In this paper, coupled Higgs field equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equation and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the coupled Higgs equation and Hamiltonian amplitude equation using $(G'/G)$-expansion methodc, where $G = G(\xi)$ satisfies a second-order linear ordinary differential equation (ODE). The travelling wave solutions expressed by hyperbolic, trigonometric and the rational functions are obtained.

In this paper, the Lie group analysis method is applied to carry out the Lie point symmetries of some space–time fractional systems including coupled Burgers equations, Ito’s system, coupled Korteweg–de-Vries(KdV) equations, Hirota–Satsuma coupled KdV equations and coupled nonlinear Hirota equations with time-dependent variable coefficients with the Riemann–Liouville derivative. Symmetry reductions are constructed using Lie symmetries of the systems. To the best of our knowledge, nobody has so far derived the invariants of space–time nonlinear fractional partial differential equations with time-dependent coefficients.

In this work, Lie symmetry analysis and one-dimensional optimal system for Pavlov equation are presented. All the possible vector fields, their commutative and adjoint relations are carried out under invariance property of Lie group theory. On the basis of optimal system, similarity reductions of Pavlov equation are obtained. A repeated process of similarity reductions transforms the Pavlov equation into ordinary differential equations, which generate invariant solutions. The obtained invariant solutions are supplemented by numerical simulation toanalyse the physical behaviour. Thus, their parabolic, multisoliton, nonlinear, kink and antikink wave profiles are traced in results and discussions sections.

In the present article, Lie group of point transformations method is successfully applied to study the invariance properties of the (2 + 1)-dimensional Pavlov equation. Applying the Lie symmetry method, we strictly obtain the infinitesimals, vector fields, commutation relation and several interesting symmetry reductions of the equation. The explicit exact solutions are derived under some limiting conditions imposed on the infinitesimals $\xi$, $\phi$, $\tau$ and $\eta$. Then, the Pavlov equation is transformed into a number of nonlinear ODEs through several symmetry reductions. These new exact solutions are more general and entirely different from the work of Kumar $et al$ (Pramana - J. Phys. 94: 28 (2020)). The obtained invariant solutions are examined analytically as well as physically through numerical simulation by giving free alternative values of arbitrary functions and constants. Consequently, graphical representations of all these solutions are studied and demonstrated in 3D-graphics and the corresponding contour plots. Interestingly, the solution profiles show the annihilation of three-dimensional parabolic profile, doubly soliton and elastic multisolitons and nonlinear wave nature form.

In the present article, our main aim is to construct abundant exact solutions for the (2+1)-dimensional Kadomtsev–Petviashvili-Benjamin–Bona–Mahony (KP-BBM) equation by using two powerful techniques, the Lie symmetry method and the generalised exponential rational function (GERF) method with the help of symbolic computations via Mathematica. Firstly, we have derived infinitesimals, geometric vector fields, commutation relations and optimal system. Therefore, the KP-BBM equation is reduced into several nonlinear ODEs under two stages of symmetry reductions. Furthermore, abundant solutions are obtained in different shapes of single solitons, solitary wave solutions, quasiperiodic wave solitons, elastic multisolitons, dark solitons and bright solitons, which are more relevant, meaningful and useful to describe physical phenomena due to the existence of free parameters and constants. All these generated exact soliton solutions are new and completely different from the previous findings. Moreover, the dynamical behaviour of the obtained exact closed-form solutions is analysed graphically by their 3D, 2D-wave profiles and the corresponding density plots by using the mathematical software, which will be comprehensively used to explain complex physical phenomena in the fields of nonlinear physics, plasma physics, optical physics, mathematical physics, nonlinear dynamics, etc.

In the present work, abundant group-invariant solutions of (2 + 1)-dimensional Bogoyavlenskii–Schieff equation have been investigated using Lie symmetry analysis. The Lie infinitesimal generators, all the geometric vector fields, their commutative and adjoint relations are provided by utilising the Lie symmetry method. The Lie symmetry method depends on the invariance criteria of Lie groups, which results in the reduction of independent variables by one. A repeated process of Lie symmetry reductions, using the double, triple and septuple combinations between the considered vectors, converts the Bogoyavlenskii–Schieff (BS) equation into nonlinear ordinary differential equations (ODEs) which furnish numerous explicit exact solutions with the help of computerised symbolic computation. The obtained group-invariant solutions are entirely new and distinct from the earlier established findings. As far as possible, a comparison of our reported results with the previous findings is given. The dynamical behaviour of solutions is discussed both analytically as well as graphically via their evolutionary wave profiles by considering suitable choices of arbitrary constants and functions. To ensure rich physical structures, the exact closed-form solutions are supplemented via numerical simulation, which produce some bright solitons, doubly solitons, parabolic waves, U-shaped solitons and asymptotic nature.

The versatile and exactly solvable Scarf II potential has been predicting, confirming and demonstrating interesting phenomena in complex PT-symmetric sector, most impressively. However, for the non-PT-symmetric sector, it has gone underutilised. Here, we present the most simple analytic forms for the scattering coefficients $(T (k), R(k), | det S(k)|)$. On the one hand, these forms demonstrate earlier effects and confirm the recent ones. On the other hand, they make new predictions – all simple and analytical. We show the possibilities of both self-dual and non-self-dual spectral singularities (NSDSS) in two non-PT sectors (potentials). The former one is not accompanied by time-reversed coherent perfect absorption (CPA) and gives rise to the parametrically controlled splitting of spectral singularity (SS) into a finite number of complex conjugate pairs of eigenvalues (CCPEs). NSDSS behave just oppositely: CPA but no splitting of SS. We demonstrate a one-sided reflectionlessness without invisibility. Most importantly, we bring out a surprising coexistence of both real discrete spectrum and a single SS in a fixed potential. Nevertheless, so far, the complex Scarf II potential is not known to be pseudo-Hermitian ($η ^{−1}Hη = H^{†})$ under a metric of the type $η(x)$.

Some new infinite-dimensional generalised Lie symmetries of Einstein’s field equations for perfect fluid distribution are found by using the Lie symmetry analysis. The reduced ordinary differential equations are solved to obtain new non-trivial exact solutions. The software MAPLE is used for computation and MAPLE code is given to facilitate the research in this field.

In this work, we apply the generalised exponential rational function (GERF) method on an extended (3+1)-dimensional Jimbo–Miwa (JM) equation which describes the modelling of water waves of long wavelength with weakly nonlinear restoring forces and frequency dispersion. This JM equation is also used to construct modelling waves in ferromagnetic media and two-dimensional matter-wave pulses in Bose–Einstein condensates. The main purpose is to construct analytical wave solutions for the (3+1)-dimensional JM equation by utilising theGERF method with the help of symbolic computations.We have also presented three-dimensional plots to observe the dynamics of obtained results. To understand physical phenomenon through different shapes of solitary waves,we discussed solitons, the interaction of multiwave solitons, lump-type solitons and kink-type solutions.

The prime objective of this paper is to obtain some new families of exact solitary wave solutions of the Klein–Gordon–Zakharov (KGZ) equations via computerised symbolic computation on Wolfram Mathematica. By applying the generalised exponential rational function method, numerous exact soliton solutions are constructed for the KGZ equations, which provide a model of the interaction between the Langmuir wave and the ion-acoustic wave in high-frequency plasma. Consequently, the exact solitary wave solutions are obtained in different forms of dynamical wave structures of solitons including multisolitons, lump-type solitons, travelling waves, kink waves, also trigonometric and hyperbolic function solutions, and rational function solutions. Moreover, the dynamical behaviour of the resulting multiple soliton solutions is discussed both analytically and graphically by using suitable values of free parameters through numerical simulation. The reported results have rich physical structures that are helpful to explain the nonlinear wave phenomena in plasma physics and soliton theory.

The dynamics of localised solitary wave solutions play an essential role in the fields of mathematical sciences such as optical physics, plasma physics, nonlinear dynamics and many others. The prime objective of this study is to obtain localised solitary wave solutions and exact closed-form solutions of the (2 + 1)-dimensionaluniversal hierarchy equation (UHE) using the Lie symmetry approach. Besides, the Lie infinitesimals, all the vector fields, commutation relations of Lie algebra and symmetry reductions are derived via the Lie transformation method. Meanwhile, the universal hierarchy equation is reduced into nonlinear ODEs through two stages of symmetry reductions. The closed-form invariant solutions are attained under some parametric conditions imposed on infinitesimal generators. Because of the presence of arbitrary independent functional parameters and other constants, these group-invariant solutions are explicitly displayed in terms of arbitrary functions that are more relevant, beneficial and useful for explaining nonlinear complex physical phenomena. Furthermore, the dynamical structures of the obtained exact solutions are illustrated for suitable values of arbitrary constants through 3D-plots based on numerical simulation. Some of these localised solitary waves are double solitons, periodic lump solitons,dark solitons, five-solitons, hemispherical solitons and lump-type solitons.

The Lie symmetry technique is utilised to obtain three stages of similarity reductions, exact invariant solutions and dynamical wave structures of multiple solitons of a (3+1)-dimensional generalised BKP–Boussinesq (gBKP-B) equation.We obtain infinitesimal vectors of the gBKP-B equation and each of these infinitesimals depends on five independent arbitrary functions and two parameters that provide us with a set of Lie algebras. Thenceforth, the commutative and adjoint tables between the examined vector fields and one-dimensional optimal system ofsymmetry subalgebras are constructed to the original equation. Based on each of the symmetry subalgebras, the Liesymmetry technique reduces the gBKP-B equation into various nonlinear ordinary differential equations through similarity reductions. Therefore, we attain closed-form invariant solutions of the governing equation by utilisingthe invariance criteria of the Lie group of transformation method. The established solutions are relatively new and more generalised in terms of functional parameter solutions compared to the previous results in the literature. All these exact explicit solutions are obtained in the form of different complex wave structures like multiwave solitons, curved-shaped periodic solitons, strip solitons, wave–wave interactions, elastic interactions between oscillating multisolitons and nonlinear waves, lump waves and kinky waves. The physical interpretation of computational wave solutions is exhibited both analytically and graphically through their three-dimensional postures by selecting relevant values of arbitrary functional parameters and constant parameters.

We point out that PT-symmetric potentials V$_{PT}$(x) having imaginary asymptotic saturation, V$_{PT}$(x = ±∞) = ±iV$_1$, V$_1$ ∈$ \mathbb{R}$ are devoid of scattering states and spectral singularity. We show the existence of real (positive and negative) discrete spectrum both with and without complex conjugate pair(s) of eigenvalues (CCPEs). If the eigenstates are arranged in the ascending order of the real part of the discrete eigenvalues, the initial states have few nodes but latter ones oscillate fast. Both real and imaginary parts of ψn(x) vanish asymptotically, and|ψn(x)| are nodeless. For the CCPEs, these are asymmetric and peaking on the left (right) and for real energies these are symmetric and peaking at the origin. For CCPEs E$_{±}$, the eigenstates ψ± follow the interesting property, |ψ+(x)| = N|ψ−(−x)|, N ∈ $ \mathbb{R$^+$}$