Wave Motion最新文献

The solution of the macroscopic fluctuation theory (MFT) equation can describe the optimal path of the process, and the Darboux transformation (DT) method can solve soliton solution of some integrable equations. In this paper, we obtained the exact solutions of the coupled macroscopic fluctuation theory (CMFT) equations using the DT method. By constructing a novel type of Lax pairs with $\sqrt{ik}$, we derive some expressions for the 1-soliton, 2-soliton, and $n$-soliton solutions of the CMFT equations, including some soliton solutions, breather solutions and rational wave solutions. Based on these solutions, we consider the elastic interactions and dynamics between two solitons in CMFT equations. These results can present some novel phenomena in the optimal path of the process.

In this study, we systematically review various ripplon solutions to the Kadomtsev–Petviashvili equation with positive dispersion (KP1 equation). We show that there are mappings that allow one to transform the horseshoe solitons and curved lump chains of the KP1 equation into circular solitons of the cylindrical Korteweg–de Vries (cKdV) equation and two-dimensional solitons of the cylindrical Kadomtsev–Petviashvili (cKP) equation. Then, we present analytical solutions that describe new nonlinear highly localized ripplons of a horseshoe shape. Ripplons are two-dimensional waves with an oscillatory structure in space and a decaying character in time; they are similar to lumps but non-stationary. In the limiting case, the horseshoe ripplons reduce to solitons decaying with time and having bent fronts. Such entities can play an important role in the description of strong turbulence in plasma and other media.

In the present work we explore the potential of models of the discrete nonlinear Schrödinger (DNLS) type to support spatially localized and temporally quasiperiodic solutions on top of a finite background. Such solutions are rigorously shown to exist in the vicinity of the anti-continuum, vanishing-coupling limit of the model. We then use numerical continuation to illustrate their persistence for finite coupling, as well as to explore their spectral stability. We obtain an intricate bifurcation diagram showing a progression of such solutions from simpler ones bearing single- and two-site excitations to more complex, multi-site ones with a direct connection of the branches of the self-focusing and self-defocusing nonlinear regime. We further probe the variation of the solutions obtained towards the limit of vanishing frequency for both signs of the nonlinearity. Our analysis is complemented by exploring the dynamics of the solutions via direct numerical simulations.

In this paper, we systematically investigate the Riemann–Hilbert (RH) approach and obtain the soliton solutions for the inhomogeneous discrete nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs). Starting from the spectral problem and introducing the uniformization variable $\kappa$ to avoid the complexity of double-valued function and Riemann surface, we deduce the analyticity, asymptotics and symmetries of the eigenfunctions and scattering coefficients, then the RH problem and reconstruction formula for the potential are successfully constructed. Under reflectionless condition and combining the time evolution of the scattering coefficients and eigenfunctions, we obtain various first-order soliton solutions with different direction of propagation caused by the change of the coefficients. Based on the analytic solution and the choice of special parameter values, we obtain the collision mechanism of two soliton solutions. Furthermore, the important advantage of the RH problem is that it can be further used to study the soliton resolution and the long-time asymptotic behavior of the solutions.

Prior knowledge of the dispersion curves and mode shapes of guided waves provides valuable information for wave mode selection and excitation in the field of non-destructive evaluation (NDE) and structural health monitoring (SHM). They are typically computed by the matrix methods, the finite element (FE) and semi-analytical finite element (SAFE) methods. However, the former is prone to numerical instability, and the latter two are limited by the refinement level of the FE mesh. In this paper, a semi-analytical wavelet finite element (SAWFE) method is presented to characterize wave propagation in rectangular rods. The piecewise polynomial interpolation functions of the SAFE method are replaced by two-dimensional scaling functions of the B-spline wavelet on the interval (BSWI). To demonstrate the accuracy of the proposed SAWFE technique, the propagation of guided waves in an aluminium plate is studied first. Then, the propagation of guided waves in rectangular rods of arbitrary aspect ratio is investigated. The results of this work clearly show that the SAWFE method presented here has higher accuracy and efficiency than the SAFE method.

In this paper, we derive a Kadomtsev-Petviashvili equation by using the multi-scale expansion and perturbation method, which is a model from the potential vorticity equation in the traditional approximation and describes the Rossby wave propagation properties. The N-soliton solutions, resonance Y-type soliton solutions, resonance X-type soliton solutions and interaction solutions of the equation are obtained with the help of dependent-variable transformation. In addition, the composite graphs are given to view the resonance phenomenon of Rossby waves. The results better enrich the research of Rossby waves in ocean dynamics and atmospheric dynamics.

In this article, we study solitary wave solutions of the generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation using higher-order shape elements in the Galerkin finite element method (FEM). Higher-order elements in FEMs are known to produce better results in solution approximations; however, these elements have received fewer studies in the literature. As a result, for the finite element analysis of the gBBMB equation, we consider Lagrange quadratic shape functions. We employ the Galerkin finite element approximation to derive a priori error estimates for semi-discrete solutions. For fully discrete solutions, we adopt the Crank–Nicolson approach, and to handle nonlinearity, we utilize a predictor–corrector scheme with Crank–Nicolson extrapolation. Additionally, we perform a stability analysis for time using the energy method. In the space, $O\left(h^3\right)$ convergence in $L^2\left(\Omega \right)$ norm and $O\left(h^2\right)$ convergence in $H^1\left(\Omega \right)$ norm are observed. Furthermore, an optimal $O\left(\Delta t^2\right)$ convergence in the maximum norm for the temporal direction is also obtained. We test the theoretical results on a few numerical examples in one- and two-dimensional spaces, including the dispersion of a single solitary wave and the interaction of double and triple solitary waves. To demonstrate the efficiency and effectiveness of the present scheme, we compute $L^2$ and $L^{\infty }$ normed errors, along with the mass, momentum, and energy invariants. The obtained results are compared with the existing literature findings both numerically and graphically. We find quadratic shape functions improve accuracy in mass, momentum, energy invariants and also give rise to a higher order of convergence for Galerkin approximations for the considered nonlinear problems.

To analyze structural systems’ oscillatory behaviors, time integration is a versatile broadly accepted time-consuming tool. In 2008, a technique, recently addressed as the SEB THAAT (Step-Enlargement-Based Time-History-Analysis-Acceleration-Technique), was proposed to accelerate the analysis when the excitation is available in digitized format. After many successful experiences regarding structural dynamic analysis, in this paper, it is tested whether the SEB THAAT can be successfully applied to seismic wave propagation analyses. As the main results; for wave propagation analyses, by using the SEB THAAT, we may be able to reduce the analysis run-time without significantly affecting the accuracy of the response; and the amount of the reduction is considerable and around those of structural dynamic analyses. Besides, when the behavior is highly oscillatory, application of the SEB THAAT may be unsuccessful, while the accuracy considerations also do not recommend using the SEB THAAT. The achievements broaden applicability of the SEB THAAT and, due to the improved efficiency, may increase the interest of structural engineers in seismic time history analysis.

We derive the rogue wave solutions of the Kuralay-II equation by applying the Darboux transformation method with the Lax pair on the periodic background. These solutions are represented using Jacobian elliptic functions: dnoidal and cnoidal. The rogue wave solutions can be obtained on the periodic background while the dnoidal travelling periodic wave and cnoidal travelling periodic wave are modulationally unstable with respect to the long-wave perturbations.

We report results of systematic investigation of dynamics featured by moving two-dimensional (2D) solitons generated by the fractional nonlinear Schrödinger equation (FNLSE) with the cubic-quintic nonlinearity. The motion of solitons is a nontrivial problem, as the fractional diffraction breaks the Galilean invariance of the underlying equation. The addition of the defocusing quintic term to the focusing cubic one is necessary to stabilize the solitons against the collapse. The setting presented here can be implemented in nonlinear optical waveguides emulating the fractional diffraction. Systematic consideration identifies parameters of moving fundamental and vortex solitons (with vorticities 0 and 1 or 2, respectively) and maximum velocities up to which stable solitons persist, for characteristic values of the Lévy index which determines the fractionality of the underlying model. Outcomes of collisions between 2D solitons moving in opposite directions are identified too. These are merger of the solitons, quasi-elastic or destructive collisions, and breakup of the two colliding solitons into a quartet of secondary ones.