Periodic Hölder waves in a class of negative-order dispersive equations

Hildrum, Fredrik; Xue, Jun

doi:10.1016/j.jde.2022.10.023

Cited by 7 publications

(12 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Smooth periodic solutions exist for c ∈ (0, c * ) where dispersion is absent, that is β = 0, we prove that small amplitude solutions are still smooth, while large amplitude solutions may exhibit singularities in the form of peaks or cusps. This result can be seen in the frame of the often observed dependence of the regularity of a traveling solution on its amplitude and propagation speed (see [14,25,5,42] for uni-directional models and [12] for bidirectional model). Eventually, we investigate the symmetry of periodic traveling wave solutions and apply our results to the Gardner-Ostrovsky equation, which is (1.1) with nonlinearity (1.2).…”

Section: Introductionmentioning

confidence: 66%

See 1 more Smart Citation

Existence, regularity and symmetry of periodic traveling waves for Gardner–Ostrovsky type equations

Bruell,

Pei

2023

Monatsh Math

View full text Add to dashboard Cite

We study the existence, regularity, and symmetry of periodic traveling solutions to a class of Gardner-Ostrovsky type equations, including the classical Gardner-Ostrovsky equation, the (modified) Ostrovsky, and the reduced (modified) Ostrovsky equation. The modified Ostrovsky equation is also known as the short pulse equation. The Garder-Ostrovsky equation is a model for internal ocean waves of large amplitude.We prove the existence of nontrivial, periodic traveling wave solutions using local bifurcation theory, where the wave speed serves as the bifurcation parameter. Moreover, we give a regularity analysis for periodic traveling solutions in the presence as well as absence of the Boussinesq dispersion. We see that the presence of Boussinesq dispersion implies smoothness of periodic traveling wave solutions, while its absence may lead to singularities in the form of peaks or cusps. Eventually, we study the symmetry of periodic traveling solutions by the method of moving planes. A novel feature of the symmetry results in the absence of Boussinesq dispersion is that we do not need to impose a traditional monotonicity condition or a recently developed reflection criterion on the wave profiles to prove the statement on the symmetry of periodic traveling waves.

show abstract

Section: Introductionmentioning

confidence: 66%

“…(a) The proof is analog to the proof of Lemma 3.3 in [25] which in turn is based on [14,12], and the analysis of F follows from Lemma 4.1 in [12]. We include details here for convenience and completeness.…”

Section: Regularity Of Periodic Traveling Wavesmentioning

confidence: 99%

Existence, regularity and symmetry of periodic traveling waves for Gardner–Ostrovsky type equations

Bruell,

Pei

2023

Monatsh Math

View full text Add to dashboard Cite

show abstract

“…A new idea would therefore be required to proceed past this value. Highest Hölder and Lipschitz waves have been constructed in a number of settings [2,4,6,20,22,26,37], and we expect a similar approach to go through for many such equations.…”

Section: Homogeneous Singularity (Whitham)mentioning

confidence: 99%

On the precise cusped behaviour of extreme solutions to Whitham-type equations

Ehrnström¹,

Mæhlen²,

Varholm³

2023

Preprint

View full text Add to dashboard Cite

We prove exact leading-order asymptotic behaviour at the origin for nontrivial solutions of two families of nonlocal equations. The equations investigated include those satisfied by the cusped highest steady waves for both the uni-and bidirectional Whitham equations. The problem is therefore analogous to that of capturing the 120°interior angle at the crests of classical Stokes' waves of greatest height. In particular, our results partially settle conjectures for such extreme waves posed in a series of recent papers [14,15,34]. Our methods may be generalised to solutions of other nonlocal equations, and can moreover be used to determine asymptotic behaviour of their derivatives to any order.

show abstract

“…However, the right‐hand side of () tends to a strictly negative value, which then leads to a contradiction. For general

p

‐nonlinearity with the form

{\left&amp;amp;#x0007C;u\right&amp;amp;#x0007C;}&amp;amp;#x0005E;p

u{\left&amp;amp;#x0007C;u\right&amp;amp;#x0007C;}&amp;amp;#x0005E;{p-1},p&amp;gt;1

, a more complicated analysis on the precise Hölder regularity for

\phi

at those nonsmooth crests or toughs can be found in Hildrum and Jun 27 …”

Section: Symmetry Of Periodic Traveling Wavesmentioning

confidence: 99%

“…Solutions belonging to (iii) do exist in some dispersive equations. In fact, a period traveling wave with both a singular crest and a singular trough simultaneously per period has been found in Hildrum and Jun 27 when the nonlinearity has a power of odd number. It is worth mentioning that the wave profile in case (ii) can have arbitrary type singularities, while the singularity type in case (iii) is narrower but still includes the usual peak, cusp, and stump type singularities.…”

Section: Introductionmentioning

confidence: 99%

On the regularity and symmetry of periodic traveling solutions to weakly dispersive equations with cubic nonlinearity

Pei

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

We consider the a priori regularity and symmetry of periodic traveling wave solutions to a class of weakly dispersive equations with cubic nonlinearity. We prove that any continuous periodic traveling wave solution of small amplitude is symmetric if it satisfies the reflection criterion proposed recently by Bruell and Pei. Moreover, we confirm that continuous periodic traveling waves with singular crests or troughs, which correspond to large‐amplitude waves, are also symmetric if some cancellation structure appears near the singular points or if the wave profile has only a single singular point per period. Included in the results as particular cases are periodic traveling waves with peak, cusp, or stump type singular structures. The argument and results also apply to dispersive equations with more general nonlinearity which is non‐smooth.

show abstract

Periodic Hölder waves in a class of negative-order dispersive equations

Cited by 7 publications

References 23 publications

Existence, regularity and symmetry of periodic traveling waves for Gardner–Ostrovsky type equations

Existence, regularity and symmetry of periodic traveling waves for Gardner–Ostrovsky type equations

On the precise cusped behaviour of extreme solutions to Whitham-type equations

On the regularity and symmetry of periodic traveling solutions to weakly dispersive equations with cubic nonlinearity

Contact Info

Product

Resources

About