High-Order Noether’s Transformation and New Densities for BBM Equation via Contact Symmetries
محورهای موضوعی : فصلنامه ریاضی
Mehdi Jafari
1
,
Mojtaba Farazi
2
1 - Payame Noor University (PNU), Department of Mathematics, P.O. Box 19395-4697, Tehran, Iran.
2 - Payame Noor University (PNU), Department of Mathematics, P.O. Box 19395-4697, Tehran, Iran.
کلید واژه: BBM equation, Generalized symmetries, Multiplier method, Contact symmetries, Conservation laws.,
چکیده مقاله :
Noether's first theorem shows how symmetry groups of one-parameter transformations lead to the generation of conservation laws for the Euler-Lagrange equations. She states in her second theorem that there is a relationship between Euler's basic equation and Lagrange's basic equation. This one-to-one correspondence leads to a type of symmetry called generalized symmetry. According to these materials, in this paper, we want to obtain these types of symmetries for the Benjamin-Bona-Mahony (BBM) equation and show that each symmetry is connected to a specific conservation law. For this purpose, we obtain the symmetries of this equation using the Lie symmetry method, and then using the adjoint operator, we provide a classification on the group invariant solutions of this equation. Then, by applying Noether's method, we obtain a new conservation law for each symmetry.
Noether's first theorem shows how symmetry groups of one-parameter transformations lead to the generation of conservation laws for the Euler-Lagrange equations. She states in her second theorem that there is a relationship between Euler's basic equation and Lagrange's basic equation. This one-to-one correspondence leads to a type of symmetry called generalized symmetry. According to these materials, in this paper, we want to obtain these types of symmetries for the Benjamin-Bona-Mahony (BBM) equation and show that each symmetry is connected to a specific conservation law. For this purpose, we obtain the symmetries of this equation using the Lie symmetry method, and then using the adjoint operator, we provide a classification on the group invariant solutions of this equation. Then, by applying Noether's method, we obtain a new conservation law for each symmetry.
