How symmetries yield non-invertible mappings of linear partial differential equations

Bluman, George W.; Yüzbaşı, Zühal Küçükarslan

doi:10.1016/j.jmaa.2020.124354

Cited by 11 publications

(27 citation statements)

References 13 publications

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“…Here, applying the theory presented in §2 is, in a way, useless since the resulting nonlocally related system is nonlinear. Nevertheless, in [20], the authors showed the usefulness of the symmetry-based method when applied to linear PDEs. In particular, they showed how to determine systematically non-invertible mappings of linear PDEs to linear PDEs.…”

Section: Special Case Of Linear Pdesmentioning

confidence: 99%

“…Nevertheless, in [20], the authors showed the usefulness of the symmetry-based method when applied to linear PDEs. In particular, they showed how to determine systematically non-invertible mappings of linear PDEs to linear PDEs.…”

Section: Special Case Of Linear Pdesmentioning

confidence: 99%

“…In particular, they showed how to determine systematically non-invertible mappings of linear PDEs to linear PDEs. Following the method presented in [20] we focus on point symmetries of a linear PDE (3.1) of the form x_∗=x_and1emu∗=u+ϵffalse(x_false),} where L f ( x ) = 0, i.e. u = f ( x ) is a solution of the linear PDE (3.1).…”

Section: Special Case Of Linear Pdesmentioning

confidence: 99%

“…In §2, we present the DI method for obtaining nonlocally related systems for ODE systems, PDE systems with two independent variables and PDE systems with at least three independent variables. In §3, we review a most interesting application of the symmetry-based method (DI method) presented in [20] for obtaining nonlocally related linear PDE systems for a given linear PDE system. In §4, through examples, we show in detail how to obtain nonlocally related systems for each of the three situations presented in §2.…”

Section: Introductionmentioning

confidence: 99%

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Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

et al. 2021

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Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it is shown that the symmetry-based method is a differential invariant-based method. It is shown that this allows one to naturally extend the symmetry-based method to ordinary differential equation (ODE) systems and to PDE systems with at least three independent variables. In particular, we present the situations for ODE systems, PDE systems with two independent variables and PDE systems with three or more independent variables, separately, and show that these three situations are directly connected. Examples are exhibited for each of the three situations.

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Section: Special Case Of Linear Pdesmentioning

confidence: 99%

Section: Special Case Of Linear Pdesmentioning

confidence: 99%

Section: Special Case Of Linear Pdesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

et al. 2021

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“…In §3, we focus on the construction of nonlocally related linear systems for linear PDE systems. In particular, we review the symmetry-based method (DI method) presented in [9] for finding nonlocally related linear PDE systems for a given linear PDE system. In §4, we consider the situation for self-adjoint linear PDE systems.…”

Section: Introductionmentioning

confidence: 99%

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. II: Connections with the conservation law method

et al. 2021

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In this paper, we show direct connections between the conservation law (CL)-based method and the differential invariant (DI)-based method for obtaining nonlocally related systems and nonlocal symmetries for a given partial differential equation (PDE) system. For a PDE system with two independent variables, we show that the CL method is a special case for the DI method. For a PDE system with at least three independent variables, we show that the CL method, for a curl-type CL, is a special case for the DI method. We also consider the situation for a self-adjoint, i.e. variational, linear PDE system. Here, a solution of the linear PDE system yields a nonlocally related system for both approaches. In particular, the resulting nonlocally related systems need not be invertibly equivalent. Through an example, we show that three distinct nonlocally related systems can be obtained from an admitted point symmetry.

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Lie symmetry analysis, exact solutions, and conservation laws to multi-component nonlinear Schrödinger equations

Bai,

Liu,

2023

Nonlinear Dyn

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In this paper, the Lie symmetry group method is used to study the multi-component nonlinear Schrödinger (MNLS) equations in nonlinear mathematical physics. The Lie point symmetries of MNLS equations are obtained. By applying the obtained symmetries, the symmetry reductions and some explicit solutions to the MNLS equations are constructed. In addition, we also find the conservation laws of the MNLS equations using Ibragimov's method.

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How symmetries yield non-invertible mappings of linear partial differential equations

Cited by 11 publications

References 13 publications

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. II: Connections with the conservation law method

Lie symmetry analysis, exact solutions, and conservation laws to multi-component nonlinear Schrödinger equations

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