The Shallow-Water Models with Cubic Nonlinearity

Spectral and linear stability of peakons in the Novikov equation

“…Proof. Proving ( 7) is done by applying 𝐿 given in (6), on 𝜙 and 𝜙 ′ and using the facts that 𝜙 ′2 = 𝜙 2 , 𝜙 ′′ = 𝜙 − 2𝛿 0 , and (1 − 𝜙 2 )𝛿 0 = 0, with 𝛿 0 being the Dirac delta distribution centered at 𝜉 = 0. □…”

“…We multiply both sides of the resolvent equation ( 16) by v and integrate over ℝ. Using the definition of 𝐿 given in (6), one finds…”

“…24 Finally, the Novikov equation was derived in the modeling of the propagation of shallow water waves of "moderately large amplitude." 6 The one-peakon solutions take the form 𝑢(𝑥, 𝑡) = √ 𝑐 𝑒 −|𝑥−𝑐𝑡| 𝑥 ∈ ℝ 𝑐 > 0.…”

“…The Novikov equation also admits smooth multisoliton solutions 24 . Finally, the Novikov equation was derived in the modeling of the propagation of shallow water waves of “moderately large amplitude.” 6 …”

Orbital Stability of Smooth Solitary Waves for the Novikov Equation

“…The equation (1.1) is known to be completely integrable via the Inverse Scattering Transform, has infinitely many symmetries and conserved quantities, and is bi-Hamiltonian [26,40]. Further, the Novikov equation has been shown to model the propagation of shallow water waves of moderately large amplitude [11]. In this regard, (1.1) can be regarded as generalization of the Camassa-Holm equation [8,9] (1.2)…”