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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.

In this paper, the time-fractional coupled Burger equation under Riemann-Liouville derivative is systematically analyzed. Firstly, the Lie point symmetry is obtained by applying the Lie symmetry an...

The aim of this work is to explore and find new closed-form nematicon solutions for different nonlinearities which occur in nematic liquid crystals (NLC) along with proposing optical system application that utilizes NLC nonlinearities. In particular, Lie point symmetry method is employed to scrupulously inspect and acquire solutions for some interesting cases of nonlinearities which have not been fully examined in literature such as quadratic, generalized dual-power law, and eighth-order nonlinearities. A variety of different nematicon dynamics are observed, including bright solitons, dark solitons and periodic behaviors. The explicit solution form for each dynamical behavior is obtained and the solution dependence on model parameters is investigated. The proposed optical system enables the flexible realization of different types of NLC nonlinearities. To the best of our knowledge, this is the first time to attain explicit exact nontrivial solutions for the particular cases of generalized dual-power law and eighth-order nonlinearities.

The aim of the research is to study the propagation of a hydraulic fracture with tortuosity due to contact areas between touching asperities on opposite crack walls. The tortuous fracture is replaced by a model symmetric partially open fracture with a hyperbolic crack law and a modified Reynolds flow law. The normal stress at the crack walls is assumed to be proportional to the half-width of the model fracture. The Lie point symmetry of the nonlinear diffusion equation for the fracture half-width is derived and the general form of the group invariant solution is obtained. It was found that the fluid flux at the fracture entry cannot be prescribed arbitrarily, because it is determined by the group invariant solution and that the exponent n in the modified Reynolds flow power law must lie in the range 2 n δ ≪ 1 to avoid singularities, to the fracture entry. The numerical results showed that the tortuosity and the pressure due to the contact regions both have the effect of increasing the fracture length. The spatial gradient of the half-width was found to be singular at the fracture tip for 3 n n = 3 and to be zero for 2 n n n < 3. The effect of the touching asperities is to decrease the width averaged fluid velocity. An approximate analytical solution for the half-width, which was found to agree well with the numerical solution, is derived by making the approximation that the width averaged fluid velocity increases linearly with distance along the fracture.

We study the early inflationary phase of the universe driven by noncanonical scalar field models using an exponential potential. The noncanonical scalar field models are represented by Lagrangian densities containing square and square-root kinetic corrections to the canonical Lagrangian density. We investigate the Lie symmetry of the homogeneous scalar field equations obtained from noncanonical Lagrangian densities and find only a one-parameter Lie point symmetry for both canonical and noncanonical scalar field equations. We use the Lie symmetry generator to obtain an exact analytical group-invariant solution of the homogeneous scalar field equations from an invariant curve condition. The solutions obtained are consistent and satisfy the Friedmann equations subject to constraint conditions on the potential parameter $$\lambda$$ for the canonical case and on the parameter $$\mu$$ for the noncanonical case. In this scenario, we obtain the values of various inflationary parameters and make useful checks on the observational constraints on the parameters from Planck data by imposing a set of bounds on the parameters $$\lambda$$ and $$\mu$$ . The results for the scalar spectral index ( $$n_{S}$$ ) and the tensor-to-scalar ratio ( $$r$$ ) are presented in the $$(n_{S},r)$$ plane in the background of Planck-2015 and Planck-2018 data for noncanonical cases and are in good agreement with cosmological observations. For theoretical completeness of the noncanonical models, we verify that the noncanonical models under consideration are free from ghosts and Laplacian instabilities. We also treat the noncanonical scalar field model equations for two power-law (kinetic) corrections by the dynamical system theory. We provide useful checks on the stability of the critical points for both cases and show that the group-invariant analytical noncanonical inflation solutions are stable attractors in phase space.

This study examines the invariance properties of the thermodynamic entropy production in its global (integral), local (differential), bilinear, and macroscopic formulations, including dimensional scaling, invariance to fixed displacements, rotations or reflections of the coordinates, time antisymmetry, Galilean invariance, and Lie point symmetry. The Lie invariance is shown to be the most general, encompassing the other invariances. In a shear-flow system involving fluid flow relative to a solid boundary at steady state, the Galilean invariance property is then shown to preference a unique pair of inertial frames of reference—here termed an entropic pair—respectively moving with the solid or the mean fluid flow. This challenges the Newtonian viewpoint that all inertial frames of reference are equivalent. Furthermore, the existence of a shear flow subsystem with an entropic pair different to that of the surrounding system, or a subsystem with one or more changing entropic pair(s), requires a source of negentropy—a power source scaled by an absolute temperature—to drive the subsystem. Through the analysis of different shear flow subsystems, we present a series of governing principles to describe their entropic pairing properties and sources of negentropy. These are unaffected by Galilean transformations, and so can be understood to “lie above” the Galilean inertial framework of Newtonian mechanics. The analyses provide a new perspective into the field of entropic mechanics, the study of the relative motions of objects with friction.

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 residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

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.

Magnetohydrodynamic (MHD) flow due to a nonlinearly stretching sheet acting on non-Newtonian Eyring-Powell fluids inside a Darcy-Forchheimer porous medium is studied. Energy equation is considered subject to 1- Prescribed Surface Temperature 2- Prescribed Surface Heat Flux, both of which with radiation consideration. The resulting similarity equations are tackled analytically by homotopy analysis method, homotopy perturbation method and numerically by 5th order Runge-Kutta and an advanced multistep MATLAB solver. A comparison reveals a remarkable agreement between the solution strategies. Finally, the influence of the involved parameters such as magnetic parameter, Eyring-Powell constitutional constants, Darcy-Forchheimer resistance factors, Prandtl number and radiation parameter is studied in a comprehensive manner. The new considerations in the present research can be summarized as: 1- Lie point symmetry analysis to simply extract the similarity problem. 2- Considering Forchheimer parameter associated with the microscopic inertial effects those leading to significant macroscopic impacts. 3- Different thermal boundary conditions, radiation effect included. 4- A comparative study on several analytical and numerical techniques. 5- A graphical demonstration of the influence of the involved parameters. Keywords: MHD Eyring-Powell fluids, Nonlinearly stretching sheet, Homotopy analysis method, Homotopy perturbation method, Numerical solutions, Darcy-Forchheimer porosity, Prescribed surface temperature, Prescribed surface heat flux

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.

Under investigation in this paper is a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation in fluid mechanics and plasma physics. We obtain the Lie point symmetry generators, Lie symmetry group and symmetry reductions via the Lie group method. Hyperbolic-function, power-series, trigonometric-function, soliton and rational solutions are derived via the power-series expansion, polynomial expansion and $$\left( \frac{G^{'}}{G}\right) $$ expansion method.

The residual symmetry for the modified JaulentMiodek equation is obtained with truncated Painleve expansion method and the form of Lie point symmetry group and the corresponding finite transformations are obtained. Furthermore, the modified Jaulent-Miodek equation also be proved to have consistent tanh expansion form. Based on this property, the interaction solutions between soliton and conoidal periodic wave are explicitly given analytically and graphically in two special cases.

We study the inflationary phase of the early universe as modeled by a non-canonical scalar field. The homogeneous scalar field equation is derived from a Lagrangian density containing a new form of non-canonical kinetic term and a general potential function. The Lie symmetry is studied and a one parameter Lie point symmetry for the homogeneous scalar field equation is found. We use Lie symmetry generator to construct the exact analytical group invariant closed-form solution of the homogeneous scalar field equation without applying any slow-roll approximation from invariant curve condition. The solution thus obtained is seen to be consistent with the Friedmann equations subject to constraint conditions on the potential parameter $$\lambda $$ . In this scenario, we obtain the values for various inflationary parameters and make useful checks on the observational constraints on the parameters from Planck data by imposing a set of bounds on the parameter $$\lambda $$ of the potential. The results for scalar spectral index ( $$n_S$$ ) and tensor-to-scalar ratio (r) presented in the $$(n_S,\,r)$$ plane in the background of Planck2015 and Planck2018 data are in good agreement with cosmological observations. We find $$r\sim 10^{-3}$$ , the targeted value of r that will be detected by the future CMB observation such as LiteBIRD. Interestingly, most significant primordial non-Gaussianity is also achieved. For theoretical completeness of our non-canonical model, we obtain the allowed parameter space in which the ghosts and Laplacian instabilities are absent. We apply the formulas for slow-roll parameter to explain exit from the inflationary phase using the general potential. We also treat the non-canonical scalar field model equation by the dynamical system theory to provide useful checks on the stability of the critical points and show that the group invariant non-canonical inflationary solution is stable attractor in the phase space.

The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation u ″ ′ = f ( x , u , u ′ , u ″ ) which admits a five-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained constant invariant is also presented. We also show how one obtains the point transformation that does the reduction to linear form. Moreover, some applications are provided.

The barotropic vorticity equation is a classical model in atmospheric sciences. In this paper, we study the symmetry invariance properties of multipliers (integrating factors) admitted by this equation. The results are classified according to the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth’s angular rotation. A plethora of conservation laws can be obtained by studying the interaction between Lie point symmetry generators and multipliers.

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 nonlocal symmetry for the Whitham-Broer-Kaup (WBK) equation is obtained by the Painleve analysis. With the introduction of two auxiliary dependent variables, the nonlocal symmetry is localized to the Lie point symmetry. The reciprocal transformation is established by the localization nonlocal symmetry procedure and can be used to construct a link between the WBK equation and the usual Burgers equation. By using the similarity reductions related with the nonlocal symmetry of the Burgers equation, we derive the symmetry invariant solutions for the WBK equation. Furthermore, the power-series solutions are computed by using the power-series theory.

In this article, we investigate a (3+1)-dimensional Schamel–Zakharov–Kuznetsov–Burgers (SZKB) equation, which describes the nonlinear plasma-dust ion acoustic waves (DIAWs) in a magnetised dusty plasma. With the aid of the Kudryashov method and symbolic computation, a set of new exact solutions for the SZKB equation are derived. By introducing two special functions, a variational principle of the SZKB equation is obtained. Conservation laws of the SZKB equation are obtained by two different approaches: Lie point symmetry and the multiplier method. Thus, the conservation laws here can be useful in enhancing the understanding of nonlinear propagation of small amplitude electrostatic structures in the dense, dissipative DIAWs’ magnetoplasmas. The properties of the shock wave solutions structures are analysed numerically with the system parameters. In addition, the electric field of this solution is investigated. Finally, we will study the physical meanings of solutions.

The paper deals with a non--minimally coupled scalar field in the background of homogeneous but anisotropic Kantowski--Sachs space--time model. The form of the coupling function of the scalar field with gravity and the potential function of the scalar field are not assumed phenomenologically, rather they are evaluated by imposing Noether symmetry to the Lagrangian of the present physical system. The physical system gets considerable mathematical simplification by a suitable transformation of the augmented variables $(a, b, \phi)\rightarrow (u, v, w)$ and by the use of the conserved quantities due to the geometrical symmetry. Finally, cosmological solutions are evaluated and analyzed from the point of view of the present evolution of the Universe., Comment: Accepted in "Annals of Physics"

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.

This paper obtains a combined optical solitary wave solution that is modeled by nonlinear Chen–Lee–Liu equation (NCLE) which arises in the context of temporal pulses along optical fibers associated with the self-steepening nonlinearity using the complex envelope function ansatz. The novel combined solitary wave describes bright and dark solitary wave properties in the same expression. The intensity and the nonlinear phase shift of the combined solitary wave solution are reported. Moreover, the Lie point symmetry generators or vector fields of a system of partial differential equations (PDEs) which is acquired by transforming the NCLE to a real and imaginary parts are derived. It is observed that the obtained system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to establish a set of conservation laws (Cls) for the system using the general Cls theorem. Numerical simulation and physical interpretations of the obtained results are demonstrated with interesting figures showing the meaning of the acquired results. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the NCLE.

Group invariant and numerical solutions for the evolution of a two-dimensional fracture with non-zero initial length in permeable rock and driven by a laminar incompressible Newtonian fluid are obtained. The fluid leak-off into the rock mass is modelled using Darcy law. With the aid of lubrication theory and the PKN approximation, a system of nonlinear partial differential equations for the fracture half-width and the extent of leak-off is derived. Since the fluid–rock interface is permeable the nonlinear diffusion equation contains a leak-off velocity sink term. Using the Lie point symmetries the problem is reduced to a boundary value problem for a system of second order ordinary differential equations.

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, we consider a (2+1)-dimensional breaking soliton equation which describe the (2+1)-dimensional interaction of the Riemann wave propagating along the y-axis with a long wave along the x-axis. By the Lie group analysis, the Lie point symmetry generators and symmetry reductions were deduced. From the viewpoint of exact solutions, we have performed two distinct methods to the equation for getting some exact solutions. Kudryashov’s simplest methods and ansatz method with the assistance of Maple were carried out. The local conservation laws are also constructed by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.

In this paper, the sine-Gordon equation expansion method (SGEM) is used to acquire the optical solitary waves to the nonlinear Schrodinger's equation (NLSE) that arises from compressional dispersive Alven (CDA) waves. As a result of the operations, dark, bright, dark–bright and singular optical solitary waves are derived. The solitary waves appear with all necessary constraint conditions which guarantee their existence. The Lie point symmetry generators of a system of partial differential equations (PDEs) obtained by transforming the equation into real and imaginary parts are derived. We prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to construct a set of conservation laws (Cls) for the system using the general Cls theorem presented by Ibragimov. Furthermore, the modulation instability (MI) is studied based on the standard linear-stability analysis and the MI gain spectrum is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

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.

In this paper, the optical solitons to the modified nonlinear Schrodinger’s equation for davydov solitons are investigate. The modified F-expansion method is the integration technique employed to achieve this task. This yielded a combined and other soliton solutions. The Lie point symmetry generators of a system of partial differential equations acquired by decomposing the equation into real and imaginary components are derived. We prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to construct a set of local conservation laws (Cls) for the system using the general Cls theorem presented by Ibragimov. Furthermore, the modulation instability (MI) is analyzed based on the standard linear-stability analysis and the MI gain spectrum is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

In this paper, the complex envelope function ansatz method is used to acquire the optical solitons to the cubic nonlinear Shrodinger's equation with repulsive delta potential (δ−NLSE). The method reveals dark and bright optical solitons. The necessary constraint conditions which guarantee the existence of the solitons are also presented. We studied the δ−NLSE by analyzing a system of partial differential equations (PDEs) obtained by decomposing the equation into real and imaginary components. We derive the Lie point symmetry generators of the system and prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to establish a set of conserved vectors for the system using the general Cls theorem presented by Ibragimov. Some interesting figures for the acquired solutions are also presented.

In this work, we investigate the soliton solutions to the resonant nonlinear Shrodinger's equation (R-NSE) with Kerr law nonlinearity. By adopting the Riccati–Bernoulli sub-ODE technique, we present the exact dark optical, dark-singular and periodic singular soliton solutions to the model. The soliton solutions appear with all necessary constraint conditions that are necessary for them to exist. We studied the R-NSE by analyzing a system of nonlinear partial differential equations (NPDEs) obtained by decomposing the equation into real and imaginary components. We derive the Lie point symmetry generators of the system, then we apply the general conservation theorem to establish a set of nontrivial and nonlocal conservation laws (Cls). Some interesting figures for the acquired solutions are Cls also presented.

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.

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.

We review the symmetry properties of differential equations involving the Schwarzian derivative. We propose and study new differential equations and systems determined by their Lie point symmetry groups, which generalize the Kummer-Schwarz equation and the Schwarzian Korteweg-de Vries equation.

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

In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.

Under investigation in this paper is a modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics/plasma physics. Painleve integrability is investigated. Soliton-cnoidal and resonant solutions are derived via the consistent tanh expansion method. The equation is reduced to certain symmetry reduction equations via the Lie point symmetry generators obtained though the Lie group method. Hyperbolic function and rational solutions are derived through those symmetry reduction equations.

A generalized Kadomtsev—Petviashvili equation is studied by nonlocal symmetry method and consistent Riccati expansion (CRE) method in this paper. Applying the truncated Painleve analysis to the generalized Kadomtsev—Petviashvili equation, some Backlund transformations (BTs) including auto-BT and non-auto-BT are obtained. The auto-BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painleve expansion. Then the nonlocal symmetry is localized to the corresponding nonlocal group by introducing two new variables. Further, by applying the Lie point symmetry method to the prolonged system, a new type of finite symmetry transformation is derived. In addition, the generalized Kadomtsev—Petviashvili equation is proved consistent Riccati expansion (CRE) solvable. As a result, the soliton-cnoidal wave interaction solutions of the equation are explicitly given, which are difficult to be found by other traditional methods. Moreover, figures are given out to show the properties of the explicit analytic interaction solutions.

A consistent tanh expansion (CTE) method is used to study the modified Boussinesq equation. It is proved that the modified Boussinesq equation is CTE solvable. The soliton-cnoidal periodic wave is explicitly given by a nonanto-BT theorem. Furthermore, the nonlocal symmetry for the modified Boussinesq equation is obtained by the Painleve analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing one auxiliary dependent variable. The finite symmetry transformation related with the nonlocal symemtry is obtained by solving the initial value problem of the prolonged systems. Thanks to the localization process, many interaction solutions among solitons and other complicated waves are computed through similarity reductions. Some special concrete soliton-cnoidal wave interaction behaviors are studied both in analytical and graphical ways.

In this paper, we consider the conservation laws for the far downstream wake equations described by eddy viscosity. A basis of conserved vectors is constructed. The well-known conserved quantities for the turbulent classical wake and the turbulent wake of a self-propelled body are obtained by integrating the corresponding conservation law across the wake and imposing the boundary conditions. For the wake of a self-propelled body the additional condition that the drag on the body is zero and is required to obtain the conserved quantity. A third conservation law, which possibly belongs to another type of wake, is discovered. The Lie point symmetry associated with the conserved vector is used to obtain the invariant solution and a typical velocity profile for this wake is provided. This wake appears to have common properties with the other two well-known wakes. We then analyse the invariant solutions to all three wake problems and prove that a simple mathematical relationship exists between them thus unifying the theory for turbulent wake flows.

In this paper, Lie point symmetry method is used to obtain the general exact solution of the second order nonlinear ordinary differential equation which governing heat transfer in rectangular fin with variable thermal conductivity. Some new forms of thermal conductivity are introduced and the associated exact solution is obtained in each case. The general relation among the fin efficiency, thermal conductivity and thermo-geometric parameter is obtained.

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.

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.

Power-law inflationary scenarios based on minimal and non-minimal coupling of a scalar field to gravity with exponential- and power-law potential, respectively, are studied using the symmetry-based approach. In particular, we obtained only one parameter Lie point symmetry for both the minimal and non-minimal coupling cases and it is interesting to note that the non-minimal coupled equation is invariant under a scale transformation. We find the exact analytical group invariant solutions from invariant curve condition for both the minimal and non-minimal cases of the power-law inflationary model. The solutions obtained are consistent with the Friedmann equations subject to constraints on the inflationary potential parameter $ \lambda$ for the minimal case and on the coupling parameter $ \zeta$ for the non-minimal case. In this scenario, we find transformation relations for various inflationary parameters e.g. amplitude of the scalar power spectrum, spectral index, slow-roll parameters (SRP), tensor-to-scalar perturbation ratio, equation of state parameters, non-Gaussianity parameter as well as the form of the potential in two different frames, namely the Jordan and Einstein frames by making use of the conformal transformation. The results for various inflationary parameters for the non-minimal case are presented in the background of Planck2015 and Planck2018 and are in good agreement with the cosmological observations if the non-minimal coupling parameter is chosen properly. We treat minimally and non-minimally coupled scalar field equations by the dynamical system theory and present critical point analysis. By checking the stability of the critical points in the phase space for both cases we have shown that the solutions obtained from the Lie symmetry approach are the stable attractor solutions.

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.

We work on the (3+1)-dimensional Zakharov-Kuznetsov-Burgers equation for the ion-acoustic waves in a collisionless magnetized electron-positron-ion plasma. We derive the Lie point symmetry generators and Lie symmetry groups. We construct certain solutions which are related to the known solutions. Using the symmetry generators, we obtain the reduction equations, through one of which we derive some power-series solutions and travelling-wave solutions including the shock solutions via the power-series and polynomial expansion methods. Shock waves are pictured out. Effects of the normalized ion gyrofrequency, $\Omega_{i}$ , the normalized kinematic viscosity, $\eta$ , the real parameter measuring the deviation from the Maxwellian equilibrium, $ \kappa$ , the ratio of the ion temperature to electron temperature, $\sigma_{1}$ , and the ratio of the electron temperature to positron temperature, $ \sigma_{2}$ , on the amplitude of the shock wave, |a|, are found: i) |a| is a hyperbolic function of $\eta$ ; ii) $\vert a\vert \rightarrow 0$ when $\Omega_{i}$ , $\kappa$ or $\eta \rightarrow 0$ ; iii) | a| goes to a constant when $ \Omega_i \rightarrow \pm \infty$ ; iv) |a| becomes a constant when $\kappa \rightarrow \pm \infty$ ; v) $\vert a\vert \rightarrow 0$ when $\sigma_{1} \rightarrow \pm \infty$ ; vi) |a| keeps unchanged when $\sigma_{2}$ varies and $\sigma_{1}= \frac{-3(2\kappa + \lambda^{2} - 2\kappa\lambda^{2})}{5(-1+2\kappa)}$ where $\lambda$ denotes the ion-acoustic wave propagation speed; vii) $\vert a\vert \rightarrow 0$ when $\sigma_{2} \rightarrow \pm \infty$ and $\sigma_{1} \neq\frac{-3(2\kappa +\lambda^{2}-2\kappa\lambda^{2})}{5(-1+2\kappa)}$ . We present the conditions of the nonlinear self-adjointness. Based on the nonlinear self-adjointness, we construct the conservation laws which are related to $\Omega_{i}$ , $\eta$ , $\kappa$ , $\sigma_{1}$ and $\sigma_{2}$ .

In this paper, we analyze the integrability of the Boussinesq equation by using the truncated Painleve expansion and the CRE method. Based on the truncated Painleve expansion, the nonlocal symmetry and Backlund transformation of this equation are obtained. A prolonged system is introduced to localize the nonlocal symmetry to the local Lie point symmetry. It is proved that the Boussinesq equation is CRE solvable. The two-solitary-wave fusion solutions, single soliton solutions and soliton-cnoidal wave solutions are presented by means of the Backlund transformations.