Narrow Results

In this paper, we present a comprehensive examination of the Lie symmetry analysis, optimal system, exact solutions, and conservation laws pertaining to the variable-coefficient higher-order Boussinesq–Burgers system. The Lie symmetry analysis method is initially employed to derive the vector field, followed by the utilization of Olver’s method to determine the optimal system of one-dimensional subalgebras for the system. By utilizing the optimal system, the Boussinesq–Burgers system is similarly reduced. Subsequently, the $(\frac{G^{\prime }}{G})$-expansion method is applied to this reduced system, yielding kink soliton solution, dark soliton solution, trigonometric-function solution and rational solution for the variable-coefficient higher-order Boussinesq–Burgers system. By assigning various functions to the coefficient functions, different types of soliton solutions are obtained, and these solutions’ structural features are visualized through the use of three-dimensional figures. Last, an examination of the conservation laws governing the system is conducted.

In this present article, the new $(2+1)$-dimensional modified Calogero-Bogoyavlenskii-Schiff (mCBS) equation is studied. Using the Lie group of transformation method, all of the vector fields, commutation table, invariant surface condition, Lie symmetry reductions, infinitesimal generators and explicit solutions are constructed. As we all know, an optimal system contains constructively important information about the various types of exact solutions and it also offers clear understandings into the exact solutions and its features. The symmetry reductions of $(2+1)$-dimensional mCBS equation is derived from an optimal system of one-dimensional subalgebra of the Lie invariance algebra. Then, the mCBS equation can further be reduced into a number of nonlinear ODEs. The generated explicit solutions have different wave structures of solitons and they are analyzed graphically and physically in order to exhibit their dynamical behavior through 3D, 2D-shapes and respective contour plots. All the produced solutions are definitely new and totally different from the earlier study of the Manukure and Zhou (Int. J. Mod. Phys. B33, (2019)). Some of these solutions are demonstrated by the means of solitary wave profiles like traveling wave, multi-solitons, doubly solitons, parabolic waves and singular soliton. The calculations show that this Lie symmetry method is highly powerful, productive and useful to study analytically other nonlinear evolution equations in acoustics physics, plasma physics, fluid dynamics, mathematical biology, mathematical physics and many other related fields of physical sciences.

Abundant hybrid solitary wave solutions have been investigated for the $(3+1)$-dimensional nonlinear evolution equation with main part mKdV equation (NLEE-mKdV) in its first study via the Lie symmetry method. The latter has been harnessed to attain 10-dimensional vector fields of symmetries from the converted NLEE-mKdV to a simpler equation under special transformation. From the derived symmetries, the corresponding one-dimensional optimal system has been constructed. Furthermore, the converted NLEE-mKdV is reduced via five subalgebras of symmetries to obtain nine novel solutions, which are shifted by the considered transformation to reach NLEE-mKdV’ solutions that possess different dynamics depending on a special ansatz in adjusting the arbitrary functions and parameters. Therefore, many significant solitary wave patterns are achieved. For example, dark, bright, periodic, dipole, damped periodic, breather, kink, and their interactions are well depicted in 3D and contour plots. Most importantly, a fascinating solitary wave solution is first explored; the intrinsic insight appears on the periodically-parabolic-periodic background, which is collided with a bright soliton solution to induce a parabolic-humps breather on its top.

The deformed KdV equation is a generalization of the classical equation that can describe the motion of the interaction between different solitary waves. In this paper, the Lie symmetry analysis is performed on the deformed KdV equation. The similarity reductions and exact solutions are obtained based on the optimal system method. The exact analytic solutions are considered by using the power series method. The conservation laws for the deformed KdV equation are presented. Finally, the analytic solutions are given and their dynamics are studied.

In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters.

Malaria is a life-threatening disease caused by parasites that are transmitted to people through the bites of infected mosquitoes. In this paper, a deterministic model for malaria transmission, that incorporates superinfection is presented. Qualitative analysis of the model reveals the presence of backward bifurcation in which a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. Optimal control theory is then applied to the model to study time-dependent treatment efforts to minimize the infected in individuals while keeping the implementation cost at a minimum.

Higher-dimensional nonlinear models can describe more complex evolutionary mechanisms. In this paper, we considered the time fractional $(3+1)$-dimensional modified extended Zakharov–Kuznetsov equation with the sense of the Riemann–Liouville fractional derivative in plasma physics. In the first place, the existence of symmetry of this studied equation through the symmetry scheme was proved. Then, the optimal system to the time fractional $(3+1)$-dimensional modified extended Zakharov–Kuznetsov equation was also constructed. Subsequently, the time fractional higher-dimensional equation was reduced into the lower-dimensional fractional differential equation with the help of the Erdélyi–Kober fractional operators. Last, some conservation laws by using a new conservation theorem were also given. These novel results provide a window for us to discover this high-dimensional nonlinear equation.

In this paper, some properties of the time fractional Boussinesq equation are presented. Group analysis of the time fractional Boussinesq equation with Riemann–Liouville derivative is performed and the corresponding optimal system of subgroups are determined. Next, we apply the obtained optimal systems for constructing reduced fractional ordinary differential equations (FODEs). Finally, we show how to derive exact solutions to time fractional Boussinesq equation via invariant subspace method.

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions.

Under study in this paper is a time fractional generalized nonlinear diffusion equation which can be better to express diffusion phenomena than diffusion equation of integer order. Firstly, we apply the symmetry analysis method to find the symmetry of this considered equation. Then some conservation laws can also be constructed through the above obtained symmetry with the help of the Noether’s theorem. Next, we reduce this equation into an ordinary differential equation of fractional order in the symmetry with Erdélyi–Kober fractional differential operator under one-dimensional subalgebras optimal system framework. Finally, some exact solutions contain the invariant solutions have found for this given equation. The results give us a new interpretation of this type diffusion process.

In this paper, we present a study of a fifth-order nonlinear partial differential equation, which was recently introduced in the literature. This equation can be used as a model for bidirectional water waves propagating in a shallow medium. Using elements of an optimal system of one-dimensional subalgebras, we perform similarity reductions culminating in analytic solutions. Rational, hyperbolic, power series and elliptic solutions are obtained. Furthermore, by using the multiple exponential function method we obtain one and two soliton solutions. Finally, local and low-order conserved quantities are derived by enlisting the multiplier approach.

In this paper, a $(2+1)$-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the $(\frac{G^{\prime }}{G})$-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.

In this paper, we consider a one-dimensional model of blood flow along the compliant arteries. With the help of the invariant function, we construct and classify the optimal system of subalgebras. Next, we reduced the given system of partial differential equations (PDEs) to the system of ordinary differential equations (ODEs) for each subalgebra and subsequently solved them exactly. Further, we investigate the evolutionary behavior of the average blood flow velocity and the cross-sectional area of the arteries; under the influence of the physical parameter $\alpha$ graphically. Furthermore, we construct nonlocally related PDEs for the given system of PDEs, consisting of inverse potential systems (IPS) and potential systems. Finally, we classify the nonlocal symmetries arising from the potential system and IPS.

Lie symmetry group method is applied to study the telegraph equation. The symmetry group and one-parameter group associated to the symmetries with the structure of the Lie algebra symmetries are determined. The reduced version of equation and its one-dimensional optimal system are given.

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.