Search

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painleve expansion. Three kinds of non-auto and auto Backlund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the n th Backlund transformation. The consistent Riccati expansion method is employed to derive a Backlund transformation. Furthermore, the soliton solutions, fusion-type N -solitary wave solutions and soliton–cnoidal wave solutions are gained through Backlund transformations.

In the present paper, some new exact solutions of the (3+1)-dimensional sinh-Gorden equation are displayed. First, based on the Lie point symmetry, basic infinitesimal generators given; also, and the Lie algebra and commutative relations are shown. Second, some exact solutions are obtained using the F-expansion method, and a great many Jacobian-elliptic function solutions are presented. Finally, other forms of solutions are derived using the traveling wave transform.

We use the method of Lie symmetry analysis to investigate the properties of a (2+1)-dimensional KdV–mKdV equation. Using the Ibragimov method, which relies only on the existence of the commutator table, we construct an optimal system of one-dimensional subalgebras of the Lie algebra and study invariant solutions and similarity reductions by considering representatives of the optimal system. To analyze some nonlocal symmetry properties, we apply the truncated Painleve expansion method and obtain two Backlund transformations that are not autotransformations and one auto-Backlund transformation. To localize the nonlocal symmetry and obtain a local Lie point symmetry, we introduce an expanded system. Using solutions of the corresponding Cauchy problems for Lie point symmetries, we prove a theorem on a finite symmetry transformation and find the $$n$$ th Backlund transformation in terms of determinants. Based on one of the obtained Backlund transformations that are not autotransformations, we derive lump-type solutions. In addition, we prove the integrability of the equation by the consistent Riccati expansion method. We present explicit soliton-cnoidal wave solutions and investigate the dynamical characteristics of the obtained solutions using numerical analysis.

This paper considers a generalized (2+1) dimensional nonlinear evolution equation depending on two nonzero arbitrary constants. We derive the Lie point symmetry generators and Lie symmetry groups. This symmetry analysis leads us the reductions equations, through one of which we obtain solutions. We also get the low-order conservation laws of the equation that have been obtained using the corresponding symmetries of the family. We will present a classification of conservation laws for this equation and we will apply Lie symmetry analysis to the equation in order to obtain exact solutions.

In this paper, nonlocal symmetries of variable coefficient Ablowitz-Kaup-Newell-Segur(AKNS) system are studied for the first time. In order to construct some new analytic solutions, a new variable is introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of closed system, we give out two types of symmetry reductions and some analytic solutions. For some interesting solutions, such as interaction solutions among solitons and other complicated waves, we give corresponding images to describe their dynamic behavior.

The truncated Painleve expansion is used to obtain the residual symmetry of the combined KdV–negative-order KdV (KdV–nKdV) equation, which can be localized to the Lie point symmetry. The multiple residual symmetries are established and localized in the prolonged system by defining more new quantities, and the corresponding finite symmetry transformation which is n th Backlund transformation is given by determinant form. In addition, we derive a second order Lax pair by introducing a transformation. Finally, we apply the consistent Riccati expansion (CRE) method to prove that the combined KdV–nKdV equation is CRE solvable and construct exact interaction solution between the soliton and the cnoidal periodic wave.

The Korteweg-de Vries (KdV)-type equations appear during the studies on the cosmic plasmas, planetary atmospheres and oceans. In this paper, via the Lie symmetry method, we investigate a three-coupled KdV system. Lie point symmetry generators and Lie symmetry groups are given. Based on the Lie point symmetry generators, the system is reduced to some ordinary differential systems. Applying the power-series, polynomial, Jacobi elliptic function and (G′/G) expansion methods on those reduced systems, we derive some group-invariant solutions including the explicit power-series, soliton, snoidal-wave and cnoidal-wave solutions. We graphically analyze the solitons and snoidal waves. Besides, we demonstrate the nonlinear self-adjointness of the system, based on which we present the conservation laws according to the Lie point symmetries.

In this article, we take into account the $$(1+1)$$ -dimensional dispersive long-wave equation which is reduced from the Wu–Zhang equation. Firstly, the B $$\ddot{\text{ a }}$$ cklund transformation and the non-local symmetry are successfully acquired from the truncated Painlev $$\acute{\text{ e }}$$ expansion. At the same time, the non-local symmetry is transformed to Lie point symmetry by a suitable prolonged system. Then some solitary solutions are derived without a hitch via new B $$\ddot{\text{ a }}$$ cklund transformation which originates from Lie point symmetry. Secondly, by utilising the consistent Riccati expansion method and the consistent tanh-expansion method, we find the interaction solutions between soliton and cnoidal wave by using the Jacobi elliptic function. Lastly, the conservation laws which are related to symmetries of the equation are successfully obtained by Ibragimov’s method.

The 1 + 2 dimensional Bogoyavlensky-Konopelchenko Equation is investigated for its solution and conservation laws using the Lie point symmetry analysis. In the recent past, certain work has been done describing the Lie point symmetries for the equation and this work seems to be incomplete (Ray S (2017) Compt. Math. Appl. 74, 1157). We obtained certain new symmetries and corresponding conservation laws. The travelling-wave solution and some other similarity solutions are studied., 12 pages. Accepted for publication in Quaestiones Mathematicae

For Lienard I-type equations it is proved the existence of a family of λ − symmetries such that any of them lets the computation by quadratures of a time-dependent first integral of the equation. This is achieved by using a solvable structure constructed out of the λ − symmetry and one Lie point symmetry. The first integral obtained by quadratures and the first integral associated to the Lie symmetry generator are always functionally independent and they can be therefore used to integrate completely the Lienard I-type equation. The method is illustrated by examples of wide classes of Lienard I-type equations. These classes contain but not limited to generalized force-free Duffing-Van der Pol oscillator equations. Analytical solutions are explicitly provided for both oscillatory and nonoscillatory types of equations.

The residual symmetry of the (2+1)-dimensional Bogoyavlenskii equation is obtained and localized to a Lie point symmetry by introducing new dependent variables to enlarge the system, then the corresponding finite transformation is obtained by using Lie’s first theorem. Furthermore, by introducing more dependent variables, the linear superposition of arbitrary number of residual symmetries is also localized and the corresponding finite transformations which are just N th Backlund transformations of the (2+1)-dimensional Bogoyavlenskii equation are obtained in determinants form with some concrete solutions explicitly given.

In this paper, the truncated Painleve expansion is employed to derive a Backlund transformation of a ( $$2+1$$ )-dimensional nonlinear system. This system can be considered as a generalization of the sine-Gordon equation to $$2+1$$ dimensions. The residual symmetry is presented, which can be localized to the Lie point symmetry by introducing a prolonged system. The multiple residual symmetries and the nth Backlund transformation in terms of determinant are obtained. Based on the Backlund transformation from the truncated Painleve expansion, lump and lump-type solutions of this system are constructed. Lump wave can be regarded as one kind of rogue wave. It is proved that this system is integrable in the sense of the consistent Riccati expansion (CRE) method. The solitary wave and soliton–cnoidal wave solutions are explicitly given by means of the Backlund transformation derived from the CRE method. The dynamical characteristics of lump solutions, lump-type solutions and soliton–cnoidal wave solutions are discussed through the graphical analysis.

We consider the generalised dispersive modified Benjamin–Bona–Mahony equation, which describes an approximation status for long surface wave existed in the non-linear dispersive media. By employing the truncated Painlevé expansion method, we derive its non-local symmetry and Bäcklund transformation. The non-local symmetry is localised by a new variable, which provides the corresponding non-local symmetry group and similarity reductions. Moreover, a direct method can be provided to construct a kind of finite symmetry transformation via the classic Lie point symmetry of the normal prolonged system. Finally, we find that the equation is a consistent Riccati expansion solvable system. With the help of the Jacobi elliptic function, we get its interaction solutions between solitary waves and cnoidal periodic waves.

In this paper, Backlund transformation and interaction solutions of the generalized coupled Klein–Gordon (cKG) equations with the variable coefficients are studied via the residual symmetry and the consistent Riccati expansion (CRE) method. From the truncated Painleve expansion, we drive the residual symmetry which can be transformed into the local Lie point symmetry. The multiple residual symmetries are established and nth Backlund transformation in terms of determinant is presented. The cKG equations are proved to be CRE solvable by using the CRE method. More importantly, we construct exact interaction solutions between soliton and cnoidal periodic wave. It is shown that when the modulus of the Jacobi elliptic function tends to one, the interaction solutions degenerate to the special resonant soliton solutions. Moreover, these reduced resonant soliton solutions exhibit the interesting properties which differ from the usual resonant soliton solutions.

The residual symmetry is derived for the negative-order Korteweg–de Vries equation from the truncated Painleve expansion. This nonlocal symmetry is transformed into the Lie point symmetry and the finite symmetry transformation is presented. The multiple residual symmetries are constructed and localized by introducing new auxiliary variables, and then n th Backlund transformation in terms of determinant is provided. With the help of the consistent tanh expansion (CTE) method, the explicit soliton-cnoidal wave interaction solutions are obtained from the last consistent differential equation.

In this paper, the constant astigmatism equation is investigated, which finds numerous applications in geometry and physics describing surfaces of constant astigmatism. Utilizing a set of non-singular local multipliers, we present a set of local conservation laws for the equation, based on which, we further find a tree of nonlocally related PDE systems. Moreover, by considering these nonlocally related PDE systems, we systematically present Lie symmetries, optimal systems and some interesting analytical solutions of the constant astigmatism equation. Here it is the first time to investigate the nonlocally related PDE systems for this model, which can be used to expand the solution space of the given PDE system. Finally, by using its local conservation laws and variational principle, we obtain four kinds of Lagrangians.

The resonant nonlinear Shrodinger's equation (RNLSE) with both spatio-temporal (STD) and inter-modal (IMD) dispersions which describes the modelling of fluids and propagation dynamics of optical solitons is studied using three analytical schemes. These are generalized projective-Riccati equation method (GPRE), Bernoulli sub-ODE method and the Riccati-Bernoulli sub-ODE. The presented problem is studied with Kerr law nonlinearity. Dark optical, singular, and combined formal solitons are acquired. The constraint conditions that naturally fall out of the solution structure guarantee the existence of these solitons. We derive the Lie point symmetry generators of a system of partial differential equations (PDEs) obtained by decomposing the underlying equation into real and imaginary components. Then we used these symmetries to construct a set of nonlocal conservation laws (Cls) using the technique introduced by Ibragimov.

For a system of two partial differential equations in two independent variables the Lie point symmetry generator has the form τ ( t , x , u , v ) ∂ t + ξ ( t , x , u , v ) ∂ x + η ( t , x , u , v ) ∂ u + μ ( t , x , u , v ) ∂ v . We consider a system of evolution equations of general class and we provide restrictions on the functional forms of the coefficient functions of the Lie generator. These restrictions reduce the amount of calculations in determining the Lie symmetries of the system under study. We show that τ = τ ( t ) except in one special case and we present further restrictions on the remaining coefficient functions.

The n th finite transformations of the fifth-order KdV equation are obtained from the Lie point symmetry approach via localisation of nonlocal symmetries to local ones of the enlarged system. Through the obtained transformations, some periodic and soliton solutions are derived.

In this article, we mainly apply the nonlocal residual symmetry analysis to a (2 + 1)-dimensional strongly coupled Burgers system, which is defined by us through taking values in a commutative subalgebra. On the basis of the general theory of Painlevé analysis, we get a residual symmetry of the strongly coupled Burgers system. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is derived by Lie’s first theorem. Further, the linear superposition of the multiple residual symmetries is localized to a Lie point symmetry, and an N-th Bäcklund transformation is also obtained.

In this article, we construct a new strongly coupled Boussinesq&ndash, Burgers system taking values in a commutative subalgebra Z 2 . A residual symmetry of the strongly coupled Boussinesq&ndash, Burgers system is achieved by a given truncated Painlev&eacute, expansion. The residue symmetry with respect to the singularity manifold is a nonlocal symmetry. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a B&auml, cklund transformation is obtained with the help of Lie&rsquo, s first theorem. Further, the linear superposition of multiple residual symmetries is localized to a Lie point symmetry, and a N-th B&auml, cklund transformation is also obtained.

In this paper, we consider the $(3+1)$ ( 3 + 1 ) -dimensional time-fractional Schamel–Zakharov–Kuznetsov–Burgers (SZKB) equation. With the help of the Riemann–Liouville derivatives, the Lie point symmetries of the $(3+1)$ ( 3 + 1 ) -dimensional time-fractional SZKB equation are derived. By applying the Lie point symmetry method as well as Erdélyi–Kober fractional operator, we get the similarity reductions of the time-fractional SZKB equation. Conservation laws of the time-fractional SZKB are constructed. Moreover, we obtain its power series solutions with the convergence analysis. In addition, the analytical solution is obtained by modified trial equation method. Finally, stability is analyzed graphically in different planes.

In this paper, we introduce the Lie-point symmetry method into backward stochastic differential equation and forward&ndash, backward stochastic differential equations, and get the corresponding deterministic equations.

The nonlocal residual symmetry for the Whitham–Broer–Kaup (WBK) equation is derived by the truncated Painleve analysis. The nonlocal residual symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Based on the consistent tanh expansion method (CTE), some exact interaction solutions among different nonlinear excitations are explicitly presented. Some special interaction solutions are investigated in both analytical and graphical ways.

The nonlocal symmetries for the $$(2+1)$$ -dimensional Konopelchenko–Dubrovsky equation are obtained with the truncated Painleve method and the Mobious (conformal) invariant form. The nonlocal symmetries are localized to the Lie point symmetries by introducing auxiliary dependent variables. The finite symmetry transformations are obtained by solving the initial value problem of the prolonged systems. The multi-solitary wave solution is presented with the finite symmetry transformations of a trivial solution. In the meanwhile, symmetry reductions in the enlarged systems are studied by the Lie point symmetry approach. Many explicit interaction solutions between solitons and cnoidal periodic waves are discussed both in analytical and in graphical ways.

The nonlocal symmetry for the potential Kadomtsev–Petviashvili (pKP) equation is derived by the truncated Painleve analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variable. Thanks to localization process, the finite symmetry transformations related with the nonlocal symmetry are obtained by solving the prolonged systems. The inelastic interactions among the multiple-front waves of the pKP equation are generated from the finite symmetry transformations. Based on the consistent tanh expansion method, a nonauto-Backlund transformation (BT) theorem of the pKP equation is constructed. We can get many new types of interaction solutions because of the existence of an arbitrary function in the nonauto-BT theorem. Some special interaction solutions are investigated both in analytical and graphical ways.

A full Lie point symmetry analysis of rational difference equations is performed. Non-trivial symmetries are derived and exact solutions using these symmetries are obtained., Comment: 19 pages

The nonlocal symmetries are derived for the Korteweg–de Vries–negative-order Korteweg–de Vries equation from the Painlevé truncation method. The nonlocal symmetries are localized to the classical Lie point symmetries for the enlarged system by introducing new dependent variables. The corresponding similarity reduction equations are obtained with different constant selections. Many explicit solutions for the integrable equation can be presented from the similarity reduction.

In this work we use Lie group theoretic methods and the theory of prolonged group actions to study two fully nonlinear partial differential equations (PDEs). First we consider a third order PDE in two spatial dimensions that arises as the analogue of the Euler-Lagrange equations from a second order variational principle in $L^{\infty}$. The equation, known as the $\infty$-Polylaplacian, is a higher order generalisation of the $\infty$-Laplacian, also known as Aronsson's equation. In studying this problem we consider a reduced equation whose relation to the $\infty$-Polylaplacian can be considered analogous to the relationship of the Eikonal to Aronsson's equation. Solutions of the reduced equation are also solutions of the $\infty$-Polylaplacian. For the first time we study the Lie symmetries admitted by these two problems and use them to characterise and construct invariant solutions under the action of one dimensional symmetry subgroups.

The residual symmetry of the (3 + 1) -dimensional Burgers system is localized to a Lie point symmetry in a prolonged system and the corresponding finite transformation is obtained by using Lie’s first theorem. By further localize the linear superposition of multiple residual symmetries, the N -th Backlund transformations (BT) of the (3 + 1) -dimensional Burgers system are also got. By applying the standard Lie symmetry method to the prolonged system, not only the Lie symmetry group but also the symmetry reduction solutions are obtained, which include abundant interaction solutions between solitons and nonlinear waves. Furthermore, the (3 + 1) -dimensional Burgers system is proved to have consistent Riccati expansion (CRE) property, based on which some new BTs are given.

Through the standard truncated Painleve expansion of the Boiti–Leon–Pempinelli equation, the residual symmetry is localized in the properly prolonged system with the Lie point symmetry vector. Some different transformation invariances are derived through the obtained symmetries. At the same time, the symmetry of the equation is also derived utilizing the Clarkson–Kruskal direct method. From which, through solving the characteristic equations, several types of the explicit reduction solutions that related the hyperbolic tangent function are obtained. Finally, some kink-type soliton excitations are depicted from one of them.

Using the standard truncated Painlevé expansion, the residual symmetry of the (2+1)-dimensional modified dispersive water-wave system is localized in the properly prolonged system with the Lie point symmetry vector. Some different transformation invariances are derived by utilizing the obtained symmetries. The symmetries of the system are also derived through the Clarkson–Kruskal direct method, and several types of explicit reduction solutions relate to the trigonometric or the hyperbolic functions are obtained. Finally, some special solitons are depicted from one of the solutions.

The purpose of this review paper is two-fold. First, we point out one more time the importance and usefulness of the S. Lie theory of symmetries of differential equations. In particular some illustrative examples are presented. Then, we discuss the Noetherian approach to Rellich–Pohozaev type identities.

A previous article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs. They have infinite-dimensional Lie point symmetry groups due to the linear superposition principle. Their symmetry algebra always contains a two-dimensional {sub}algebra realized by linearly connected vector fields. We identify all classes of linear first-order DODSs that have additional symmetries, not due to linearity alone. We present representatives of each class. These additional symmetries are then used to construct exact analytical particular solutions using symmetry reduction.

We introduce an inhomogeneous term, f(t,x), into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions f(t,x) which depend nontrivially on both t and x, we find that there is just one symmetry. If f is a function of only x, there are three symmetries with the algebra s l(2,R). When f is a function of only t, there are five symmetries with the algebra s l(2,R) ⊕ s 2A 1. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.

Starting from the Lax pairs, the nonlocal symmetries of the coupled variable-coefficient Newell-Whitehead system are obtained. By introducing an appropriate auxiliary dependent variable, the nonloc...

We find the Lie point symmetries of a nonlinear population model, i.e. a second-order reaction-diffusion equation with a variable coefficient b ( x ) and classify the model into three kinds. Then, with the help of the Lie point symmetries and self-adjointness of each kind, using a general theorem on conservation law (Ibragimov, 2007), we establish the conservation laws corresponding to every Lie point symmetry obtained. In addition, some exact solutions are constructed.

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.

In this article, we construct novel explicit solutions for nonlinear Schr\"odinger systems with spatially inhomogeneous nonlinearity by means of the Lie symmetry method. We focus the attention to solutions with non-trivial phase, which have been scarcely considered in the related literature. To get started, the theoretical method based on Lie symmetries is exposed, thus reducing the problem to the integrability of an ODE. The non-trivial phase introduces a singular term into the ODE. Then, the method is used to construct new families of analytical solutions. Some illustrative examples are provided., Comment: 21 pages, 11 figures, some corrections made and 5 new figures added

Some exact solutions of KdV-Burgers-Kuramoto (KBK) equation are derived by the anzas and tanh methods. Also, the most general Lie point symmetry group of the KBK equation are presented using the basic Lie symmetry method. As well as, the non-classical and weak symmetries of this equation, as well as the corresponding similarity reductions, are investigated. Finally, the classical and non-classical symmetries of KBK and KdV-Burgers (KB) equations are compared.

In this letter, based on the Lax pair of the Ablowitz–Kaup–Newell–Segur (AKNS) system, the nonlocal symmetry is obtained and successfully localized to a Lie point symmetry by introducing an appropriate auxiliary dependent variable. For the closed prolongation, the construction for the one-dimensional optimal system is presented in detail. Furthermore, using the obtained optimal system, we give the reductions and the explicit analytic interaction solutions between cnoidal waves and solitary waves. For some interesting solutions, the figures are given to show their properties.

In this paper, the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation (HSE) is analyzed. By applying the basic Lie symmetry method for the HSE, the classical Lie point symmetry operators are obtained. Also, the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of one-dimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed. Particularly, the Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. Mainly, the conservation laws of the HSE are computed via three different methods including Boyer's generalization of Noether's theorem, first homotopy method and second homotopy method.

Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell-Bloch equations. The nonlocal symmetries of the reduced Maxwell-Bloch equations are obtained by the truncated Painleve expansion approach and the Mobious invariant property. The nonlocal symmetries are localized to a prolonged system by introducing suitable auxiliary dependent variables. The extended system can be closed and a novel Lie point symmetry system is constructed. By solving the initial value problems, a new type of finite symmetry transformations is obtained to derive periodic waves, Ma breathers and breathers travelling on the background of periodic line waves. Then rich exact interactional solutions are derived between solitary waves and other waves including cnoidal waves, rational waves, Painleve waves, and periodic waves through similarity reductions. In particular, several new types of localized excitations including rogue waves are found, which stem from the arbitrary function generated in the process of similarity reduction. By computer numerical simulation, the dynamics of these localized excitations and interactional solutions are discussed, which exhibit meaningful structures., Comment: 19 pages, 10 figures

The present work deals with a quintom model of dark energy in the framework of a spatially flat isotropic and homogeneous Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. At first, Lie point symmetry is imposed to the system and the unknown coupled potential of the model is determined. Then Noether symmetry, which is also a point like symmetry of the Lagrangian, is imposed on the physical system and the potential takes a general form. It is shown that the Lie algebra of Noether symmetry is a sub-algebra of the corresponding Lie algebra of the Lie symmetry. Finally, a point transformation in the three dimensional augmented space is performed suitably so that one of the variables become cyclic and as a result there is considerable simplification to the physical system. Hence conserved quantities (i.e, constants of motion) are expressed in a compact form and cosmological solutions are evaluated and analyzed in the present context., 14 pages, 6 figures, accepted in IJMPD

The $$(1+2)$$ -dimensional heat equation on the plane ( $$\mathbb R ^{2}$$ ) and sphere ( $$\mathbb S ^{2}$$ ) is considered respectively. For each surface a class of functions is presented for which considered equation has nontrivial symmetries. We consider whether the background metric ( $$\mathbb S ^{2}$$ ) or the nonlinearity have the dominant role in the infinitesimal generators of the considered equation. Then the time dependent Ginzburg–Landau equation (TDGL model) is considered on $$\mathbb S ^{2}$$ . Lie point symmetry generators are calculated and optimal systems of its subalgebras up to conjugacy classes are obtained. Similarity reductions for each class are performed.

We utilize the Lie–Tresse linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painleve–Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painleve–Gambier equations. It is also shown that those Painleve–Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painleve–Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.

With the aid of the classical Lie group method and nonclassical Lie group method, we derive the classical Lie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS) equation. Using the symmetries, we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation. Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.

The results obtained previously by Nadjafikhah [M. Nadjafikhah, Lie symmetries of inviscid Burgers equation, Adv. Appl. Clifford Alg., vol. 19, 101–112, (2009)] are complemented since a new projectable point symmetry of the inviscid Burgers equation is found.

Lie point symmetries of the first-order inviscid Burgers equation in a general setting are studied. Some new and interesting solutions are presented.

In this paper, the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system: the two-dimensional damped Kuramoto—Sivashinsky ((2D) DKS) equation is studied. By applying the basic Lie symmetry method for the (2D) DKS equation, the classical Lie point symmetry operators are obtained. Also, the optimal system of one-dimensional subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The nonclassical symmetries of the (2D) DKS equation are also investigated.