Serre-type Equations in Deep Water

Dutykh, Denys; Clamond, Didier; Chhay, Marx

doi:10.1051/mmnp/201712103

Cited by 3 publications

(4 citation statements)

References 48 publications

(55 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Hamiltonian form of the fourth-order NLSE derived in this paper does not contain the non-Hamiltonian term that violates the conservation of energy. With the development of structure-preserving integrators, the conserved Hamiltonian (energy) makes such PDEs more attractive from the computational point of view [3,7,15,20]. Further reading on the numerical integration of high-order NLSEs can be found amongst others in Refs.…”

Section: Discussionmentioning

confidence: 99%

“…[12]. The same expansion can be written for the operator √ ω in formula (15) for the Hamiltonian density…”

Section: Operator Expansions For the Slow Envelopementioning

confidence: 97%

“…The Hamiltonian density H is a sum of two parts, H 2 and H int (see formula ( 9)). The Hamiltonian density H 2 can be calculated by formula (15) with taking into account relation (19) and rule (21) applied for the operator √ ω, namely…”

Section: Hamilton's Equation For the Amplitude Umentioning

confidence: 99%

See 2 more Smart Citations

Hamiltonian form of an Extended Nonlinear Schrödinger Equation for Modelling the Wave field in a System with Quadratic and Cubic Nonlinearities

Sedletsky¹,

Gandzha²

2022

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

We derive a Hamiltonian form of the fourth-order (extended) nonlinear Schrödinger equation (NLSE) in a nonlinear Klein-Gordon model with quadratic and cubic nonlinearities. This equation describes the propagation of the envelope of slowly modulated wave packets approximated by a superposition of the fundamental, second, and zeroth harmonics. Although extended NLSEs are not generally Hamiltonian PDEs, the equation derived here is a Hamiltonian PDE that preserves the Hamiltonian structure of the original nonlinear Klein-Gordon equation. This could be achieved by expressing the fundamental harmonic and its first derivative in symplectic form, with the second and zeroth harmonics calculated from the variational principle. We demonstrate that the non-Hamiltonian form of the extended NLSE under discussion can be retrieved by a simple transformation of variables.

show abstract

Section: Discussionmentioning

confidence: 99%

“…[12]. The same expansion can be written for the operator √ ω in formula (15) for the Hamiltonian density…”

Section: Operator Expansions For the Slow Envelopementioning

confidence: 97%

See 1 more Smart Citation

Hamiltonian form of an Extended Nonlinear Schrödinger Equation for Modelling the Wave field in a System with Quadratic and Cubic Nonlinearities

Sedletsky¹,

Gandzha²

2022

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

show abstract

“…Indeed, a large class of models for water waves inherits a Hamiltonian structure of infinite dimension. Thus a multisymplectic structure can be exhibited, for example, for Serre-type equations in deep water configuration [172], and for the Serre-Green-Naghdi equations in shallow water configuration [173]. Therefore multisymplectic schemes appear as natural structure preserving integrators applied to these models.…”

Section: Multisymplectic Integratorsmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

View full text Add to dashboard Cite

Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

show abstract