arXiv subjects

Saleh Baqer

Publications and source records attributed to Saleh Baqer.

9 recordsLinked to original sources

arXiv2026

The Extended KdV Equation: Augmented Lagrangian and Variational Solitary Waves with Applications to Dispersive Hydrodynamics

In this work, we extend the method of averaged Lagrangian to the study of the general second-order (non-conservative) extended Korteweg--de Vries equation, known as the eKdV equation. Building on the framework introduced in [18], we construct a master (augmented) Lagrangian, modeled on Luke's Lagrangian, that incorporates the governing constraints at the appropriate asymptotic orders via the method of Lagrange multipliers. Averaging the resulting Euler-Lagrange equations in the traveling wave setting yields the existence of a (single) solitary wave solution with a $\operatorname{sech}^2$ profile. Explicit second-order formulas are obtained for the height of the solitary wave, together with the solitary wave velocity and inverse width, in terms of a fixed amplitude parameter. A key feature of the derived expressions is their asymptotic reduction to the classical KdV results when the first-order terms are retained. To assess the robustness and utility of the variational solitonic solutions, the derived formulas are subsequently applied, via the dispersive shock equal amplitude approximation method, to estimate the height and velocity of the leading solitary wave edge of dispersive shock waves governed by the eKdV Riemann problem. Theoretical predictions for the relevant wave parameters in both the eKdV solitary wave and dispersive shock wave problems are compared with direct numerical simulations and found to be in strong agreement.

physics.flu-dyn↗

arXiv2026

A study of variational single solitary waves governed by the conservative-extended KdV equation with applications to shallow water dispersive shocks

The extended KdV equation is a nonlinear dispersive wave model that is asymptotically or variationally derived from the full dispersive Euler shallow water waves equations when gravity-capillary and higher order nonlinear effects are taken into account, under weakly nonlinear and long-wave approximations. This reduction introduces four additional terms beyond the classical KdV equation: a nonlinear term (quadratic nonlinearity), two nonlinear-dispersive terms, and a fully dispersive term (fifth order dispersion). In this paper, we employ a variational approach based on averaged Lagrangians to analyze the accuracy of single solitary wave solutions governed by a particular extended KdV equation where energy conservation is a key feature. Compared with solitary wave solutions previously obtained through higher order asymptotics and algebraic methods, the present variational solutions are notably simpler and more readily applicable to practical problems. The solitary wave solutions obtained through this method are then systematically compared with direct numerical simulations, and the corresponding results are critically discussed. We further demonstrate the applicability of these single solitary waves to problems in the field of non-convex dispersive hydrodynamics. These problems include shallow water classical undular bores, commonly known as dispersive shock waves, and non-classical (resonant) dispersive shocks which are additionally analyzed using the concept of Whitham shocks. Theoretical predictions show excellent agreement with numerical simulations.

nlin.PS↗

arXiv2025

On shallow water non-convex dispersive hydrodynamics: the extended KdV model

In this work, we investigate non-classical wavetrain formations, and particularly dispersive shock waves (DSWs), or undular bores, in systems exhibiting non-convex dispersion. Our prototypical model, which arises in shallow water wave theory, is the extended Korteweg-de Vries (eKdV) equation. The higher-order dispersive and nonlinear terms of the latter, lead to resonance between dispersive radiation and solitary waves, and notably, the individual waves comprising DSWs, due to non-convex dispersion. This resonance manifests as a resonant wavetrain propagating ahead of the dispersive shock wave. We present a succinct overview of the fundamental principles and characteristics of DSWs and explore analytical methods for their analysis. Wherever applicable, we demonstrate these concepts and techniques using both the classical KdV equation and its higher-order eKdV counterpart. We extend the application of the dispersive shock fitting method and the equal amplitude approximation to investigate radiating DSWs governed by the eKdV equation. We also introduce Whitham shock solutions for the regime of traveling DSWs of the eKdV model. Theoretical predictions are subsequently validated against direct numerical solutions, revealing a high degree of agreement.

nlin.PS↗

arXiv2024

Solitons, dispersive shock waves and Noel Fredrick Smyth

Noel Frederick Smyth (NFS), a Fellow of the Australian Mathematical Society and a Professor of Nonlinear Waves in the School of Mathematics at the University of Edinburgh, passed away on February 5, 2023. NFS was a prominent figure among applied mathematicians who worked on nonlinear wave theory in a broad range of areas. Throughout his academic career, which spanned nearly forty years, NFS developed mathematical models, ideas, and techniques that have had a large impact on the understanding of wave motion in diverse media. His major research emphasis primarily involved the propagation of solitary waves, or solitons, and dispersive shock waves, or undular bores, in various media, including optical fibers, liquid crystals, shallow waters and atmosphere. Several approaches he developed have proven effective in analyzing the dynamics and modulations of related wave phenomena. This tribute in the journal of Wave Motion aims to provide a brief biographical sketch of NFS, discuss his major research achievements, showcase his scientific competence, untiring mentorship and unwavering dedication, as well as share final thoughts from his former students, colleagues, friends, and family. The authors had a special connection with NFS on both on personal and professional levels and hold deep gratitude for him and his invaluable work. In recognition of his achievements in applied mathematics, Wave Motion hosts a Special Issue entitled "Modelling Nonlinear Wave Phenomena: From Theory to Applications," which presents the recent advancements in this field.

physics.hist-ph↗

arXiv2023

Whitham Shocks and Resonant Dispersive Shock Waves Governed by the Higher Order Korteweg-de Vries Equation

The addition of higher order asymptotic corrections to the Korteweg-de Vries equation results in the extended Korteweg-de Vries equation. These higher order terms destabilise the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and result in the emission of resonant radiation. In broad terms, there are three possible dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dispersive shock wave (CDSW) and travelling dispersive shock wave (TDSW). While there are existing solutions for the RDSW and TDSW regimes obtained using modulation theory, there is no existing solution for the CDSW regime. Modulation theory and the associated concept of a Whitham shock are used to obtain this CDSW solution. In addition, it is found that the resonant wavetrain emitted by the extended Korteweg-de Vries equation with water wave coefficients has a minimal amplitude. This minimal amplitude is explained based on the developed Whitham modulation theory.

nlin.PS↗

arXiv2021

Nematic dispersive shock waves from nonlocal to local

The structure of optical dispersive shock waves in nematic liquid crystals is investigated as the power of the optical beam is varied, with six regimes identified, which complements previous work pertinent to low power beams only. It is found that the dispersive shock wave structure depends critically on the input beam power. In addition, it is known that nematic dispersive shock waves are resonant and the structure of this resonant is also critically dependent on the beam power. Whitham modulation theory is used to find solutions for the six regimes with the existence intervals for each identified. These dispersive shock wave solutions are compared with full numerical solutions of the nematic equations and excellent agreement is found.

nlin.PS↗

arXiv2020

Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals

A full analysis of all regimes for optical dispersive shock wave (DSW) propagation in nematic liquid crystals is undertaken. These dispersive shock waves are generated from step initial conditions for the optical field and are resonant in that linear diffractive waves are in resonance with the DSW, resulting in a resonant linear wavetrain propagating ahead of it. It is found that there are six regimes, which are distinct and require different solution methods. In previous studies, the same solution method was used for all regimes, which does not yield solutions in full agreement with numerical solutions. Indeed, the standard DSW structure disappears for sufficiently large initial jumps. Asymptotic theory, approximate methods or Whitham modulation theory are used to find solutions for these resonant dispersive shock waves in a given regime. The solutions are found to be in excellent agreement with numerical solutions of the nematic equations in all regimes. It is found that for small initial jumps, the resonant wavetrain is unstable, but that it stabilises above a critical jump height. It is additionally found that the DSW is unstable, except for small jump heights for which there is no resonance and large jump heights for which there is no standard DSW structure.

nlin.PS↗

arXiv2017

Fractional Schrödinger Equation with Zero and Linear Potentials

This paper is about the fractional Schrödinger equation expressed in terms of the Caputo time-fractional and quantum Riesz-Feller space fractional derivatives for particle moving in a potential field. The cases of free particle (zero potential) and a linear potential are considered. For free particle, the solution is obtained in terms of the Fox $H$-function. For the case of a linear potential, the separation of variables method allows the fractional Schrödinger equation to be split into space fractional and time fractional ones. By using the Fourier and Mellin transforms for the space equation and the contour integrals technique for the time equation, the solutions are obtained also in terms of the Fox $H$-function. Moreover, some recent results related to the time fractional equation have been revised and reconsidered. The results obtained in this paper contain as particular cases already known results for the fractional Schrödinger equation in terms of the quantum Riesz space-fractional derivative and standard Laplace operator.

math-ph↗

arXiv2017

On The Space-Time Fractional Schrödinger Equation with time independent potentials

This paper is about the fractional Schrödinger equation (FSE) expressed in terms of the quantum Riesz-Feller space fractional and the Caputo time fractional derivatives. The main focus is on the case of time independent potential fields as a Dirac-delta potential and a linear potential. For such type of potential fields the separation of variables method allows to split the FSE into space fractional equation and time fractional one. The results obtained in this paper contain as particular cases already known results for FSE in terms of the quantum Riesz space fractional derivative and standard Laplace operator.

math-ph↗