Efficient computation of capillary–gravity generalised solitary waves

Dutykh, Denys; Clamond, Didier; Durán, Ángel

doi:10.1016/j.wavemoti.2016.04.007

Cited by 5 publications

(5 citation statements)

References 37 publications

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“…Notice that LMA can find exact solutions, in case that they exist, as it is the case of the results presented, e.g., in Ref. [23].…”

Section: Appendix: the Levenberg-marquardt Algorithmsupporting

confidence: 56%

“…With these methods, one can seek for local minima of the norm of F[u] instead of zeros of that function. In our problem, we have made use of the Levenberg-Marquardt algorithm (see Appendix A for more details), which has been successfully used for computing solitary gravity-capillary water waves [23], and established a tolerance of ||F[u]|| < 10 −3 with ||F[u]|| being the L 2 -norm of F[u]:…”

Section: Stationary Statesmentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

Cuevas–Maraver

Kevrekidis

Frantzeskakis

et al. 2018

Springer Tracts in Modern Physics

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The subject of PT -symmetry and its areas of application have been blossoming over the past decade. Here, we consider a nonlinear Schrödinger model with a complex potential that can be tuned controllably away from being PTsymmetric, as it might be the case in realistic applications. We utilize two parameters: the first one breaks PT -symmetry but retains a proportionality between the imaginary and the derivative of the real part of the potential; the second one, detunes from this latter proportionality. It is shown that the departure of the potential from the PT -symmetric form does not allow for the numerical identification of exact stationary solutions. Nevertheless, it is of crucial importance to consider the dynamical evolution of initial beam profiles. In that light, we define a suitable notion of optimization and find that even for non PT -symmetric cases, the beam dynamics, both in 1D and 2D -although prone to weak growth or decay-suggests that the optimized profiles do not change significantly under propagation for specific parameter regimes.

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“…Notice that LMA can find exact solutions, in case that they exist, as it is the case of the results presented, e.g., in Ref. [23].…”

Section: Appendix: the Levenberg-marquardt Algorithmsupporting

confidence: 56%

Section: Stationary Statesmentioning

confidence: 99%

Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

Cuevas–Maraver

Kevrekidis

Frantzeskakis

et al. 2018

Springer Tracts in Modern Physics

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show abstract

“…Thus, alternative methods must be used. First of all, we considered the Levenberg-Marquardt algorithm (LMA) which, for instance, has proved to be useful for finding capillary solitary waves [56]. The main drawback of such an algorithm (which is detailed in [40]) lies in the fact that it is an optimization method, and consequently, it looks for local minima of the residual which do not necessarily have to be zero.…”

Section: Existence and Propagation Properties Of Solitary Wavesmentioning

confidence: 99%

Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach

Kominis

Cuevas–Maraver

Kevrekidis

et al. 2019

Chaos, Solitons & Fractals

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The existence of stationary solitary waves in symmetric and non-symmetric complex potentials is studied by means of Melnikov's perturbation method. The latter provides analytical conditions for the existence of such waves that bifurcate from the homogeneous nonlinear modes of the system and are located at specific positions with respect to the underlying potential. It is shown that the necessary conditions for the existence of continuous families of stationary solitary waves, as they arise from Melnikov theory, provide general constraints for the real and imaginary part of the potential, that are not restricted to symmetry conditions or specific types of potentials. Direct simulations are used to compare numerical results with the analytical predictions, as well as to investigate the propagation dynamics of the solitary waves.

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“…Rienecker and Fenton [22] compared the results of wave speed squared by the Fourier method with different N values with those of Cokelet [16] and Vanden-Broeck and Schwartz [38] for a moderately long wave and examined the accuracy of the Fourier method with N = 16, finding that the approximation did not converge to a solution for the limiting wave; however, it was still accurate up to 97% as high as the largest. An efficient and fast computation was developed for pure solitary waves by Clamond and Dutykh [39] and Dutykh and Clamond [40] or for generalized solitary gravity-capillary water waves by Clamond et al [41] and Dutykh et al [42]. Zhong and Liao [43] used the homotopy analysis method (HAM) to perform a convergent computation for the limiting Stokes waves in an arbitrary water depth and solitary waves in extremely shallow water.…”

Section: Introductionmentioning

confidence: 99%

Accuracy Examination of the Fourier Series Approximation for Almost Limiting Gravity Waves on Deep Water

Chen,

Chang

2024

MCA

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A permanent gravity wave propagating on deep water is a classic mathematical problem. However, the Fourier series approximation (FSA) based on the physical plane was examined to be valid for almost waves at all depths. The accuracy of the FSA for almost-limiting gravity waves remains unevaluated, which is the purpose of this study. We calculate some physical properties of almost-limiting waves on deep water using the FSA and compare them with other studies on the complex plane. The comparison results show that the closer the wave is, the greater the difference. We find that the main reason for this difference is that the wave profile in the FSA retains an original implicit form and is not represented by Fourier series. Therefore, the kinematic and dynamic conditions of the free surface around the wave crest cannot be satisfied at the same time.

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Efficient computation of capillary–gravity generalised solitary waves

Cited by 5 publications

References 37 publications

Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach

Accuracy Examination of the Fourier Series Approximation for Almost Limiting Gravity Waves on Deep Water

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