Lie symmetry analysis is applied to a(3+1)-dimensional combined potential Kadomtsev-Petviashvili equation with B-type Kadomtsev-Petviashvili equation(pKP-BKP equation)and the corresponding similarity reduction equatio...Lie symmetry analysis is applied to a(3+1)-dimensional combined potential Kadomtsev-Petviashvili equation with B-type Kadomtsev-Petviashvili equation(pKP-BKP equation)and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions and constants for the(3+1)-dimensional pKP-BKP equation,including the lump solution,the periodic-lump solution,the two-kink solution,the breather solution and the lump-two-kink solution,have been studied analytically and graphically.展开更多
We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inho...We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.展开更多
The generalized one-dimensional Fokker-Planck equation is analyzed via potential symmetry method and the invariant solutions under potential symmetries are obtained. Among those solutions, some are new and first repor...The generalized one-dimensional Fokker-Planck equation is analyzed via potential symmetry method and the invariant solutions under potential symmetries are obtained. Among those solutions, some are new and first reported.展开更多
Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dy...Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained.Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore,the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.展开更多
Lie symmetry analysis is applied to a(3+1)-dimensional Virasoro integrable model and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with a...Lie symmetry analysis is applied to a(3+1)-dimensional Virasoro integrable model and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions for the(3+1)-dimensional Virasoro integrable model,including the interaction solution between a kink and a soliton,the lump-type solution and periodic solutions,have been studied analytically and graphically.展开更多
On the basis of Lie group theory,(1 + N)-dimensional time-fractional partial differential equations are studied and the expression of η_α~0 is given. As applications, two special forms of nonlinear time-fractional d...On the basis of Lie group theory,(1 + N)-dimensional time-fractional partial differential equations are studied and the expression of η_α~0 is given. As applications, two special forms of nonlinear time-fractional diffusionconvection equations are investigated by Lie group analysis method. Then the equations are reduced into fractional ordinary differential equations under group transformations. Therefore, the invariant solutions and some exact solutions are obtained.展开更多
Lie group analysis method is applied to the extended(3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation and the corresponding similarity reduction equations are obtained with various infinitesimal generator...Lie group analysis method is applied to the extended(3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation and the corresponding similarity reduction equations are obtained with various infinitesimal generators.By selecting suitable arbitrary functions in the similarity reduction solutions,we obtain abundant invariant solutions,including the trigonometric solution,the kink-lump interaction solution,the interaction solution between lump wave and triangular periodic wave,the two-kink solution,the lump solution,the interaction between a lump and two-kink and the periodic lump solution in different planes.These exact solutions are also given graphically to show the detailed structures of this high dimensional integrable system.展开更多
The dynamics of atmosphere and ocean can be examined under different circumstances of shallow water waves like shallow water gravity waves,Kelvin waves,Rossby waves and inertio-gravity waves.The influences of these wa...The dynamics of atmosphere and ocean can be examined under different circumstances of shallow water waves like shallow water gravity waves,Kelvin waves,Rossby waves and inertio-gravity waves.The influences of these waves describe the climate change adaptation on marine environment and planet.Therefore,the present work aims to derive symmetry reductions of Broer-Kaup-Kupershmidt equation in shallow water of uniform depth and then a variety of exact solutions are constructed.It represents the propagation of nonlinear and dispersive long gravity waves in two horizontal directions in shallow water.The invariance of test equations under one parameter transformation leads to reduction of independent variable.Therefore,twice implementations of symmetry method result into equivalent system of ordinary differential equations.Eventually,the exact solutions of these ODEs are computed under parametric constraints.The derive results entail several arbitrary constants and functions,which make the findings more admirable.Based on the appropriate choice of existing parameters,these solutions are supplemented numerically and show parabolic nature,intensive and non-intensive behavior of solitons.展开更多
Based on the theory of Lie group analysis,the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied...Based on the theory of Lie group analysis,the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied.Full symmetry group admitted by the equilibrium equation and the yield criterion is a finitely generated Lie group with ten parameters.Several subgroups of the full symmetry group are used to generate invariants and group invariant solutions.Moreover,physical explanations of each group invariant solution are discussed by all appropriate transformations.The methodology and solution techniques used belong to the analytical realm.展开更多
Invariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoreticalmethod.The second order partial differential equations of elastodynamics are redu...Invariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoreticalmethod.The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators.Three invariant solutions are constructed.Their graphical figures are presented and physical meanings are elucidated in some cases.展开更多
Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to seco...Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations （ODEs） are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times singularities, we also obtain solutions which Along with solutions with time-dependent do not exhibit time-dependent singularities.展开更多
This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applic...This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applications in various scientific fields,including fluid dynamics,plasma physics,biological systems,and electricity-electronics.The study adopts Lie symmetry analysis as the primary framework for exploration.This analysis involves the identification of Lie point symmetries that are admitted by the differential equation.By leveraging these Lie point symmetries,symmetry reductions are performed,leading to the discovery of group invariant solutions.To obtain explicit solutions,several mathematical methods are applied,including Kudryashov's method,the extended Jacobi elliptic function expansion method,the power series method,and the simplest equation method.These methods yield solutions characterized by exponential,hyperbolic,and elliptic functions.The obtained solutions are visually represented through 3D,2D,and density plots,which effectively illustrate the nature of the solutions.These plots depict various patterns,such as kink-shaped,singular kink-shaped,bell-shaped,and periodic solutions.Finally,the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors.These conserved vectors play a crucial role in the study of physical quantities,such as the conservation of energy and momentum,and contribute to the understanding of the underlying physics of the system.展开更多
By Lie symmetry method, the Lie point symmetries and its Kac-Moody-Virasoro (KMV) symmetry algebra of (2+1)-dimensional dispersive long-wave equation (DLWE) are obtained, and the finite transformation of DLWE is given...By Lie symmetry method, the Lie point symmetries and its Kac-Moody-Virasoro (KMV) symmetry algebra of (2+1)-dimensional dispersive long-wave equation (DLWE) are obtained, and the finite transformation of DLWE is given by symmetry group direct method, which can recover Lie point symmetries. Then KMV symmetry algebra of DLWE with arbitrary order invariant is also obtained. On basis of this algebra the group invariant solutions and similarity reductions are also derived.展开更多
This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admi...This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admit certain types of AGCSs is derived. Some approximate invariant solutions to the resulting equations can also be obtained.展开更多
This paper studies the asymptotic behavior of solutions for nonlinear RLCnetwoiks which have the following form The function p i,v,t),called tile mired potential function,call be used to construct Liapunov functions t...This paper studies the asymptotic behavior of solutions for nonlinear RLCnetwoiks which have the following form The function p i,v,t),called tile mired potential function,call be used to construct Liapunov functions to prove the convergence of solutions under certain conditiolts. Under the assumption that every element value involving voltage source is asymptotically constallt, we establish four creteria for all solutiolls of such a system to converge to the set of equilibria of its limiting equations via LaSalle invariant principle.We also present two theorems on the existence of periodic solutions for periodically excited uonliltear circuits.This results generalize those of Brayton and Moser[1,2].展开更多
For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each...For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.展开更多
By means of the classical symmetry method,we investigate two types of the(2+1)-dimensional nonlinearKlein-Gorden equation.For the wave equation,we give out its symmetry group analysis in detail.For the secondtype of t...By means of the classical symmetry method,we investigate two types of the(2+1)-dimensional nonlinearKlein-Gorden equation.For the wave equation,we give out its symmetry group analysis in detail.For the secondtype of the(2+1)-dimensional nonlinear Klein-Gorden equation,an optimal system of its one-dimensional subalgebrasis constructed and some corresponding two-dimensional symmetry reductions are obtained.展开更多
We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous <span style="white-space:nowrap;">Monge-Ampère equation to first-order ...We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous <span style="white-space:nowrap;">Monge-Ampère equation to first-order ODEs. Some classes of the invariant solutions are constructed.展开更多
The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the se...The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.展开更多
The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via th...The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.展开更多
文摘Lie symmetry analysis is applied to a(3+1)-dimensional combined potential Kadomtsev-Petviashvili equation with B-type Kadomtsev-Petviashvili equation(pKP-BKP equation)and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions and constants for the(3+1)-dimensional pKP-BKP equation,including the lump solution,the periodic-lump solution,the two-kink solution,the breather solution and the lump-two-kink solution,have been studied analytically and graphically.
文摘We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.
文摘The generalized one-dimensional Fokker-Planck equation is analyzed via potential symmetry method and the invariant solutions under potential symmetries are obtained. Among those solutions, some are new and first reported.
文摘Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained.Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore,the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.
文摘Lie symmetry analysis is applied to a(3+1)-dimensional Virasoro integrable model and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions for the(3+1)-dimensional Virasoro integrable model,including the interaction solution between a kink and a soliton,the lump-type solution and periodic solutions,have been studied analytically and graphically.
基金Supported by the Natural Science Foundation of China under Grant Nos.11371287 and 61663043
文摘On the basis of Lie group theory,(1 + N)-dimensional time-fractional partial differential equations are studied and the expression of η_α~0 is given. As applications, two special forms of nonlinear time-fractional diffusionconvection equations are investigated by Lie group analysis method. Then the equations are reduced into fractional ordinary differential equations under group transformations. Therefore, the invariant solutions and some exact solutions are obtained.
文摘Lie group analysis method is applied to the extended(3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation and the corresponding similarity reduction equations are obtained with various infinitesimal generators.By selecting suitable arbitrary functions in the similarity reduction solutions,we obtain abundant invariant solutions,including the trigonometric solution,the kink-lump interaction solution,the interaction solution between lump wave and triangular periodic wave,the two-kink solution,the lump solution,the interaction between a lump and two-kink and the periodic lump solution in different planes.These exact solutions are also given graphically to show the detailed structures of this high dimensional integrable system.
文摘The dynamics of atmosphere and ocean can be examined under different circumstances of shallow water waves like shallow water gravity waves,Kelvin waves,Rossby waves and inertio-gravity waves.The influences of these waves describe the climate change adaptation on marine environment and planet.Therefore,the present work aims to derive symmetry reductions of Broer-Kaup-Kupershmidt equation in shallow water of uniform depth and then a variety of exact solutions are constructed.It represents the propagation of nonlinear and dispersive long gravity waves in two horizontal directions in shallow water.The invariance of test equations under one parameter transformation leads to reduction of independent variable.Therefore,twice implementations of symmetry method result into equivalent system of ordinary differential equations.Eventually,the exact solutions of these ODEs are computed under parametric constraints.The derive results entail several arbitrary constants and functions,which make the findings more admirable.Based on the appropriate choice of existing parameters,these solutions are supplemented numerically and show parabolic nature,intensive and non-intensive behavior of solitons.
文摘Based on the theory of Lie group analysis,the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied.Full symmetry group admitted by the equilibrium equation and the yield criterion is a finitely generated Lie group with ten parameters.Several subgroups of the full symmetry group are used to generate invariants and group invariant solutions.Moreover,physical explanations of each group invariant solution are discussed by all appropriate transformations.The methodology and solution techniques used belong to the analytical realm.
文摘Invariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoreticalmethod.The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators.Three invariant solutions are constructed.Their graphical figures are presented and physical meanings are elucidated in some cases.
文摘Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations （ODEs） are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times singularities, we also obtain solutions which Along with solutions with time-dependent do not exhibit time-dependent singularities.
基金the South African National Space Agency (SANSA) for funding this work
文摘This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation(LGHe),which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves.The LGHe finds applications in various scientific fields,including fluid dynamics,plasma physics,biological systems,and electricity-electronics.The study adopts Lie symmetry analysis as the primary framework for exploration.This analysis involves the identification of Lie point symmetries that are admitted by the differential equation.By leveraging these Lie point symmetries,symmetry reductions are performed,leading to the discovery of group invariant solutions.To obtain explicit solutions,several mathematical methods are applied,including Kudryashov's method,the extended Jacobi elliptic function expansion method,the power series method,and the simplest equation method.These methods yield solutions characterized by exponential,hyperbolic,and elliptic functions.The obtained solutions are visually represented through 3D,2D,and density plots,which effectively illustrate the nature of the solutions.These plots depict various patterns,such as kink-shaped,singular kink-shaped,bell-shaped,and periodic solutions.Finally,the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors.These conserved vectors play a crucial role in the study of physical quantities,such as the conservation of energy and momentum,and contribute to the understanding of the underlying physics of the system.
基金Supported by the Natural Science Foundation of China under Grant No. 10735030Ningbo Natural Science Foundation under Grant No. 2008A610017+3 种基金National Basic Research Program of China (973 Program 2007CB814800)Shanghai Leading Academic Discipline Project under Grant No. B412Program for Changjiang Scholars and Innovative Research Team in University (IRT0734)K.C. Wong Magna Fund in Ningbo University
文摘By Lie symmetry method, the Lie point symmetries and its Kac-Moody-Virasoro (KMV) symmetry algebra of (2+1)-dimensional dispersive long-wave equation (DLWE) are obtained, and the finite transformation of DLWE is given by symmetry group direct method, which can recover Lie point symmetries. Then KMV symmetry algebra of DLWE with arbitrary order invariant is also obtained. On basis of this algebra the group invariant solutions and similarity reductions are also derived.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10371098 and 10447007the Natural Science Foundation of Shanxi Province of China under Grant No.2005A13
文摘This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admit certain types of AGCSs is derived. Some approximate invariant solutions to the resulting equations can also be obtained.
文摘This paper studies the asymptotic behavior of solutions for nonlinear RLCnetwoiks which have the following form The function p i,v,t),called tile mired potential function,call be used to construct Liapunov functions to prove the convergence of solutions under certain conditiolts. Under the assumption that every element value involving voltage source is asymptotically constallt, we establish four creteria for all solutiolls of such a system to converge to the set of equilibria of its limiting equations via LaSalle invariant principle.We also present two theorems on the existence of periodic solutions for periodically excited uonliltear circuits.This results generalize those of Brayton and Moser[1,2].
基金supported by the Natural science foundation of China(NSF),under grand number 11071159.
文摘For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10735030 and 90718041Shanghai Leading Academic Discipline Project under Grant No.B412+1 种基金Program for Changjiang Scholars and Innovative Research Team in University (IRT0734)K.C.Wong Magna Fund in Ningbo University
文摘By means of the classical symmetry method,we investigate two types of the(2+1)-dimensional nonlinearKlein-Gorden equation.For the wave equation,we give out its symmetry group analysis in detail.For the secondtype of the(2+1)-dimensional nonlinear Klein-Gorden equation,an optimal system of its one-dimensional subalgebrasis constructed and some corresponding two-dimensional symmetry reductions are obtained.
文摘We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous <span style="white-space:nowrap;">Monge-Ampère equation to first-order ODEs. Some classes of the invariant solutions are constructed.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10671156)the Program for New Century Excellent Talents in Universities (Grant No. NCET-04-0968)
文摘The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.
基金The project supported by National Natural Science Foundation of China under Grant No. 10447007, the China Postdoctoral Science Foundation, and the Natural Science Foundation of Shanxi Province under Grant No. 2005A13
文摘The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.
