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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 this work, we emphasize the use of singularity analysis in obtaining analytic solutions for equations for which standard Lie point symmetry analysis fails to make any lucid decision. We study the higher-dimensional Kadomtsev–Petviashvili, Boussinesq, and Kaup–Kupershmidt equations in a more general sense. With higher-order equations, there can be a commensurate number of resonances and when consistency for the full equation is examined at each resonance the constant of integration is supposed to vanish from the expression so that it remains arbitrary, but if there is an instance of this not happening, the consistency can be partially established by giving the offending constant the value from the defining equation. If consistency is otherwise not compromised, the equation can be said to be partially integrable, i.e., integrable on a surface of the complex space. Furthermore, we propose an approach that is meant to magnify the scope of singularity analysis for equations admitting higher values for resonances or positive leading-order exponent.

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...

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.

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.

Solving nonlinear ordinary differential equations is one of the fundamental and practically important research challenges in mathematics. However, the problem of their algorithmic linearizability so far remained unsolved. In this contribution, we propose a solution of this problem for a wide class of nonlinear ordinary differential equation of arbitrary order. We develop two algorithms to check if a nonlinear differential equation can be reduced to a linear one by a point transformation of the dependent and independent variables. In this regard, we have restricted ourselves to quasi-linear equations with a rational dependence on the occurring variables and to point transformations. While the first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra, the second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of our algorithms is discussed and evaluated using several examples.

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.

Рассматривается нелинейное двумерное ортотропное уравнение фильтрации с дробной производной Римана-Лиувилля по времени. Доказывается, что такое уравнение может допускать группы точечных преобразований только линейно-автономного типа. Решается задача групповой классификации рассматриваемого уравнения по его точечным симметриям относительно коэффициентов пьезопроводности, являющихся функциями квадрата модуля градиента давления. Доказывается, что если порядок дробного дифференцирования меньше единицы, основная допускаемая уравнением группа точечных преобразований является четырехпараметрической и расширяется до пятипараметрической в изотропном случае. Для степенных зависимостей коэффициентов пьезопроводности допускаемая группа становится пятипараметрической в ортотропном случае и шестипараметрической в изотропном случае. Также выделяется специальный вид степенной зависимости коэффициентов, не имеющий аналога в случае классического уравнения фильтрации, при котором происходит дополнительное расширение группы оператором проективного преобразования. Для уравнения с порядком дробного дифференцирования $\alpha \in (1,2)$ размерности всех допускаемых групп оказываются на единицу больше за счет допускаемого оператора, соответствующего сдвигу решения на дополнительное частное решение этого уравнения. На основе проведенной групповой классификации для соответствующих алгебр Ли инфинитезимальных операторов групп точечных преобразований, допускаемых различными видами рассматриваемого нелинейного ортотропного дробно-дифференциального уравнения, выписываются представления инвариантно-групповых решений, соответствующие оптимальным системам двумерных подалгебр. Приводятся примеры уравнений, получающихся в результате симметрийной редукции исходного дробно-дифференциального уравнения, а также некоторые их решения. Доказывается, что рассматриваемое дробно-дифференциальное уравнение фильтрации является нелинейно самосопряженным, что дает возможность строить его законы сохранения. Соответствующие компоненты всех найденных сохраняющихся векторов приводятся в явном виде.

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 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 paper, the Fujimoto-Watanabe equation is studied with the help of the classical Lie point symmetry analysis method. Infinitesimal generators, the en­tire geometric vector fields and symmetr...

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.

Orientador: Igor Leite Freire Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica Resumo: Neste trabalho, é estudada uma generalização da equação de Burgers do ponto de vista da teoria de simetrias de Lie Abstract: In this work, a generalization of Burgers equation is studied from the point of view of Lie point symmetry theory Mestrado Matemática Aplicada Mestre em Matemática Aplicada

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 present paper considers the group analysis of the Newell–Whitehead–Segel equation of fractional order. The important advantage of the article is that for the first time we analyzed the equation for time, space and space–time derivatives of Riemann fractional together. This is based on similarity variables obtained from Lie point symmetry operators of the equation in triple cases. These variables help us to find the reduced equations. Finally, we use the generalized Noether’s theorem of fractional order for finding conservation laws of the equations.

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.

In this paper, we study Lie symmetry analysis of the time fractional fifth-order modified Sawada-Kotera equation (FMSK) with Riemann-Liouville derivative. Applying the adapted the Lie group theory to the equation under study, two dimensional Lie algebra is deduced. Using the obtained nontrivial Lie point symmetry, it is shown that the equation can be converted into a nonlinear fifth order ordinary differential equation of fractional order in the meaning of the Erdelyi-Kober fractional derivative operator. In addition, we construct some exact traveling solutions for the FMSK using the sub-equation method.

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.

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 paper, Lie point symmetry group of the Harry-Dym type equation with Riemann-Liouville fractional derivative is constructed. Then complete subgroup classification is obtained by means of the optimal system method. Finally, corresponding group-invariant solutions with reduced fractional ordinary differential equations are presented via similarity reductions.

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.

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.

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.

The enhanced group classification of a semi linear generalization of a general bond-pricing equation is carried out by harnessing the underlying equivalence and additional equivalence transformations. We employ that classification to unearth the particular cases with a larger Lie algebra than the general case and use them to find non trivial invariant solutions under the terminal and the barrier option condition.

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 Lie symmetry analysis method is applied to study nonlinear generalized Zakharov system which is the coupled nonlinear system of Schrodinger equations. With the aid of Lie point symmetry, nonlinear generalized Zakharov system is reduced into the ODEs and some group invariant solutions are obtained where some solutions are new, which are not reported in literatures. Then the bifurcation theory and qualitative theory are employed to investigate nonlinear generalized Zakharov system. Through the analysis of phase portraits, some Jacobi-elliptic function solutions are found, such as the periodic-wave solutions, kink-shaped and bell-shaped solitary-wave 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.

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 article, we discuss Lie point symmetry of stochastic differential equations driven by Wiener and Poisson processes. The symmetry is obtained by considering infinitesimals involving not only spatial and temporal variables but also that of vector Wiener process variable W(t). This work leads to the derivation of the random time-change formula of Ito Brownian motion in Lie transformation context.

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.

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.

In this paper, we obtain some new exact solutions of the modified Korteweg de Vries equation using Lie point symmetry method. The new obtained solutions are doubly periodic and periodic solutions. We confirm that this equation possesses many unusual features of distinct solutions. To our knowledge, the obtained solutions are new.

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.