On cyclic symmetries ofn-dimensional nonlinear wave equations

This research investigates the bifurcation theory of a generalized (3 + 1)-dimensional P-type nonlinear wave equation with an M-fractional derivative (M-fGP-NWE) and develops its soliton solutions. The model is initially transformed into an ordinary differential equation form using a wave variable. By employing a Galilean transformation, a dynamical system of equations is obtained. The phase portrait and Hamiltonian function are examined under different parametric conditions, facilitating the identification of homoclinic and heteroclinic orbits. These orbits illustrate solitary, bell-shaped, periodic wave solutions for certain parameter values. The modified simple equation (MSE) method is employed to derive soliton solutions for the M-fractional generalized (3 + 1)-dimensional P-type nonlinear wave equation. The resultant solutions are articulated in hyperbolic, trigonometric, and exponential forms under parametric circumstances. Complex wave phenomena are further exemplified through detailed 3D, 2D, and density graphs for certain parameter values. Additionally, we also analyse the modulation Instability of the proposed model. The computational results and visual depictions validate the efficiency and dependability of the MSE method, highlighting its efficacy as a flexible solution for complex fractional differential equations.

Abundant soliton wave solutions and the linear superposition principle for generalized (3+1)-D nonlinear wave equation in liquid with gas bubbles by bilinear analysis

In this manuscript, we investigate the (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. It describes the dynamics of nonlinear waves in hydrodynamics. Liquids with gas bubbles are commonly seen in natural science, engineering, medical science and daily life. The Lie symmetry analysis is used to investigate its vector fields and optimal systems. Moreover, the symmetry reductions and invariant solutions of the equation are obtained based on the optimal systems. Finally, we obtained the exact solutions of (3+1)-dimensional nonlinear wave equation. We showed the physical explanation of obtained solutions by three-dimensional graphically. These newly constructed solutions play an important role in mathematical analysis and other various branches of applied sciences.

This paper introduces the structure-preserving numerical methods for the two dimensional non-linear fractional wave equation. By using the variational principle with fractional Laplace, the equation is transformed into a Hamiltonian system with symplectic and multi-symplectic structure, and they show the corresponding conservation laws. Then a numerical method is proposed with the Fourier pseudospectral method in space and midpoint method in time. It is proved that the proposed numerical method preserves the corresponding conservation laws in the discrete sense. Furthermore, one investigates energy errors of fully discrete schemes, and discusses convergence of the proposed schemes which are second-order accuracy in time and spectral accuracy in space. Finally, the validity and accuracy of the theoretical results are verified by several numerical examples.

Based on a direct variable transformation, we obtain multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including first-order, two-order and three-order rogue wave solutions. Their dynamic behaviors are shown by some 3D plots. Compared with Zha’s symbolic computation approach, we do not need to resort to Hirota bilinear form, and it can be used to deal with variable-coefficient integrable equations. Interaction solution between rogue wave and periodic wave is obtained by using the Hirota bilinear form. Abundant breather wave solutions are presented by a direct test function.

We develop a generalized conditional symmetry approach for the functional separation of variables in a nonlinear wave equation with a nonlinear wave speed. We use it to obtain a number of new (1+1)-dimensional nonlinear wave equations with variable wave speeds admitting a functionally separable solution. As a consequence, we obtain exact solutions of the resulting equations.

The Lie group method is a powerful technique for obtaining analytical solutions for various nonlinear differential equations. This study aimed to explore the behavior of nonlinear elastic wave equations and their underlying physical properties using Lie group invariants. We derived eight-dimensional symmetry algebra for the (3+1)-dimensional nonlinear elastic wave equation, which was used to obtain the optimal system. Group-invariant solutions were obtained using this optimal system. The same analysis was conducted for the damped version of this equation. For the conservation laws, we applied Noether’s theorem to the nonlinear elastic wave equations owing to the availability of a classical Lagrangian. However, for the damped version, we cannot obtain a classical Lagrangian, which makes Noether’s theorem inapplicable. Instead, we used an extended approach based on the concept of a partial Lagrangian to uncover conservation laws. Both techniques account for the conservation laws of linear momentum and energy within the model. These novel approaches add an application of variational calculus to the existing literature. This offers valuable insights and potential avenues for further exploration of the elastic wave equations.

Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations

The (2+1)-dimensional conformable nonlinear shallow water wave equation is examined in this work. Initially, definitions and properties of suitable derivatives are presented. Subsequently, exact solutions to this equation are derived using the exp(–ϕ(ξ))-expansion and the modified extended tanh function methods. Then, a numerical method, namely the residual power series method, is utilized to obtain approximate solutions. The interplay between analytical and numerical approaches is explored to validate the solutions. This study fills a gap in the literature on fractional shallow water models, particularly in (2+1) dimensions, and offers new insights into wave dynamics governed by fractional derivatives. The physical implications of the findings are illustrated through 3D and 2D contour surfaces of some obtained data, offering insight into the physical interpretation of geometric structures. A table is also presented to compare the obtained results. These solutions highlight the practical uses of the investigated model and other nonlinear models in applied sciences. These techniques can potentially yield significant results in solving various fractional differential equations.

Recently some (1+1)-dimensional nonlinear wave equations with linearlydispersive terms were shown to possess compacton-like and solitarypattern-like solutions. In this paper, with the aid of Maple,new solutions of (2+1)-dimensional generalizationof mKdV equation, which is of only linearly dispersive terms, are investigatedusing three new transformations. As a consequence, it is shownthat this (2+1)-dimensional equation also possesses newcompacton-like solutions and solitary pattern-like solutions.

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

New forms of different-periodic travelling wave solutions for the (2+1)-dimensional Zakharov–Kuznetsov (ZK) equation and the Davey–Stewartson (DS) equation are obtained by the linear superposition approach of Jacobi elliptic function. A sequence of cyclic identities plays an important role in these procedures.

Author response for "Rogue-wave structures for a generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles"

A WAS neural network framework for computing and analyzing solutions of a generalized $ (3+1) $-dimensional nonlinear Wave equation: Stability analysis

The Integrability and Several Localized Wave Solutions of a Generalized (2+1)-Dimensional Nonlinear Wave Equation