Conservation laws and symmetries of time-dependent generalized KdV equations

Anco, Stephen C.; Rosa, María; Gandarias, M. L.

doi:10.3934/dcdss.2018035

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…For the KdV equation, there are three such local conserved quantities: where c 1 and c 2 are often identified as “momentum” and “energy”, respectively 3 . These local conserved quantities also have direct analogues in generalized KdV-type equations, hinting at their robustness 30 .…”

Section: Resultsmentioning

confidence: 97%

Discovering conservation laws using optimal transport and manifold learning

Dangovski

Soljačić

2023

Nat Commun

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Conservation laws are key theoretical and practical tools for understanding, characterizing, and modeling nonlinear dynamical systems. However, for many complex systems, the corresponding conserved quantities are difficult to identify, making it hard to analyze their dynamics and build stable predictive models. Current approaches for discovering conservation laws often depend on detailed dynamical information or rely on black box parametric deep learning methods. We instead reformulate this task as a manifold learning problem and propose a non-parametric approach for discovering conserved quantities. We test this new approach on a variety of physical systems and demonstrate that our method is able to both identify the number of conserved quantities and extract their values. Using tools from optimal transport theory and manifold learning, our proposed method provides a direct geometric approach to identifying conservation laws that is both robust and interpretable without requiring an explicit model of the system nor accurate time information.

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Section: Resultsmentioning

confidence: 97%

Discovering conservation laws using optimal transport and manifold learning

Dangovski

Soljačić

2023

Nat Commun

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“…has conservation law Example 2 (Time-dependent evolution). The generalized KdV equation [3] u t + f (t, u)u x + u xxx = 0, (4.22) where f (t, u) = at −1/3 u + bu + cu 2 for a, b, c constant, has explicit time dependence if a = 0. It has a conservation law D t T + D x X= 0 with conserved density…”

Section: Critical Points and Symmetriesmentioning

confidence: 99%

“…and characteristic (note that [3] has a typo in the u xx term) which solves (4.22) for c = 0. Suppose now that c = 0.…”

Section: Critical Points and Symmetriesmentioning

confidence: 99%

Quasi-Noether Systems and Quasi-Lagrangians

Shankar

2019

Symmetry

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We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same conservation laws as Lagrange (Green-Lagrange) identity. We discuss quasi-Noether systems, and some of their properties, and generate classes of quasi-Noether differential equations of the second order. We next introduce a more general version of quasi-Lagrangians which allows us to extend Noether theorem. Here, variational symmetries are only sub-symmetries, not true symmetries. We finally introduce the critical point condition for evolution equations with a conserved integral, demonstrate examples of its compatibility, and compare the invariant submanifolds of quasi-Lagrangian systems with those of Hamiltonian systems.be the i-th total derivative, 1 ≤ i ≤ p, the sum extending over all (unordered) multi-indices J = (j 1 , j 2 , ..., j k ) for k ≥ 0 and 1 ≤ j k ≤ p. Two conservation laws K andK are equivalent if they differ by a trivial conservation law [11]. A conservation law D i P i . = 0 is trivial if a linear combination of two kinds of triviality

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“…This method called the multiplier method allows finding all local conservation laws admitted by any evolution equation. Many papers have been published in the last few years using this method [11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation

Márquez

Bruzón

2019

Symmetry

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In this paper, we study a generalization of the well-known Kelvin-Voigt viscoelasticity equation describing the mechanical behaviour of viscoelasticity. We perform a Lie symmetry analysis. Hence, we obtain the Lie point symmetries of the equation, allowing us to transform the partial differential equation into an ordinary differential equation by using the symmetry reductions. Furthermore, we determine the conservation laws of this equation by applying the multiplier method.

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Conservation laws and symmetries of time-dependent generalized KdV equations

Cited by 3 publications

References 23 publications

Discovering conservation laws using optimal transport and manifold learning

Discovering conservation laws using optimal transport and manifold learning

Quasi-Noether Systems and Quasi-Lagrangians

Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation

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