The Kadomtsev-Petviashvili equation is a fundamental nonlinear model with applications ranging from plasma physics and shallow water dynamics to ferromagnetism, Bose-Einstein condensation, and string theory. This manuscript investigates a newly formulated (3+1)-dimensional Kadomtsev-Petviashvili equation and presents its novel solution structures. By imposing constraints on the N-soliton solutions, resonant X-type, Y-type, XX-type, YY-type and parallel soliton solutions are derived. Furthermore, M-lump solutions are systematically constructed via the Long-wave limit method, enabling the exploration of higher-order lump dynamics. In addition, the interactions among lumps, soliton waves, and breathers are analyzed to reveal complex nonlinear behaviors. Finally, the physical properties and dynamical behaviors of these localized waveforms are illustrated through three-dimensional surface plots and corresponding contour representations. These results reveal new nonlinear phenomena and extend the theoretical understanding of soliton theory in higher dimensions.
The paper is a strict exploration of the existence, dynamic behavior, and stability of optical soliton states of a (3+1)-dimensional Boussinesq equation system, which is conventionally used to describe fluid mechanics systems but is here applied to the dynamics of waves in nonlinear optical materials. Through the application of two powerful analytic procedures, the expansion method of the ϕ ^6 -model expansion method and the modified exp(-ϕ (ϱ )) -expansion function method, we have been able to obtain a spectrum of new exact soliton solutions that explain the fine balance between higher-order dispersion and nonlinearity in the multidimensional setting. One of the most important contributions of the given work is the sensitivity analysis of the relevant dynamical system in which the transition between regular soliton behavior and the singular or chaotic one is determined systematically based on the values of the main physical parameters, including stratification and nonlinear gain, which determine the parametric regimes of stable propagation. The results do not merely add to the mathematical physics of the Boussinesq equation, but they provide a new theoretical basis to develop robust high-dimensional optical communications in which interactions between complex spatiotemporal dynamics are common. The physical relevance of the solutions is further confirmed by graphical simulations, which provide a clear picture of the underlying wave phenomena.
This study investigates the pressure distribution in porous media while accounting for the memory effects in fluid flow. A mathematical model is developed to describe fluid flow in both homogeneous and heterogeneous porous media under various boundary conditions. The presence of porous media can lead to anomalous diffusion, which preserves the history of the interactions between the fluid and medium. To capture this behavior, the Caputo-type fractional derivative is incorporated into Darcy’s law. The problem is analyzed numerically using a spectral method based on orthogonal polynomials. In this approach, the derivatives of orthogonal polynomials are represented in matrix form, which is known as the operational matrices. These matrices provide a systematic framework for handling polynomials, facilitating their application to fluid flow problems. By employing operational matrices, the original fractional differential equation is transformed into a system of linear algebraic equations. A closed-form approximation of the fluid pressure is then obtained. The convergence analysis of the proposed method is presented, and its stability is demonstrated numerically using a perturbation approach. The accuracy of the method is validated by comparing the results of the proposed operational matrix scheme with existing methods in the literature. Numerical results indicate that the influence of the fractional-order derivative on pressure distribution depends on both the boundary conditions and the permeability of the medium.
This paper introduces Lineal, a new linear algebra library optimized for runtime and especially memory efficiency to solve large sparse linear systems arising from PDE discretization on attainable hardware. Lineal supports general matrix-based linear systems using Compressed Sparse Row (CSR) matrices and matrix-free linear systems for stencil-based problems, which can store a single value per cell to minimize memory use. Both can be paired with MAMGO, Lineal’s Algebraic Multi-Grid implementation based on DUNE ISTL’s algorithm, which can use a matrix-free linear system on the finest, most memory-intensive level and CSR matrices on coarser levels, greatly reducing memory consumption—a novel approach, to our knowledge. Lineal also offers very flexible support for mixed precision to minimize memory use and the effect of reduced precision on convergence. Almost all components are fully multithreaded and many support explicit SIMD operations and/or tiling. Tests solving the stationary diffusion problem using two sets of high-resolution scans of soil samples show Lineal’s very good strong scaling and high performance. On the smaller dataset, Lineal outperforms hypre’s BoomerAMG, DUNE ISTL, and Ginkgo, requiring under half the time and memory with CSR matrices only and less than a quarter with the hybrid matrix-free approach. Unlike the other libraries, Lineal can handle the much larger dataset on the test system due to its low memory requirements. On the SPE10 Model 2 benchmark, Lineal is somewhat slower than BoomerAMG but uses much less memory, and it is faster than the other libraries.
Examining the exact soliton solutions of the modified Korteweg-de Vries-Kadomtsev-Petviashvili (KDV-KP) equation is the subject of this article. The studied equation, which manages the physical propagation of waves, explicitly describes how nonlinearity and dispersion can generate complex and captivating wave phenomena, such as solitary waves and wave turbulence. The equations discussed herein find practical applications in fluid dynamics when applied to simulate shallow water waves, including those encountered in rivers and tsunamis; astrophysical plasmas and fusion reactors; and nonlinear optics, where they are employed to analyse the propagation of light through nonlinear media, including optical fibres. Solitons find application in diverse domains, such as optical communications (for misdirected transmission of information over vast distances) and fluid dynamics (to simulate the behavior of long-lasting ocean waves). A wide range of exact analytical wave solutions with mixed, bright, dark, singular, and hybrid solitonic forms has been obtained using the application of the modified Sardar subequation approach and modified generalized Riccati equation mapping method. The novelty of this work lies in the application of the our proposed advanced methods to obtain diverse exact traveling wave solutions of the considered model. Using these techniques, several new families of analytical solutions–including bright, dark, singular, mixed, and hybrid soliton structures–are constructed and verified, demonstrating the effectiveness of the proposed approaches for solving nonlinear evolution equations. The accuracy of all derived solutions is checked using a direct substitution into the field equation using a Mathematica software package. Moreover, contour plots and 2D and 3D plots will be shown for a variety of parameter choices to further identify and explain the nature and behavior of these derived solutions, thus emphasizing and validating the accuracy and efficiency of this technique. We anticipate that our work will be helpful for a large number of engineering models and other related problems.
The exact wave structures of a novel integrable (3+1) -dimensional wave equation related to shallow water wave phenomena are investigated. The Bell polynomial approach is employed to construct the Hirota bilinear form, Bäcklund transformation, and Lax pair of the model. Exact solutions, including two-wave, mixed lump–kink, transformed breather molecule, and solitary wave structures, are derived by using different analytical techniques. In addition, the Lie symmetry method is applied to reduce the governing equation to an ordinary differential equation, for which solitary wave solutions are obtained through the generalized logistic equation framework. The graphical features of the obtained solutions are illustrated through three-dimensional, contour, and line plots.
We consider the topographic bias in gravimetric geoid determination when analytically downward continuing the topographic potential from the Earth’s surface or above down to sea level. The total bias is subdivided into those of the Bouguer shell or plate and the terrain. In this process the potential of the Bouguer shell always causes a bias, which increases with the square of the topographic height and typically exceeds 1–2 cm for elevations higher than 1 km. The main understanding today is that the terrain does not provide a potential bias except possibly for masses located inside a dome of height ( √(2) - 1)H_P , centered along the vertical through the computation point P at height H_P with a lateral base radius s_0 = H_P at sea-level. This result implies that the potential of all terrain masses of arbitrary density located exterior to the dome are unbiasedly downward continued to sea level. In this article a homogenous cylinder and a cone are used to verify the above results. If the computation point P is located at or above the cylinder along its axis and the radius of the cylinder is larger than the height of P, there is no terrain bias. Similar result is obtained for the cone with base radius larger than the height H_P , except when P is located at the vertex of the cone, in which case necessary vertical derivatives in the harmonic continuation process formally does not exist, but a solution can still be handled in the numerical/practical application.
This study aims to extract closed-form soliton solutions and illustrate the evolving forms of solitons for a significant class of nonlinear evolution equations, focusing specifically on the (3+1)-dimensional Boussinesq equation. This equation is a mathematical model for describing complicated wave processes in fluid dynamics, oceanography, plasma physics, and earth and atmospheric sciences. Due to its multidimensional nature and strong nonlinearity, finding exact solutions to this equation poses a considerable analytical challenge. To address this, we implement two methods, namely, the Generalized Exponential Rational Function (GERF) method and the Modified Sardar Sub-equation (MSSE) method. These analytical techniques are known for their ability to generate closed-form solutions for higher-order nonlinear partial differential equations. Through these methods, we obtain various types of accurate soliton solutions, including dark and bright solitons, kink and anti-kink profiles, periodic waves, W-shaped structures, and multi-soliton solutions. The dynamic aspects of the solitonic profiles, as well as newly formed exact solutions, are graphically represented in 2D and 3D with density graphs. By tuning the free parameters present in the solution expressions, various types of wave patterns can be outlined, which highlight the rich dynamics of the system. The findings of our study enable us to gain a thorough understanding of the evolving features of higher-order nonlinear evolution equations in Earth and atmospheric sciences. They also demonstrate how modern mathematical methods, such as the GERF and MSSE methods, can effectively represent complex physical processes in various physical and engineering systems.
This problem concerns the propagation of horizontally polarised magnetoelastic shear waves on an inhomogeneous fibre-reinforced layer overlying a fibre-reinforced half space under gravity. The assumption is that the lower layer is an FRC half-space subject to gravity, and the upper layer is a heterogeneous FRC medium with initial stress. Exponential (case I) and linear (case II) variations in the elastic property cause inhomogeneity in the upper layer. The layered structure is separated by corrugation and a loosely bonded interface. It is evident from the graphical representation that the inhomogeneous parameters, including the corrugation parameter, loosely bonding, and the Magnetoelastic parameter, have a notable effect on the phase velocity and wave number. The findings may be utilized in geophysics and civil engineering.
In this study, we develop various soliton solutions to the (3 + 1)-dimensional time fractional Kadomtsev–Petviashvili-I equation, which is relevant to fluid dynamics under strong surface tension and provides a comprehensive dynamical analysis of the system. Using the ( G^'/G,1/G )-expansion scheme and the improved F-expansion method together with the beta fractional derivative, hyperbolic, trigonometric, and algebraic soliton solutions are obtained. The obtained solutions represent diverse wave structures, including bell-shaped, kink-shaped, compacton, and singular periodic solitons. It is shown that the soliton profiles and dynamical behavior vary significantly with changes in the fractional-order parameter. The dynamical analysis identifies both stable center points and unstable saddle equilibrium states. We also demonstrate that the system’s behavior is highly sensitive to the choice of initial conditions. This work improves the theoretical understanding of the phenomena by extending the class of soliton solutions and revealing new dynamical features. It also shows that the adopted methods are effective for studying fractional nonlinear wave models.
The Boussinesq type equation is a versatile nonlinear model for describing wave propagation in dispersive media, including shallow-water dynamics, tsunami evolution, and internal ocean waves. This study applies the Hirota bilinear method (HBM) to the compact form of the modified Boussinesq-type equation to construct an extensive family of exact analytical solutions, encompassing single- and multi-solitons, first- and second-order breathers, lump waves, and hybrid soliton–breather interactions. These solutions exhibit rich nonlinear phenomena such as phase shifts, amplitude modulation, resonance-induced fusion and fission, and long-range algebraic decay. High-resolution 3D, contour, and discrete visualizations confirm the spatiotemporal localization, oscillatory behaviors, and interaction patterns predicted by the analysis. Physically, the results correspond to real-world geophysical processes, including solitary tsunami fronts, modulated wave packets in coastal waters, and slowly decaying disturbances relevant to early tsunami detection. The findings expand the known solution space of the Boussinesq-type system and provide a rigorous mathematical framework for modeling complex wave phenomena in geophysical and environmental fluid contexts.
Understanding earthquake dynamics is essential for seismic hazard assessment and risk mitigation. In this context, Bayesian inference provides valuable insights into model parameters by combining simulation models with real-world data. Such Bayesian parameter inference with uncertainty quantification (UQ) requires numerous simulation runs and is therefore often computationally infeasible. Already, a single high-fidelity earthquake simulation - governed by a linear hyperbolic seismic wave equation coupled nonlinearly to a friction law and plastic deformation - is computationally expensive. In this study, we investigate the use of fused ensemble simulations to accelerate large earthquake simulation workflows and UQ studies. We implement and evaluate this approach in SeisSol, a high-performance computing software for the simulation of complex earthquake events that uses an Arbitrary high-order DERivative Discontinuous Galerkin (ADER-DG) scheme. Via fused ensembles, we turn the element-local small sparse/dense matrix operations into tensor contractions working on a dense rank-3 tensor and sparse matrices. These are again executed via loops of small, sparse/dense matrix operations, but with better computational efficiency, due to better exploitation of SIMD instructions on CPUs. We also compare two implementation variants (with different implementation effort) for kernels modeling non-linear dynamic rupture and effects of material plasticity. Our results demonstrate that fused simulations can be up to 5.54 times faster than a single execution - though this depends strongly on the discretization order, the problem size, and the compute architecture. For a full UQ example workflow, we demonstrate a speedup of 1.6, resulting in 36% savings in node hours for the entire workflow.
The present study explores wave propagation in an isotropic, homogeneous thermoelastic half-space governed by the Moore–Gibson–Thompson (MGT) heat conduction model, incorporating both non-local effects and the influence of hyperbolic two-temperature (HTT) theory. A modified system of governing equations is developed, transformed into a two-dimensional, dimensionless form, and subsequently analyzed using the method of reflections. Reflection coefficients for distinct wave modes namely the longitudinal (P-wave), thermal (T-wave), and shear vertical (SV-wave) are derived at the impedance interface. These amplitude ratios are evaluated numerically and presented graphically to illustrate the individual and combined effects of non-locality, HTT parameters, and impedance conditions. Some special cases and particular cases also derived from the present investigation.
Salinization of coastal aquifers is a current problem in many regions worldwide. Simulation of this process provides an important tool for the forecast of drinking water resources. The consideration of the unsaturated phreatic zone has an essential influence on the accuracy of the prediction. However, for the large temporal and spatial scales of the aquifers, the presence of the unsaturated subdomains creates challenging difficulties for the numerical methods. In this work, we investigate two approaches for simulating haline density-driven flow in partially saturated aquifers. The first approach, based on the Richards equation, uses a representation of the saturation field and employs an adaptive linearly implicit time discretization scheme. The second approach explicitly represents the water table using a level-set method and relies on weak coupling combined with a standard implicit Euler scheme. Both approaches employ a finite-volume discretization for spatial representation of fluid flow and salt transport. The implementation is based on the UG4 toolkit together with the parallel groundwater flow simulator d3f++. The resulting linear systems are solved using a geometric multigrid method. We compare the aforementioned approaches with respect to theoretical and numerical aspects. The performance of both methods is evaluated in three numerical experiments. In particular we demonstrate robustness of the linearly implicit scheme and introduce a stabilization for the level-set method. The high-performance computing (HPC) potential of the approaches is assessed and demonstrated as well.
In this article, we obtain various newly formed closed-form soliton solutions to the Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation by taking advantage of the application of the Jacobi elliptic function expansion method (JEFEM). Jacobi elliptic functions form a well-known fundamental class of special functions in atmospheric and oceanic sciences that have garnered significant attention for their applications in various fields. The dynamics of different evolving forms of the resulting solutions are shown utilizing graphics such as 3D surfaces, 2D graphs, and contour plots through software Mathematica 14.3 with appropriate parametric values. These new solitary wave solutions include singular, dark, bright, and periodic multisolitons. Significantly, all such solutions have varied but potential applications in various fields, including mathematical physics and engineering. The findings of this work demonstrate that this method can significantly help find exact traveling wave solutions to atmospheric sciences. This study introduces a vital way to understand dynamic waveforms of soliton-form solutions and depict the intricate physical processes in various fields of nonlinear research field.
We consider a compressible Stokes problem in the quasi-stationary case coupled with a time dependent advection-diffusion equation with special emphasis on high viscosity contrast geophysical mantle convection applications. In space, we use a P2-P1 Taylor–Hood element which is generated by a blending approach to account for the non-planar domain boundary without compromising the stencil data structure of uniformly refined elements. In time, we apply an operator splitting approach for the temperature equation combining the BDF2 method for diffusion and a particle method for advection, resulting in an overall second order scheme. Within each time step, a stationary Stokes problem with a high viscosity contrast has to be solved for which we propose a matrix-free, robust and scalable iterative solver based on Uzawa type block preconditioners, polynomial Chebyshev smoothers and a BFBT type Schur complement approximation. Our implementation is using a hybrid hierarchical grid approach allowing for massively parallel, high resolution Earth convection simulations.
High order methods have shown great potential to overcome performance issues of simulations of partial differential equations (PDEs) on modern hardware, still many users stick to low-order, matrix-based simulations, in particular in porous media applications. Heterogeneous coefficients and low regularity of the solution are reasons not to employ high order discretizations. We present a new approach for the simulation of instationary PDEs that allows to partially mitigate the performance problems. By reformulating the original problem we derive a parallel in time integrator that increases the arithmetic intensity and introduces additional structure into the problem. By this it helps accelerate matrix-based simulations on modern hardware architectures. Based on a system for multiple time steps we will formulate a matrix equation that can be solved using vectorized solvers like Block Krylov methods. The structure of this approach makes it applicable for a wide range of linear and nonlinear problems. In our numerical experiments we present some first results for three different PDEs, a linear convection-diffusion equation, a nonlinear diffusion-reaction equation and a realistic example based on the Richards’ equation.
Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in z-direction and low Mach numbers, making the system stiff. Thus, implicit time integration is required and for this a fast, highly parallel, low-memory iterative solver for the resulting algebraic systems. As a basic framework, we use inexact Jacobian-Free Newton-GMRES with a preconditioner. For low order finite volume discretizations, multigrid methods have been successfully applied to steady and unsteady fluid flows. However, for high order DG methods, such solvers are currently lacking. This motivates our research to construct a Jacobian-free preconditioner for high order DG discretizations. The preconditioner is based on a multigrid method constructed for a low order finite volume discretization defined on a subgrid of the DG mesh. We design a computationally efficient and mass conservative mapping between the grids. As smoothers, explicit Runge-Kutta pseudo time iterations are used, which can be implemented in parallel in a Jacobian-free low-memory manner. We consider DG Methods for the Euler equations and for viscous flow equations in 2D, both with gravity, in a well balanced formulation. Numerical experiments in the software framework DUNE-FEM on atmospheric flow problems show the benefit of this approach.
Focal mechanism data are essential for estimating tectonic stress tensors, but heterogeneous data quality and unequal observation precision can bias inversions when uniform weights are assumed. We introduce a weighted least-squares inversion combined with variance component estimation (VCE; BIQUE in a Gauss–Helmert framework) that infers observation variances directly from residuals and assigns statistically consistent weights. In our implementation, one variance component is estimated per mechanism and applied jointly to its strike, dip, and slip angles, providing a parsimonious and catalog-consistent stochastic model. We validate the method on three datasets: a 40-event synthetic test, the 2008 West Bohemia swarm (167 mechanisms), and Agia Varvara in central Crete (31 mechanisms). Compared with uniform weighting, VCE produces sharper and more stable principal stress orientations, with azimuth and plunge uncertainties reduced by up to an order of magnitude. Variance–trace metrics decrease substantially, by roughly one order of magnitude in the synthetic test and two orders in West Bohemia. In Central Crete, the uncertainties shrink by a factor of 2–3, and the stress–shape ratio converges toward R≈0.6 . Shape-ratio distributions become narrower and more tectonically consistent, and Mohr–Coulomb instability analysis shows clearer separation between stable and near-failure planes. These results demonstrate that data-driven stochastic weighting via VCE materially improves focal-mechanism stress tensor inversion and offers a reproducible framework for reliable stress tensor analyses in complex tectonic settings.
