitons. Several other important properties of solutions and symmetry group transformations of the system are also demonstrated. As the system of Schrödinger equations under study turns out to be of Lagrange type, conservation laws associated with all variational symmetries of $L_p$ are constructed and interpreted. Some symmetry reductions of the system are also derived.

In the present work we revisit the shock wave dynamics in a granular chain with precompression. By approximating the model by an $\alpha$-Fermi–Pasta–Ulam–Tsingou chain, we leverage the connection of the latter in the strain variable formulation to two separate integrable models, one continuum, namely the KdV equation, and one discrete, namely the Toda lattice. We bring to bear the Whitham modulation theory analysis of such integrable systems and the analytical approximation of their dispersive shock waves in order to provide, through the lens of the reductive connection to the granular crystal, an approximation to the shock wave of the granular problem. A detailed numerical comparison of the original granular chain and its approximate integrable-system-based dispersive shocks proves very favorable in a wide parametric range. The gradual deviations between (approximate) theory and numerical computation, as amplitude parameters of the solution increase are quantified and discussed.

In this paper, a $(3+1)$-dimensional sine-Gordon equation is systematically investigated. Firstly, the integrability of the equation is demonstrated by Painlevé analysis. Secondly, based on the Hirota bilinear method, the $N$-soliton solution of the $(3+1)$-dimensional sine-Gordon equation is derived. Then, by selecting and establishing conjugate relationships between parameters, the kink solutions, the breather solutions and their hybrid solutions were obtained. Finally, the lump solutions of equation are derived by selecting appropriate functions in the solution. In addition, the dynamic behavior of these solutions is systematically analyzed by their respective density profile plots and three-dimensional diagrams.

In this paper, we investigate the data-driven localized solutions of Kadomtsev–Petviashvili (KP) and modified KP equation. Through the two-dimensional Miura transformation, the solutions of modified KP equation can be converted into the solutions of KP equation, but the process is not invertible in mathematics. Based on the neural network, the localized waves of modified KP equation are obtained under an unsupervised training with the aid of two-dimensional Miura transformation and the initial and boundary conditions of solution of the KP equation. As the result of the different hyperparameters, three types of localized waves are found after the training, including the shape of kink, dark and kink-bell. The evolution and error dynamics of the predicted solutions are analyzed through the graphics.

A numerical factorization method of the unidirectional propagation operators induced by the linearized Euler and Navier–Stokes equations is performed to construct a new One-Way approach to compute the sound radiation in a partially lined duct. The complex phenomena of reflection and transmission of incident waves originating from discontinuities in the duct wall are accurately taken into account by an iterative domain decomposition method. Furthermore, the proposed factorization approach allows both the derivation of the lined duct scattering matrix and an in-depth understanding of the impact of the acoustic liner on left- and right-going waves coming from a transmission or a reflection. Finally, the efficiency of the numerical method is shown based on a classical benchmark with two types of baseflows (laminar and turbulent Poiseuille flows) and by comparison with other numerical and experimental results. An unstable surface mode is observed at 1000 Hz, and we find good agreement with experimental data for the turbulent mean flow associated with the One-Way Navier–Stokes equations.

A new generalized complex modified Korteweg–de Vries (mKdV) equation is studied by using Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a 3 × 3 matrix spectral problem for the generalized complex mKdV equation. Then, we can formulate the Riemann-Hilbert problem via the spectral analysis of the $x$-part of the Lax pair. According to the symmetry properties of the potential matrix, we find two cases of zero structures for the Riemann-Hilbert problem. By solving the particular Riemann-Hilbert problem and using the inverse scattering transformation, we obtain the unified formulas of the $N$-soliton solutions for the generalized complex mKdV equation. In addition, the dynamical behaviors of the single-soliton solution and the two-soliton solution are analyzed by choosing appropriate parameters.

We study dynamics of dark solitons in the theory of the derivative nonlinear Schrödinger equations by the method based on imposing the condition that this dynamics must be Hamiltonian. Combining this condition with Stokes’ remark that relationships for harmonic linear waves and small-amplitude soliton tails satisfy the same linearized equations, so the corresponding solutions can be converted one into the other by replacement of the packet’s wave number $k$ by $i\kappa$, $\kappa$ being the soliton’s inverse half-width, we find the Hamiltonian and the canonical momentum of the soliton’s motion. The Hamilton equations are reduced to the Newton equation whose solutions for some typical situations are compared with exact numerical solutions of the Kaup-Newell DNLS equation.

In this study, we report a numerical design and observation of bound states in the continuum (BICs) for Lamb waves. BICs for elastic-wave systems, especially in non-periodic configurations, are difficult to obtain due to their intricate polarization states. However, the study in this matter has become very important, especially in the field of opto-mechanics or other multi-field couplings at micro or nanoscales. To illustrate the design concept, we simulate the introduction of a piece of silica (SiO₂) into a thin infinite Si plate and show that, for specific aspect ratios, BICs for elastic waves can be predicted. We present numerical results for both two-dimensional (2D) rectangular plates and three-dimensional (3D) disk structures. Moreover, we also investigate the modal contributions of both the background and inclusion media during the occurrence of BICs, further verifying the physical background of our design strategy. Although we have focused our work on asymmetric Lamb modes, the current method can also be applied to construct other types of elastic-wave BICs, providing a powerful tool for metamaterial device prototyping based on the control or guiding of elastic waves.

The coupled modified nonlinear Schrödinger equation, which can be regarded as a combination of nonlinear Schrödinger and derivative nonlinear Schrödinger equations, is investigated by Darboux transformation (DT) method. Based on the vector modified derivative nonlinear Schrödinger equation spectral problem, the related Lax pair and DT in compact determinant form are all successfully constructed to ensure integrability of the coupled modified nonlinear Schrödinger equation. According to DT method and the limiting technique, two main types of solutions that soliton molecules (SMs) and breather positon (B-P) solutions are systematically discussed. Beginning form zero plane wave backgrounds, the general $M-$SM-$(N-M)$-soliton solutions ($0\le M\le N$), including $M$ SMs and $N-M$ solitons, are subtly derived by the received DT. In particular, two degenerate cases can be reduced from the above general solutions, i.e., $N$-SM solutions ($M=N$) and $N$-soliton solutions ($M=0$). From the nonzero plane wave backgrounds, the higher-order B-P solutions are constructed via both DT and the limiting technique. It is interestingly shown that the central region of B-P solutions exhibit the patterns of rogue waves, and thus they are suggested to explain the generating mechanism of rogue waves. Finally, the corresponding dynamics of these received solutions are discussed in detail.