Search Results (6)

In a recent paper, denoted by MG24 in this text, we used a modified Korteweg–de Vries (KdV) equation to describe the evolution of wind-driven water wave packets in shallow water. The modifications were several forcing/friction terms describing wave growth due to critical-level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave growth due to turbulent wave stress in the air near the interface, and wave decay due to a turbulent bottom boundary layer. The outcome was a KdV–Burgers type of equation that can be a stable or unstable model depending on the forcing/friction parameters. In most cases that we examined, many solitary waves are generated, suggesting the formation of a soliton gas. In this paper, we extend that model in the horizontal direction transverse to the wind forcing to produce a similarly modified Kadomtsev–Petviashvili equation (KPII for water waves in the absence of surface tension). A modulation theory is described for the cnoidal and solitary wave solutions of the unforced KP equation, focusing on the forcing/friction terms and the transverse dependence. Then, using similar initial conditions to those used in MG24, that is a sinusoidal wave with a slowly varying envelope, but supplemented here with a transverse sinusoidal term, we find through numerical simulations that the radiation field upstream is enhanced, but that a soliton gas still emerges downstream as in MG24. Full article

This paper deals with the exact wave results of the (1+1)-dimensional nonlinear compound Korteweg–De Vries and Burgers (KdVB) equation with a truncated M-fractional derivative. This model represents the generalization of Korteweg–De Vries-modified Korteweg–De Vries and Burgers equations. We obtained periodic, combo singular, dark–bright, and other wave results with the use of the extended sinh-Gordon equation expansion (EShGEE) and modified $(G^{\prime }/G^2)$-expansion techniques. The use of the effective fractional derivative makes our results much better than the existing results. The obtained solutions are useful as well as applicable in various fields, including mathematical physics, plasma physics, ocean engineering, optics, etc. The obtained solutions are demonstrated by 2D, 3D, and contour plots. The achieved results will be fruitful for future research on this equation. Stability analysis is used to check that the results are precise as well as exact. Modulation instability (MI) analysis is performed to find stable steady-state solutions of the abovementioned model. In the end, it is concluded that the methods used are easy and reliable. Full article

The quest to reveal the physical essence of the infinitely many symmetries and/or conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of n-wave solutions, such as the n-soliton solutions, multiple breathers, complexitons, and the n-periodic wave solutions (the algebro-geometric solutions with genus n), wherein n denotes an arbitrary integer that can potentially approach infinity. Each subwave comprising the n-wave solution may possess free parameters, including center parameters $c_i$, width parameters (wave number) $k_i$, and periodic parameters (the Riemann parameters) $m_i$. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinitely many symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. This conjecture appears to hold true for all integrable systems with n-wave solutions. The conjecture intimates that the currently known infinitely many symmetries is not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned Korteweg–de Vries (KdV) equation and the Burgers equation as simple examples, the conjecture is substantiated for the n-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative, which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework. Notably, a ren-symmetric integrable Burgers hierarchy is explicitly derived. Both the supersymmetric and the classical integrable hierarchies are encompassed within the ren-symmetric integrable hierarchy. The results of this paper will make further progresses in nonlinear science: to find more infinitely many symmetries, to establish novel methods to solve nonlinear systems via symmetries, to find more novel exact solutions and new physics, and to open novel integrable theories such as the ren-symmetric integrable systems and the possible relations to fractional integrable systems. Full article

The example of two families of finite-difference schemes shows that, in general, the numerical solution of the Riemann problem for the generalized Hopf equation depends on the finite-difference scheme. The numerical solution may differ both quantitatively and qualitatively. The reason for this is the nonuniqueness of the solution to the Riemann problem for the generalized Hopf equation. The numerical solution is unique in the case of a flow function with two inflection points if artificial dissipation and dispersion are introduced, i.e., the generalized Korteweg–de Vries-Burgers equation is considered. We propose a method for selecting coefficients of dissipation and dispersion. The method makes it possible to obtain a physically justified unique numerical solution. This solution is independent of the difference scheme. Full article

In this study, the evolution of surface water solitary waves under the action of Jeffreys’ wind–wave amplification mechanism in shallow water is analytically investigated. The analytic approach is essential for numerical investigations due to the scale of energy dissipation near coasts. Although many works have been conducted based on the Jeffreys’ approach, only some studies have been carried out on finite depth. We show that nonlinearity, dispersion, and anti-dissipation are the dominating phenomena, obeying an anti-diffusive and fully nonlinear Serre–Green–Naghdi (SGN) equation. Applying an appropriate perturbation method, the current research yields a Korteweg–de Vries–Burger-type equation (KdV-B), combining weak nonlinearity, dispersion, and anti-dissipation. This derivation is novel. We show that the continuous transfer of energy from wind to water results in the growth over time of the KdV-B soliton’s amplitude, velocity, acceleration, and energy, while its effective wavelength decreases. This phenomenon differs from the classical results of Jeffreys’ approach and is due to finite depth. In this study, it is shown that expansion and breaking occur in finite time. These times are calculated and expressed with respect to soliton- and wind-appropriate parameters and values. The obtained values are measurable in experimental facilities. A detailed analysis of the breaking time is conducted with regard to various criteria. By comparing these times to the experimental results, the validity of these criteria are examined. Full article

Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf–Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers–Huxley, generalized equation of Camassa–Holm, generalized equation of Swift–Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods. Full article