Darboux transformation,exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

Nonlinear Dynamics - In this paper, we firstly deduce a reverse space-time Fokas–Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of...     

An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws

In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.     

Solitons and conservation laws of Klein–Gordon equation with power law and log law nonlinearities

This paper obtains the conservation laws of the Klein–Gordon equation with power law and log law nonlinearities. The multiplier approach with Lie symmetry analysis is employed to obtain the conserved densities. The 1-soliton solutions are subsequently used to compute the conserved quantities from the conserved densities. Later the perturbation terms are added and the conservation laws of the perturbed Klein–Gordon equation are studied.     

Solutions and conservation laws of a (3+1)-dimensional Zakharov–Kuznetsov equation

In this paper, we study a nonlinear evolution partial differential equation, namely the (3+1)-dimensional Zakharov–Kuznetsov equation. Kudryashov method together with Jacobi elliptic function method is used to obtain the exact solutions of the (3+1)-dimensional Zakharov–Kuznetsov equation. Furthermore, the conservation laws of the (3+1)-dimensional Zakharov–Kuznetsov equation are obtained by using the multiplier method.     

Generalized coupled Fokas–Lenells equation: modulation instability,conservation laws,and interaction solutions

Nonlinear Dynamics - This paper focuses on the modulation instability, conservation laws, and interaction solutions of the generalized coupled Fokas–Lenells equation. Based on the theory of...     

Exact solutions,conservation laws,bifurcation of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver equation

In the present work, we observe the dynamical behavior of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver (STO) equation. Exact solutions are derived using \({1}/{G^{^{\prime }}}\) expansion and modified Kudryashov methods. The wave transformation is used to transform STO equation into an ordinary differential equation. Combining Runge–Kutta fourth-order and Fourier spectral technique, we use a mixed scheme for the numerical study of STO equation. Since spectral methods expand the solution in trigonometric series resulting into higher-order technique and Runge–Kutta produces improved accuracy, we extract these qualities for a mixed scheme. Results so produced are presented graphically which provide a useful information about the dynamical behavior. Bifurcation behavior of nonlinear and supernonlinear traveling waves of STO equation is studied with the help of bifurcation theory of planar dynamical systems. It is observed that STO equation supports nonlinear solitary wave, periodic wave, shock wave, stable oscillatory wave and most important supernonlinear periodic wave.     

Painlevé integrability and N-soliton solution for the variable-coefficient Zakharov–Kuznetsov equation from plasmas

With the help of symbolic computation, this paper investigates the variable-coefficient Zakharov–Kuznetsov equation which governs the two-dimensional ion-acoustic waves obliquely propagating in an inhomogeneous magnetized two-ion-temperature dusty plasma. The integrability of this model is examined through the Painlevé analysis. Via the Hirota method, the bilinear form of such model is derived. Based on the obtained bilinear form, the N-soliton solution is constructed. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis.     

Modulation instability,conservation laws and soliton solutions for an inhomogeneous discrete nonlinear Schrödinger equation

In this paper, an inhomogeneous discrete nonlinear Schrödinger equation is analytically investigated. The modulation instability condition and conservation laws are derived. By virtue of the discrete Darboux transformation, two types of explicit solutions on the vanishing and non-vanishing backgrounds are generated. Those results might be useful in the study of solitons propagation in discrete optical fibers.     

On the Lie symmetry analysis,analytic series solutions,and conservation laws of the time fractional Belousov–Zhabotinskii system

Nonlinear Dynamics - In this study, a reaction mechanism proposed by Belousov and Zhabotinskii, which corresponds to many physical phenomena, from the complex wave behavior of the heart and various...     

The simplest galilean‐invariant and thermodynamically consistent conservation laws

This paper gives an introduction to formalization of Galileaninvariant and thermodynamically consistent equations of mathematical physics in which unknowns are transformed in rotations by irreducible representations of integer weights. This formalization is based on the theory of representations of the group SO(3).     

General high-order localized waves to the Bogoyavlenskii–Kadomtsev–Petviashvili equation

Nonlinear Dynamics - In this work, the Bogoyavlenskii–Kadomtsev–Petviashvili equation is investigated. By means of the Hirota bilinear system and Pfaffian, we demonstrate that the...     

Novel characteristics of lump and lump–soliton interaction solutions to the generalized variable-coefficient Kadomtsev–Petviashvili equation

Nonlinear Dynamics - With the inhomogeneities of media taken into account, a generalized variable-coefficient Kadomtsev–Petviashvili (vcKP) equation is proposed to model nonlinear waves in...     

Constructing two powerful methods to solve the Thomas–Fermi equation

Nonlinear Dynamics - We implement the reproducing kernel method and SL(2, R)-shooting method to solve the Thomas–Fermi equation. Powerful techniques are demonstrated by reproducing...     

The Variational Approach to Fracture Mechanics. A Practical Application to the French Panthéon in Paris

Recently, Francfort and Marigo (J. Mech. Phys. Solids 46, 1319–1342, 1998) have proposed a novel approach to fracture mechanics based upon the global minimization of a Griffith-like functional, composed of a bulk and a surface energy term. Later on the same authors, together with Bourdin, introduced (in J. Mech. Phys. Solids 48, 797–826, 2000) a variational approximation (in the sense of Γ-convergence) of such functional, essentially for computational purposes. Here, we utilize this new variational approach to show how it might be altered to incorporate the idea of less brittle, “deviatoric-type fracture” and apply to materials such as confined stone. To do so, we modify the original formulation of Francfort and Marigo, in particular its approximation of Bourdin, Francfort and Marigo, to only allow for discontinuities in the deviatoric part of the strain. We apply such modified model to gain insight on the deterioration and cracking in the ashlar masonry work of the French Panthéon, which are so repetitious and particular to be a distinguishable symptom of ongoing damage. Numerical experiments are performed and the results compared to those obtained using the original Francfort-Marigo model and to actual crack patterns from the Panthéon. The modified formulation allows one to reproduce fracture paths surprisingly similar to that observed in situ, to sort out the possible causes of damage, and to confirm, with a quantitative analysis, the main structural deficiencies in the French monument. This practical example enhances the importance of this promising new theory based in the mathematical sciences.     

A scaling law near the primary resonance of the quasiperiodic Mathieu equation

We analyze the quasiperiodic damped Mathieu equation

Application of the Riemann–Hilbert method to the vector modified Korteweg-de Vries equation

Nonlinear Dynamics - Under investigation in this paper is the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation, which can be reduced to several integrable...