Narrow Results

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

In this paper, we mainly investigate the time fractional nonlinear wave equation which can be usually used to express nonlinear phenomena appearing in shallow water waves by using group analysis scheme. First, the symmetry can be obtained make uses of the group analysis to the time fractional nonlinear wave equation. Based on the above-found symmetry, this equation was able to reduce into an ordinary differential equation of fractional order. As a result, some new invariant solutions were also constructed for this considered equation. Second, the scaling transformation was also obtained by introducing new independent and dependent variables. Finally, the conservation laws were also found to satisfy the time fractional nonlinear wave equation with the help of the Ibragimov theorem. These novel results show unique nonlinear phenomena.

This paper studies the potential form of the 3D potential Yu–Toda–Sasa–Fukuyama equation through the perspective of Lie symmetry analysis. This technique combined with symbolic computations does prove that the general Lie operator depends on five parameters and six independent arbitrary functions that are variable in respect to time. The group invariant solutions associated to some 1D subalgebras are systematically construct and they do involve arbitrary functions. When these functions are expressed under several specific forms, the associated wave solutions possess multiple structures. Graphical representations of some particular solutions are as well provided. As far as we know such general solutions are presented here for the first time and do indicate the symmetry method to be applied in order to solve other multidimensional, integrable, or nonintegrable nonlinear dynamical models.

In this article, the resonance nonlinear Schrödinger equation is studied, which elucidates the propagation of one-dimensional long magnetoacoustic waves in a cold plasma, dynamic of solitons and Madelung fluids in various nonlinear systems. The Lie symmetry analysis is used to achieve the invariant solution and similarity reduction of the resonance nonlinear Schrödinger equation. The infinitesimal generators, symmetry groups, commutator table and adjoint table have been obtained by the aid of invariance criterion of Lie symmetry. Also, one-dimensional system of subalgebra is constructed with the help of adjoint representation of a Lie group on its Lie algebra. By one-dimensional optimal subalgebra, the main equations are reduced to ordinary differential equations and their invariant solutions are provided. The general conservation theorem has been used to establish a set of non-local and non-trivial conservation laws.

Here attention is focused on the (1+1)-dimensional Sawada–Kotera (SK) model that is prominent in mathematical physics and engineering to analyze plasmas and coherent systems for communication. Our techniques offer fresh perspectives on the model’s attributes and structure, deepening our comprehension of the underlying dynamics. First, the SK model is reduced to ordinary differential equations by constructing the Lie symmetries and using the associated transformation. Graphs are used to build and display invariant solutions. This strategy has caused the revelation of novel constant solutions that have not been found in the previous works. We offer new understandings of nature and changes observed during the SK derivation by taking advantage of the Lie symmetries powerful tools. Next, the fluctuating layout of proposed framework is examined from several perspectives such as sensitivity and bifurcation analysis. We examined the bifurcation analysis of planar dynamical system by using bifurcation theory. We also include an external periodic perturbation term that breaks regular patterns in the perturbed dynamical system. Graphical structures are provided to display the invariant solutions. The sensitivity of the SK model is determined to be strong after sensitivity analysis under different initial conditions. These results are fascinating, fresh, and conceptually useful for understanding the suggested framework. In mathematics and the applied sciences, forecasting and learning about new technologies are greatly aided by the dynamic aspect of system processing.

In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large x and/or t) invariant under a group G which is not a symmetry of the equation. After recalling the geometrical meaning of symmetries of differential equations — and solution-preserving maps — we provide a precise definition of asymptotic symmetries of PDEs; we deal in particular, for ease of discussion and physical relevance, with scaling and translation symmetries of scalar equations. We apply the general discussion to a class of "Richardson-like" anomalous diffusion and reaction-diffusion equations, whose solution are known by numerical experiments to be asymptotically scale invariant; we obtain an analytical explanation of the numerically observed asymptotic scaling properties. We also apply our method to a different class of anomalous diffusion equations, relevant in optical lattices. The methods developed here can be applied to more general equations, as shown by their geometrical construction.

In this work we perform the symmetry classification of the Klein–Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein–Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also Noether symmetries for the Klein–Gordon equation. We use these results in order to determine all the potentials in which the Klein–Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein–Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.

In this paper, we search the existence of invariant solutions of Bianchi type I space-time in the context of $f(R,T)$ gravity with special case $f(R,T)=R+2f(T)$. The exact solution of the Einstein’s field equations are derived by using Lie point symmetry analysis method that yield two models of invariant universe for symmetries $X^{(1)}$ and $X^{(3)}$. The model with symmetries $X^{(1)}$ begins with big bang singularity while the model with symmetries $X^{(3)}$ does not favor the big bang singularity. Under this specification, we find out at set of singular and nonsingular solution of Bianchi type I model which present several other physically valid features within the framework of $f(R,T)$ gravity.

In this paper, some new invariant solutions of breaking soliton (BS) equation have been derived by using similarity transformations method. The system represents the interaction of Riemann waves propagating along $y$-axis and long waves along $x$-axis. The commutative relation and symmetry analysis of BS equation are derived using Lie group theory. Meanwhile, the method reduces the number of independent variables by one in each step. A repeated application of similarity transformations method reduces the BS equation into overdetermined equations, which provide invariant solutions. The derived results are more general than previous findings. The obtained solutions are supplemented by numerical simulation, which makes this research physically meaningful. Eventually, doubly soliton, multisoliton and asymptotic profiles of solutions are analyzed in the analysis and discussion section.

This paper investigates the $(3+1)$-dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the $(3+1)$-dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie’s continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver’s standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov’s generalized theorem and quasi-self-adjointness condition.

Nonlinear n-dimensional second-order diffusion equations admitting maximal Lie algebras of point symmetries are considered. Examples of invariant solutions, as well as of solutions on invariant subspaces for some nonlinear operators, are constructed for arbitrary n. A complete description of all possible types of invariant solutions is given in the case n = 2 for the equation possessing an infinitely dimensional symmetry algebra. The results obtained are generalized for the hyperbolic and other fourth-order parabolic equations of thin film and nonlinear dispersion type.

We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry (GCS) method is applied in order to obtain the conditions that enable the equation to admit a special class of second-order GCSs. For the case of quadratic nonlinearities, we outline a new class of invariant solutions.

In this paper, the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system in mathematical physics, the Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation, is studied. By applying the basic Lie symmetry method for the KdV–ZK equation, the classical Lie point symmetry operators are obtained. Also, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The non-classical symmetries of the KdV–ZK equation are also investigated.

We summarize our results related with mathematical modeling of Aedes aegypti and its Lie symmetries. Moreover, some explicit, group-invariant solutions are also shown. Weak equivalence transformations of more general reaction diffusion systems are also considered. New classes of solutions are obtained.

Invariant solutions of PDEs are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so–called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. Various examples of physical relevance are provided.

Lie point symmetries admitted by PDEs can be used to determine invariant solutions by solving a reduced system involving one independent variable less. For a system of PDEs with m independent variables we may consider a k–dimensional subalgebra of the Lie algebra of the admitted symmetries and get a reduced system with m – k independent variables. Also, if we have a m–dimensional Abelian Lie subalgebra we may write the PDEs in an equivalent form for which some solutions can be found explicitly. An application to the Navier–Stokes–Fourier equations is considered.

The axi–symmetric equations of a viscous and heat conducting monatomic gas are considered with various constitutive relations for relaxation times. The optimal systems of 1D and 2D subalgebras are determined and used to obtain some invariant solutions.