Conformable mass, momentum, energy, and Galilean conservation laws of the conformable modified Korteweg-de Vries equation

Nan Wangyu; Qiao Xin; Yang Jiang; Nan Wangyu; Qiao Xin; Yang Jiang

doi:10.3934/math.20261291

AIMS Mathematics

2026, Volume 11, Issue 10: 32920-32944. doi: 10.3934/math.20261291

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Research article

Conformable mass, momentum, energy, and Galilean conservation laws of the conformable modified Korteweg-de Vries equation

1.
School of Mathematics and Statistics, Yili Normal University, 448 Jiefang Road, Yining, Xinjiang, 835000, China
2.
Institute of Applied Mathematics, Yili Normal University, 448 Jiefang Road, Yining, Xinjiang, 835000, China

Received: 04 July 2026 Revised: 03 September 2026 Accepted: 21 September 2026 Published: 10 October 2026
MSC : 35B06, 35Q53

We investigate the conservation laws of the conformable fractional modified Korteweg–de Vries (mKdV) equation, in which the temporal derivative of order $ \alpha $ and the spatial derivatives of orders $ \beta_0, \beta_1, \beta_2, \beta_3 $ are all taken in the conformable sense. Our main result provides a complete classification of all conservation laws of this equation whose characteristics have order at most $ 3 $. We show that the four most fundamental conservation laws of the classical mKdV equation, i.e., the mass, momentum, energy, and Galilean conservation laws, admit conformable counterparts, and we determine the precise conditions on the spatial orders $ \beta_0, \beta_1, \beta_2, \beta_3 $ under which each counterpart holds. We also establish that, under these conditions, the conformable conservation laws provide a complete classification in the same sense as the classical case. The conserved densities and fluxes are written out explicitly, and their physical interpretations as conformable mass, momentum, energy, and centre of momentum are discussed.
- conformable modified KdV equation,
- conservation law
Citation: Nan Wangyu, Qiao Xin, Yang Jiang. Conformable mass, momentum, energy, and Galilean conservation laws of the conformable modified Korteweg-de Vries equation[J]. AIMS Mathematics, 2026, 11(10): 32920-32944. doi: 10.3934/math.20261291

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Abstract

We investigate the conservation laws of the conformable fractional modified Korteweg–de Vries (mKdV) equation, in which the temporal derivative of order $ \alpha $ and the spatial derivatives of orders $ \beta_0, \beta_1, \beta_2, \beta_3 $ are all taken in the conformable sense. Our main result provides a complete classification of all conservation laws of this equation whose characteristics have order at most $ 3 $. We show that the four most fundamental conservation laws of the classical mKdV equation, i.e., the mass, momentum, energy, and Galilean conservation laws, admit conformable counterparts, and we determine the precise conditions on the spatial orders $ \beta_0, \beta_1, \beta_2, \beta_3 $ under which each counterpart holds. We also establish that, under these conditions, the conformable conservation laws provide a complete classification in the same sense as the classical case. The conserved densities and fluxes are written out explicitly, and their physical interpretations as conformable mass, momentum, energy, and centre of momentum are discussed.

References

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