Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model

Ehrnström, Mats; Johnson, Mathew A.; Claassen, Kyle M.

doi:10.1007/s00205-018-1306-5

Cited by 32 publications

(83 citation statements)

References 39 publications

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“…When continuing from constant solutions, i.e., solutions with height zero, the breakdown of smoothness occurs when

\frac{3}{2} φ^{2} - 3 c φ + c^{2} = 0

or, more precisely, when the bifurcation branch intersects the curve

φ = c (1 - \frac{1}{\sqrt{3}})

. Thus, we expect (as is shown in ) that periodic solutions of along the bifurcation branch with fixed period will have a maximum amplitude of

max φ false(x false) = γ : = c (1 - \frac{1}{\sqrt{3}}) .

See for details of the above arguments.…”

Section: Numerical Resultsmentioning

confidence: 89%

“…We begin by studying the bifurcation of

2 π / κ

‐periodic traveling wave solutions of bifurcating from the trivial state

φ = 0

. To this end, using the profile equation in combination with the following local bifurcation formulas (see [, proposition 5.1])

\begin{matrix} true \\ right φ false(x; ε false) & left : = ε prefixcos (κ x) + \frac{3 c_{κ} ε^{2}}{4} (() \frac{1}{c_{κ}^{2} - 1} + \frac{cos (2 κ x)}{c_{κ}^{2} - c_{2 κ}^{2}}) + scriptO (ε^{3}) \end{matrix}

\begin{matrix} true \\ right c false(ε false) & left : = c_{κ} + \frac{3 ε^{2}}{8} ([] - \frac{1}{2 c_{κ}} + 3 c_{κ} (\frac{1}{c_{κ}^{2} - 1} + \frac{1}{2 (() c_{κ}^{2} - c_{2 κ}^{2})})) + scriptO (ε^{4}), \\ right c_{ℓ} & left : = \sqrt{\frac{tanh (ℓ)}{ℓ}}, \end{matrix}

we apply the pseudo‐arclength method as discussed in Section 2.2 to compute approximations

φ_{N} false(x_{i} false)

of the profile

φ (x_{i})

at the collocation points

x_{i} = \frac{false(2 i - 1 false) π}{κ N}

, continuing while

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…A global bifurcation plot of waveheight versus wavespeed for various wavenumbers κ is shown in Fig. (b), where we define

waveheight : = \underset{x \in [0, π / κ]}{trueprefixmax} φ false(x false) - \underset{x \in [0, π / κ]}{trueprefixmin} φ false(x false) = φ false(0 false) - φ false(π / κ false),

which is well defined due to the monotonicity properties of even solutions along the global bifurcation branch as demonstrated in .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…(a), the crest at the maximum of the smooth profiles along the global bifurcation branch becomes sharper as the waveheight increases, appearing to converge to a nontrivial profile with a singularity at the origin. The existence and qualitative properties of this highest singular wave have been studied analytically in , where it was shown the highest wave has a logarithmic cusp of order

| x ln | x | |

near the top of the bifurcation curve: see Fig. (b).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Though this result does not prove that the system is ill‐posed for initial data having negative infimum, numerical evidence suggests that this is indeed the case: see Section 3.1.2 for further details. Furthermore, in , the existence of 2π‐periodic traveling wave solutions of was constructed, and it was proven that there exists a highest wave with a logarithimically cusped singularity. Concerning the stability of wavetrains, it has recently been shown that there exists a critical wavenumber

κ_{c} \approx 1.008

such that all

2 π / κ

‐periodic wavetrains of of sufficiently small amplitude are modulationally unstable provided that

κ > κ_{c}

: see .…”

Section: Introductionmentioning

confidence: 99%

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Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

Claassen

Johnson

2018

Stud Appl Math

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We consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combines the full two‐way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity, providing nonlocal model equations that may be expected to exhibit some of the interesting high‐frequency phenomena present in the Euler equations that standard “long‐wave” theories fail to capture. Of particular interest here is the existence and stability of periodic traveling wave solutions in such models. Using numerical bifurcation techniques, we construct global bifurcation diagrams for each system and compare the global structure of branches, together with the possibility of bifurcation branches terminating in a “highest” singular (peaked/cusped) wave. We also numerically approximate the stability spectrum along these bifurcation branches and compare the stability predictions of these models. Our results confirm a number of analytical results concerning the stability of asymptotically small waves in these models and provide new insights into the existence and stability of large amplitude waves.

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“…When continuing from constant solutions, i.e., solutions with height zero, the breakdown of smoothness occurs when

\frac{3}{2} φ^{2} - 3 c φ + c^{2} = 0

or, more precisely, when the bifurcation branch intersects the curve

φ = c (1 - \frac{1}{\sqrt{3}})

. Thus, we expect (as is shown in ) that periodic solutions of along the bifurcation branch with fixed period will have a maximum amplitude of

max φ false(x false) = γ : = c (1 - \frac{1}{\sqrt{3}}) .

See for details of the above arguments.…”

Section: Numerical Resultsmentioning

confidence: 89%

“…We begin by studying the bifurcation of

2 π / κ

‐periodic traveling wave solutions of bifurcating from the trivial state

φ = 0

. To this end, using the profile equation in combination with the following local bifurcation formulas (see [, proposition 5.1])

\begin{matrix} true \\ right φ false(x; ε false) & left : = ε prefixcos (κ x) + \frac{3 c_{κ} ε^{2}}{4} (() \frac{1}{c_{κ}^{2} - 1} + \frac{cos (2 κ x)}{c_{κ}^{2} - c_{2 κ}^{2}}) + scriptO (ε^{3}) \end{matrix}

\begin{matrix} true \\ right c false(ε false) & left : = c_{κ} + \frac{3 ε^{2}}{8} ([] - \frac{1}{2 c_{κ}} + 3 c_{κ} (\frac{1}{c_{κ}^{2} - 1} + \frac{1}{2 (() c_{κ}^{2} - c_{2 κ}^{2})})) + scriptO (ε^{4}), \\ right c_{ℓ} & left : = \sqrt{\frac{tanh (ℓ)}{ℓ}}, \end{matrix}

we apply the pseudo‐arclength method as discussed in Section 2.2 to compute approximations

φ_{N} false(x_{i} false)

of the profile

φ (x_{i})

at the collocation points

x_{i} = \frac{false(2 i - 1 false) π}{κ N}

, continuing while

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…A global bifurcation plot of waveheight versus wavespeed for various wavenumbers κ is shown in Fig. (b), where we define

waveheight : = \underset{x \in [0, π / κ]}{trueprefixmax} φ false(x false) - \underset{x \in [0, π / κ]}{trueprefixmin} φ false(x false) = φ false(0 false) - φ false(π / κ false),

which is well defined due to the monotonicity properties of even solutions along the global bifurcation branch as demonstrated in .…”

Section: Numerical Resultsmentioning

confidence: 99%

| x ln | x | |

near the top of the bifurcation curve: see Fig. (b).…”

Section: Numerical Resultsmentioning

confidence: 99%

κ_{c} \approx 1.008

such that all

2 π / κ

‐periodic wavetrains of of sufficiently small amplitude are modulationally unstable provided that

κ > κ_{c}

: see .…”

Section: Introductionmentioning

confidence: 99%

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Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

Claassen

Johnson

2018

Stud Appl Math

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

Pei

2022

Math Methods in App Sciences

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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.

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Solitary wave solutions to a class of Whitham–Boussinesq systems

Nilsson

Wang

2019

Z. Angew. Math. Phys.

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In this note we study solitary wave solutions of a class of Whitham-Boussinesq systems which includes the bi-directional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnström, Groves and Wahlén [10]. In that paper the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni [3], Groves and Wahlén [15], together with the concentration-compactness principle.2010 Mathematics Subject Classification. 76B15; 76B25, 35S30, 35A20.

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Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model

Cited by 32 publications

References 39 publications

Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

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

Solitary wave solutions to a class of Whitham–Boussinesq systems

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