Search Results (27)

The variable-coefficient Korteweg–de Vries equation $u_t+f\left(t\right)u{u}_x+g\left(t\right)u_{xxx}+h\left(t\right)u_x+\sigma \left(t\right)u=0$ is bilinearized by the Hirota method after gauge and Galilean reductions, and exact N-soliton solutions are shown to exist precisely under the integrability condition $\sigma +{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}ln(g/f)=0$, equivalent to reducibility to the constant-coefficient equation. Closed-form laws are obtained for the soliton amplitude $A=3k_1^{2}g/f$, width, velocity, and trajectory; the two-soliton collision is proved strictly elastic, with a coefficient-independent phase shift and no fusion or fission; the conservation laws are recast as exactly modulated invariants; and a constructive coefficient-programming design realizes prescribed-amplitude and shape-preserving solitons. The physically derived variable-depth equation satisfies the condition only at constant depth, so that a generic sloping bottom admits only the adiabatic single-soliton regime ($A\propto D^{-1}$), compared structurally with the classical shoaling laws. Every solution is verified by direct symbolic substitution. Full article

The study of turbulent Prandtl number is important for turbulent flows with heat transfer. Despite its importance, no universal model exists that is able to capture its behavior for a range of molecular Prandtl numbers. In this study, new turbulent Prandtl number models were developed using multi-gene gene expression programming (MGGEP) based on direct numerical simulation (DNS) data for heated periodic channel flows. DNS datasets covering both medium- and low-Prandtl-number fluids were employed to construct more universal closures suitable for Reynolds-averaged Navier–Stokes (RANS) simulations. Two case studies were conducted. In the first case study, turbulent Prandtl number models optimized for air (Pr = 0.71) were obtained using the friction Reynolds number and normalized wall distance as the physical inputs. In the second case study, generalized models applicable to both medium- and low-Prandtl-number fluids (down to Pr = 0.025) were developed. A novel Galilean-invariant local Reynolds number parameter was introduced to accurately capture the near-wall behavior and spatial variations in turbulent heat transfer. The resulting models demonstrated mean percent relative errors below 3% for the turbulent Prandtl number compared with the DNS data, while existing models in the literature show errors of up to 26.7%. In terms of root mean square error for the periodic channel flow, medium Prandtl number studies showed root mean square error reduction from 0.0596 to 0.0302, and low-Prandtl-number studies exhibited root mean square error reduction from 0.3128 to 0.0256 when MGGEP models and models from the literature are compared with respect to turbulent Prandtl number. The models were also validated using the turbulent periodic pipe flow problem, where the mean percent relative error for the turbulent Prandtl number decreased from 31.0% to 5.8%. The developed models were subsequently implemented in RANS simulations, showing that the proposed turbulent Prandtl number models lead to highly accurate temperature predictions for the periodic channel flow problem. Full article

This work places the invariant $d{s}^2$ at the center of the gravitational interaction, interpreting it not as a purely geometric object but as the differential of proper time, endowed with direct physical meaning. Starting from the extension of Fermat’s principle to massive particles—namely, the requirement that freely falling bodies follow trajectories that extremize proper time, which for timelike motion corresponds to a local maximum—and invoking the universality of Galilean free fall, we derive the form of $d{s}^2$ in a static gravitational field. Lorentz invariance then provides the natural framework to extend this result to systems involving moving matter. The invariant derived through this procedure matches the weak-field limit of General Relativity formulated in the harmonic gauge. Within this linearized regime, we show that the structure of the theory already contains the seeds of its nonlinear completion: any dynamically consistent extension to strong gravitational fields necessarily involves the Ricci tensor. From this viewpoint, Einstein’s field equations appear not as a postulated geometric law but as the unique covariant closure required to ensure energy–momentum conservation and the self-consistency of the gravitational interaction. Full article

Objectivity is a fundamental requirement for vortex identification, ensuring that vortex structures observed remain invariant under changes in the reference frame. However, although most conventional vortex identification methods, including Liutex, are Galilean invariant, they are not objective. Since the accelerated motion of the observer does not affect the velocity gradient tensor at an instant of time, the rotational motion is only considered for the non-inertial frame. This paper proposes a method to recover the angular velocity of a rotating observer directly from flow field data measured in the rotating frame. The approach exploits the observation that, in an inertial frame, zero-vorticity points tend to dominate the region with an almost identical nonzero vorticity in the observer’s non-inertial coordinate system. By identifying the most frequently occurring vorticity within the domain, the observer’s angular velocity can be uniquely determined, enabling reconstruction of the objective velocity gradient tensor and, consequently, the objective Liutex. The key issue is to find a reference point (RP). The RP should have zero vorticity in the inertial coordinate system, and then the RP has the same angular speed as the observer. The RP can be found by comparing the vorticity of all points in the computational domain and the RP will correspond to the vorticity vector with the highest percentage in the non-inertial coordinate system. The proposed method is validated using DNS data of the boundary layer transition over a flat plate with an artificially imposed angular velocity. The recovered angular velocity agrees closely with the true value within an acceptable margin of error. Furthermore, the objective Liutex reconstructed from the rotating frame data is visually indistinguishable from the original inertial frame Liutex. These results demonstrate that the method provides a simple and accurate way to restore objectivity for Liutex and other vortex identification techniques. The objective Liutex will be equal to the original Liutex in an inertial coordinate system when the observer does not have rotational motion. Full article

This manuscript presents a comprehensive Lie symmetry analysis of the KdV-Burgers equation, a prototypical model for nonlinear wave dynamics incorporating dissipation and dispersion. We systematically derive its six-dimensional Lie algebra and construct an optimal system of one-dimensional subalgebras. This framework is used to perform a symmetry reduction, transforming the governing partial differential equation into a set of ordinary differential equations. A key contribution of this work is the identification and analysis of several non-trivial invariant solutions, including a new Galilean-boost-invariant solution related to an accelerating reference frame, which extends beyond standard traveling waves. Through a detailed physical interpretation supported by phase plane analysis and asymptotic methods, we elucidate how the mathematical symmetries directly manifest as fundamental physical behaviors. This reveals a clear classification of distinct wave regimes—from monotonic and oscillatory shocks to solitary wave trains governed by the interplay between nonlinearity, dissipation and dispersion. The numerical validation verify the accuracy and physical relevance of the derived invariant solutions, with errors less than $0.5\%$ in the Burgers limit and $3.2\%$ in the weak dissipation regime. Our work establishes a direct link between the model’s symmetry structure and its observable dynamics, providing a unified framework validated both analytically and through the examination of universal scaling laws. The results offer profound insights applicable to fields ranging from plasma physics and hydrodynamics to nonlinear acoustics. Full article

We propose a Ramsey approach to the dimensional analysis of physical systems, which complements the seminal Buckingham theorem. Dimensionless constants describing a given physical system are represented as vertices of a graph, referred to as a dimensions graph. Two vertices are connected by an aqua-colored edge if they share at least one common dimensional physical quantity and by a brown edge if they do not. In this way, a bi-colored complete Ramsey graph is obtained. The relations introduced between the vertices of the dimensions graph are non-transitive. According to the Ramsey theorem, a monochromatic triangle must necessarily appear in a dimensions graph constructed from six vertices, regardless of the order of the vertices. Mantel–Turán analysis is applied to study these graphs. The proposed Ramsey approach is extended to graphs constructed from fundamental physical constants. A physical interpretation of the Ramsey analysis of dimensions graphs is suggested. A generalization of the proposed Ramsey scheme to multi-colored Ramsey graphs is also discussed, along with an extension to infinite sets of dimensionless constants. The introduced dimensions graphs are invariant under rotations of reference frames, but they are sensitive to Galilean and Lorentz transformations. The coloring of the dimensions graph is independent of the chosen system of units. The number of vertices in a dimensions graph is relativistically invariant and independent of the system of units. Full article

The graph theory-based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting the vertices. The bi-colored, complete graph emerges. This graph is called the “momenta graph”. According to the Ramsey theorem, this graph contains at least one mono-chromatic triangle. This is true even for chaotic motion of three bodies; thus, illustrating the idea supplied by the Ramsey theory, total chaos is impossible. Coloring of the graph is independent on the rotation of frames; however, it is sensitive to Galilean transformations. The coloring of the momenta graph remains the same for general linear transformations of vectors with a positive-definite matrix. For a given motion, changing the order of the vertices does not change the number and distribution of monochromatic triangles. Symmetry of the momenta graph is addressed. The symmetry group remains the same for general linear transformation of vectors of the linear and angular momenta with a positive-definite matrix. Conditions defining conservation of the coloring of the momenta graph are addressed. The notion of the stereographic momenta graph is introduced. Shannon entropy of the momenta graph is calculated. The particular configurations of bodies are addressed, including the Lagrange configuration and the figure eight-shaped motion. The suggested approach is generalized for the quantum field theory with the Pauli–Lubanski pseudo-vector. The suggested coloring procedure is the Lorenz invariant. Full article

The pairing of non-relativistic and relativistic Lagrangians within the context of fluid mechanics, advancing methodologies for constructing Poincare-invariant Lagrangians, is explored. Through leveraging symmetries and Noether’s theorem in an inverse framework, three primary cases are investigated: potential flow, barotropic flow expressed in terms of Clebsch variables, and an extended Clebsch Lagrangian incorporating thermodynamic effects. To ensure physical correctness, the eigenvalue relation of the energy–momentum tensor, together with velocity normalisation, are applied as key criteria. The findings confirm that the relativistic Lagrangians successfully reduce to their non-relativistic counterparts in the limit $c\to \infty$. These results demonstrate a systematic approach that enhances the relationship between symmetries and variational formulations, providing the advantage of deriving Lagrangians that unify non-relativistic and relativistic theories. Full article

We propose a Ramsey-theory-based approach for the analysis of the behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by a Ramsey-theory-based, bi-colored, complete graph. Vectors of momenta of the particles ${\overrightarrow{p}}_i$ serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, where the momenta of the particles in this system are ${\overrightarrow{p}}_{cmi}.$ If ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))\ge 0$ is true, the vectors of momenta of the particles numbered i and j are connected with a red link; if ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))<0$ takes place, the vectors of momenta are connected with a green link. Thus, the complete, bi-colored graph emerges. Considering an isolated system built of six interacting particles, according to the Ramsey theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relative to the rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of the frames emerging from the rotation/translation of the original frame. This gives rise to a novel kind of mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed. Full article

A model of quasilinear differential equations is derived in the context of Rational Extended Thermodynamics to investigate some non-equilibrium phenomena in nanofluids. Following the classical Buongiorno approach, the model assumes nanofluids to be suspensions of two phases: nanoparticles and the base fluid. The field variables are the classical ones and, in addition, the stress tensors and the heat fluxes of both constituents. Balance laws for all field variables are assumed. The obtained system is not closed; therefore, universal physical principles, such as Galilean Invariance and the Entropy Principles, are invoked to close the set of field equations. The obtained model is also written in terms of the whole nanofluid and compared with the classical Buongiorno model. This allowed also the identifications of some parameters in terms of experimental data. The obtained set of field equations has the advantage to recover the Buongiorno model when the phenomena are near equilibrium. At the same time it consists of a hyperbolic set of field equations. Hyperbolicity guarantees finite speeds of propagation and more suitable descriptions of transient regimes. The present model can be used in order to investigate waves, shocks and other phenomena that can be easily described in hyperbolic systems. Furthermore, as a first application and in order to show the potential of the model, stationary 1D solutions are determined and some thermal properties of nanofluids are studied. The solution exhibits, already in the simplest case herein considered, a more accurate evaluation of some fields like the stress tensor components. Full article

A rigorous analysis is undertaken based on the analysis of both Galilean and Lorentz (Poincaré) invariance in field theories in general in order to (i) identify a unique analytical scheme for canonical pairs of Lagrangians, one of them having Galilean, the other one Poincaré invariance; and (ii) to obtain transition conditions for the state function purely from Hamilton’s principle and extended Noether’s theorem applied to the aforementioned symmetries. The general analysis is applied on Schrödinger and Klein–Gordon theory, identifying them as a canonical pair in the sense of (i) and exemplified for the scattering problem for both theories for a particle beam against a potential step in order to show that the transition conditions that result according to (ii) in a ‘natural’ way reproduce the well-known ‘methodical’ continuity conditions commonly found in the literature, at least in relevant cases, closing a relevant argumentation gap in quantum mechanical scattering problems. Full article

A nonlocally perturbed linear Schrödinger equation with a small parameter was derived under the assumption of low-level fractionality by using one of the known general nonlocal wave equations with an infinite power-law memory. The problem of finding approximate symmetries for the equation is studied here. It has been shown that the perturbed Schrödinger equation inherits all symmetries of the classical linear equation. It has also been proven that approximate symmetries corresponding to Galilean transformations and projective transformations of the unperturbed equation are nonlocal. In addition, a special class of nonlinear, nonlocally perturbed Schrödinger equations that admits an approximate nonlocal extension of the Galilei group is derived. An example of constructing an approximately invariant solution for the linear equation using approximate scaling symmetry is presented. Full article

In the past few years, distant retrograde orbits (DROs) have become increasingly popular due to their conspicuous stability. Nevertheless, it is this characteristic that results in the challenge to the design of transfer orbits into/out of DROs. This paper investigates the DROs around Ganymede in order to utilize their dynamical characteristics for Jupiter system exploration. In particular, the DRO family is calculated by numerical integration and numerical continuation, higher-period orbits near the DROs are detected using bifurcation theory, and characteristics including orbital stability and invariant manifolds of these orbits are investigated through stability indices and manifold theory. The stability of DROs and the higher-period orbits are first investigated in the circular restricted three-body problem and are then verified in a third-body gravitation perturbation model. The results show that the strong stability of DROs makes it possible to observe the Galilean moons for long periods and that the higher-period orbits that bifurcate from the DROs offer additional insight into the motion of probes approaching/departing from the vicinities of the DROs. Further investigation of the invariant manifolds around higher-period orbits reveals the feasibility of utilizing the DRO family and the nearby unstable structures for multi-target exploration and low-energy transfer between the Galilean moons. Full article

In this paper, symmetry properties of the basic equations of atmospheric motion are proposed. The results on symmetries show that the basic equations of atmospheric motion are invariant under space-time translation transformation, Galilean translation transformations and scaling transformations. Eight one-parameter invariant subgroups and eight one-parameter group invariant solutions are demonstrated. Three types of nontrivial similarity solutions and group invariants are proposed. With the help of perturbation method, we derive the second-order approximate equations for the large-scale atmospheric motion equations, including the non-dimensional equations and the dimensional equations. The second-order approximate equations of the large-scale atmospheric motion equations not only show the characteristics of physical quantities changing with time, but also describe the characteristics of large-scale atmospheric vertical motion. Full article

This study examines the invariance properties of the thermodynamic entropy production in its global (integral), local (differential), bilinear, and macroscopic formulations, including dimensional scaling, invariance to fixed displacements, rotations or reflections of the coordinates, time antisymmetry, Galilean invariance, and Lie point symmetry. The Lie invariance is shown to be the most general, encompassing the other invariances. In a shear-flow system involving fluid flow relative to a solid boundary at steady state, the Galilean invariance property is then shown to preference a unique pair of inertial frames of reference—here termed an entropic pair—respectively moving with the solid or the mean fluid flow. This challenges the Newtonian viewpoint that all inertial frames of reference are equivalent. Furthermore, the existence of a shear flow subsystem with an entropic pair different to that of the surrounding system, or a subsystem with one or more changing entropic pair(s), requires a source of negentropy—a power source scaled by an absolute temperature—to drive the subsystem. Through the analysis of different shear flow subsystems, we present a series of governing principles to describe their entropic pairing properties and sources of negentropy. These are unaffected by Galilean transformations, and so can be understood to “lie above” the Galilean inertial framework of Newtonian mechanics. The analyses provide a new perspective into the field of entropic mechanics, the study of the relative motions of objects with friction. Full article