Modified Korteweg-de Vries Equation Research Articles

Multiple and Singular Soliton Solutions for Space–Time Fractional Coupled Modified Korteweg–De Vries Equations

The focus of this paper is on the nonlinear coupled evolution equations, specifically within the context of the fractional coupled modified Korteweg–de Vries (mKdV) equation, employing the conformable fractional derivative (CFD) approach. The primary objective of this paper is to thoroughly investigate the applicability of the Hirota bilinear method for deriving analytical solutions to the fractional mKdV equations. A range of exact analytical solutions for the fractional coupled mKdV equations is obtained. The findings in general indicate that the Hirota bilinear method is an effective approach for resolving the complexities associated with the fractional coupled mKdV equations.

Homotopy Analysis with Shehu Transform Method for Time-Fractional Modified KdV Equation in Dusty Plasma

Homotopy Analysis with Shehu Transform Method for Time-Fractional Modified KdV Equation in Dusty Plasma

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries (KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.

MKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere

We investigate geometric evolution equations for Legendrian curves in the 3-sphere which are invariant under the action of the unitary group ${\rm U}(2)$. We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, are realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions which evolve by rigid motions in ${\rm U}(2)$. Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.

The focusing coupled modified Korteweg–de Vries equation with nonzero boundary conditions: the Riemann–Hilbert problem and soliton classification

The focusing coupled modified Korteweg–de Vries equation with nonzero boundary conditions: the Riemann–Hilbert problem and soliton classification

تخامد لانداو للأمواج السوليتونية الصوتية الغبارية في بلازما غبارية غير شاملة

تم دراسة الأمواج السوليتونية الصوتية الغبارية تحت تأثير تخامد لانداو للبلازما الغبارية الفضائية تحوي أيونات خاضعة لتوزع سرعة غير شامل. قمنا بالتحقيق في تأثير تخامد لانداو والبارمتر غير الشامل على البنى السوليتونية الصوتية الغبارية من خلال الحل العددي لمعادلة كورديفيك- دي فريس المعدلة بحد تخامد لانداو. باستخدام طريقة الاضطراب الاختزالية قمنا باستنتاج معادلة كورتيفيك – دي فريس مع إضافة حد تخامد لانداو. أظهرت هذه الدراسة أنً كثافة حبيبات الغبار تلعب دوراَ أساسياً في ظهور أو عدم ظهور الأمواج السوليتونية DA في البلازما الغبارية، والتأثير الواضح للخاصية غير الشاملة للأيونات على ظاهرة تخامد لانداو للبنى اللاخطية المتشكلة في البلازما الغبارية. تفيد نتائج هذه الدراسة في محاولة فهم الآلية الفيزيائية للانتشار الأمواج السوليتونية تحت تأثير تخامد لانداو في بيئات البلازما المغبرة غير الشاملة كالمناطق الكوكبية، والأغلفة المغناطيسية، وبيئات البلازما الفضائية.

Four-component combined integrable equations possessing bi-Hamiltonian formulations

This paper aims to generate a Liouville integrable Hamiltonian hierarchy by introducing a specific matrix eigenvalue problem with four components. The adopted approach is the zero curvature formulation. A bi-Hamiltonian formulation is furnished through applying the trace identity, which shows the Liouville integrability of the resulting hierarchy. Two examples of generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations are presented.

Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation

Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation

A Generalized Hierarchy of Combined Integrable Bi-Hamiltonian Equations from a Specific Fourth-Order Matrix Spectral Problem

The aim of this paper is to analyze a specific fourth-order matrix spectral problem involving four potentials and two free nonzero parameters and construct an associated integrable hierarchy of bi-Hamiltonian equations within the zero curvature formulation. A hereditary recursion operator is explicitly computed, and the corresponding bi-Hamiltonian formulation is established by the so-called trace identity, showing the Liouville integrability of the obtained hierarchy. Two illustrative examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations with four components and two adjustable parameters.

Evolution of dispersive shock waves to the complex modified Korteweg–de Vries equation with higher-order effects

In this paper, new dispersive shock waves (DSWs) in step-like initial value problems to the complex modified Korteweg–de Vries (cmKdV) equation with higher-order effects are found via Whitham modulation theory. For the aforementioned equation, the 1-genus and 2-genus periodic solutions and the associated Whitham equations which are used to describe DSWs are firstly given by the finite-gap integration method, and we also analyze nine types of rarefaction waves appearing before DSWs under the 0-genus Whitham equations. Subsequently, the DSW solutions with step-like initial data are discussed, where we acquire some DSW structures that have not been previously proposed. These notable new results include 1-genus DSW satisfying that one Riemann invariant is constant and the other three are variables and 2-genus DSW in the DSW solutions with one step-like initial data, as well as 3-genus DSW resulting from the collision to 1-genus and 2-genus or two 2-genus DSWs propagating toward each other in the possible DSW solutions with two step-like initial data. Ultimately, the dam break problem is explored to demonstrate the significant physical application of the theoretical findings.

Soliton solutions of a novel nonlocal Hirota system and a nonlocal complex modified Korteweg–de Vries equation

This paper proposes a novel nonlocal Hirota system by adding constraints to the coupled Hirota equation, which can be transformed into a nonlocal complex modified Korteweg–de Vries (mKdV) equation with a Galilean transformation. Since the nonlocal Hirota equation and nonlocal complex mKdV equation can be converted into each other, we only need to solve one of them. Starting from the eigenfunctions of the Lax pair and the adjoint Lax pair of the nonlocal complex mKdV equation, the N-fold Darboux transformation (DT) of this equation is constructed. Subsequently, bright soliton solutions, double-hump soliton solutions, breather-like solutions under zero background, and multi-breather solutions under nonzero constant background are obtained for the nonlocal complex mKdV equation. Besides that, the interaction behaviors of the multi-soliton waves are also studied. There are some interesting phenomenons. The breather-II soliton wave, that can be seen as a combination of bright wave and dark wave, will appear. It does not appear alone but in the multi-breather solution.

Nonlinear behavior of dust acoustic periodic soliton structures of nonlinear damped modified Korteweg–de Vries equation in dusty plasma

In this article, the nonlinear plasma model known the damped modified Korteweg–de Vries equation under consideration on the base of analytical technique to know the novel behavior of this nonlinear model. We formulated the nonlinear plasma model by utilizing the RP (reductive perturbation) technique with basic set of fluid system which consisted on cold negative charge dust fluid, immobile background neutral charges, positive charge ion fluid, fast (ionized) electrons and thermally electrons. We also constructed various kinds of novel soliton solutions including kink wave solitons, periodic solitary waves, anti-kink wave solitons for the damped modified KdV equation. The determined soliton solutions are novel, interested and did not determined in the previous research. The physical interpretation of demonstrated solutions represent in contour, two-dim and three-dim with computational software. The study prove that our proposed approach is powerful, efficient and can also be fruitful for study of other nonlinear models.

Evolution of initial discontinuities in a particular case of two-step initial problem for the defocusing complex modified KdV equation

In this paper, the complete classification of solutions of defocusing complex modified KdV equation with a particular case of two-step initial condition is investigated by the finite-gap integration approach and Whitham modulation theory. The periodic wave solution and corresponding Whitham modulation equations for zero-phase, one-phase, two-phase are found. The self-similar wave structures in each case are composed of different building blocks: the plateau, vacuum, rarefaction wave, dispersive shock wave and two-genus region, which are studied in detail. The results of direct numerical simulations for the defocusing complex modified KdV equation agree well with those of Whitham modulation theory.

A four-component hierarchy of combined integrable equations with bi-Hamiltonian formulations

Upon introducing a specific 4 × 4 matrix eigenvalue problem with four components, we would like to construct a Liouville integrable Hamiltonian hierarchy, within the zero curvature formulation. Bi-Hamiltonian formulations are furnished via the trace identity, through which the Liouville integrability of the resulting hierarchy is explored. The first two nonlinear examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations.

Lax Pairs for the Modified KdV Equation

Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations, converting the defined Lax pairs to some simpler forms, are found. The direct method and its possible applications to other types of evolution equations are discussed.

Observation of non-planar dust acoustic solitary wave in a strongly coupled dusty plasma

The nonlinear evolution and propagation of a stable dust acoustic solitary wave (DASW) in a non-planar geometry is investigated here. The experiment is performed in a strongly coupled dusty plasma consisting of monodisperse micrometer sized particles levitated in the sheath of a capacitively coupled radio frequency argon plasma. The non-planar waves are generated with the help of a cylindrical conducting exciter pin placed at the center of the homogeneous dust cloud. A negative excitation pulse is used to create a dust void and a dust density perturbation simultaneously around the exciter. From the edge of the void, the density perturbation propagates as a nonlinear (cylindrical) non-planar DASW. The characteristics of the solitary wave are measured using image analysis of the recorded video of wave propagation. The numerical solution of the modified Korteweg–de Vries equation with an additional term to take care of the non-planar geometry is compared with the experimental observation. The wave amplitude and width are measured as a function of time and compared with the theoretical predictions.

Electron acoustic counterpropagating multi-solitons and rogue waves collision in an unmagnetized plasma in the presence of critical density ratios

In this paper, the nonlinear propagation of electrostatic collisional among multi-solitons around the critical values along with their corresponding phase shifts and collision between two rouge waves propagating toward each other is studied in an unmagnetized plasma environment. Using the concept of Hirota’s bilinear method, the useful forms of multi-solitons solutions of the coupled modified Korteweg–de Vries equations (mKdVEs) are determined. Furthermore, the coupled nonlinear Schrödinger equations (NLSEs) are derived from mKdVEs using the appropriate starching coordinates. The analytic solutions of different orders for the coupled NLSEs are also presented. The effects of the parameters related to the plasma environment on the electron acoustic scattered solitons, phase shifts and scattered rouge waves are analyzed. The proposed results provide the theoretical guidance to understand the propagation characteristics of collisional solitons, and their phase shifts around the critical values and collisional rouge waves in the modulated ranges.

Parallel Physics-Informed Neural Networks Method with Regularization Strategies for the Forward-Inverse Problems of the Variable Coefficient Modified KdV Equation

Parallel Physics-Informed Neural Networks Method with Regularization Strategies for the Forward-Inverse Problems of the Variable Coefficient Modified KdV Equation

Brief Communication: A modified Korteweg–de Vries equation for Rossby–Khantadze waves in a sheared zonal flow of the ionospheric E layer

Abstract. The system of non-linear equations for electromagnetic Rossby–Khantadze waves in a weakly ionized conductive ionospheric E-layer plasma with sheared zonal flow is given. Use of multiple-scale analysis allows reduction of an obtained set of equations to a (1+1)D non-linear modified KdV (mKdV) equation with cubic non-linearity describing the propagation of solitary Rossby–Khantadze solitons.

Exact solutions to the $$(3 + 1)$$-dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients

Exact solutions to the $$(3 + 1)$$-dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients