sG Equation Research Articles

Multiloop soliton solutions and compound WKI–SP hierarchy

AbstractIn this paper, a compound equation which is a mix of the Wadati–Konno–Ichikawa (WKI) equation and the short‐pulse (SP) equation is first studied. By transforming both the independent and dependent variables in the equation, we introduce a novel hodograph transformation to convert the compound WKI–SP equation into the mKdV–SG (modified Korteweg–de Vries and sine‐Gordon) equation. The multiloop soliton solutions in the form of the parametric representation are found. It is shown that the ‐loop soliton solution may be decomposed exactly into separate soliton elements by using a Moloney–Hodnett‐type decomposition. By virtue of the decomposed soliton solutions, the asymptotic behaviors of and are investigated in detail. The corresponding phase shifts of each loop or antiloop soliton caused by its interaction with the other ones are calculated. Furthermore, a new hierarchy of WKI–SP‐type equations possessing multiloop soliton solutions is constructed. These deduced equations are all with time‐varying coefficients and the corresponding dispersion relation will have a time‐dependent velocity. The whole hierarchy of equations which include the WKI‐type equations, the SP‐type equations, and the compound generalized WKI–SP equations, are illustrated Lax integrable. The specific equation in the hierarchy is labeled as equation so that its Lax pairs can be directly written out with the help of and . A unified hodograph transformation is established to relate the compound WKI–SP hierarchy with the mKdV–SG hierarchy.

Some special exact solutions in nonlocal Alice-Bob sine-Gordon systems ∗ ∗ The work was sponsored by the National Natural Science Foundation of China (Nos. 11435005, 11975131, 12235007) and K. C. Wong Magna Fund in Ningbo University.

Three nonlocal Alice-Bob sine-Gordon (ABSG) systems with the parity and time reversal nonlocality and/or space-time exchange nonlocality are investigated. For the common local SG equation, two types of N-soliton solutions and three types of periodic solutions are presented. The multiple solutions, breather solution, double kink solution, and periodic solutions of the ABSG systems are obtained from the symmetry reductions of a coupled local sine-Gordon system.

Modulated periodic wavetrains in the spherical Gardner equation

The spherical Gardner (sG) equation is derived by reducing the (3+1)-dimensional Gardner–Kadomtsev–Petviashvili (Gardner–KP) equation with a similarity reduction. As a special case, the step-like initial condition is considered along a paraboloid front. By applying a multiple-scale expansion, a system of first order partial differential equations for the slowly varying parameters of a periodic wavetrain is obtained. The corresponding modulation system is transformed into a simpler form with the help of Riemann type variables. This basic form is important to investigate the dispersive shock wave (DSW) phenomena in the sG equation. DSW solution is also compared with the numerical solution of the sG equation and good agreement is found between these solutions.

Universality Near the Gradient Catastrophe Point in the Semiclassical Sine‐Gordon Equation

AbstractWe study the semiclassical limit of the sine‐Gordon (sG) equation with below threshold pure impulse initial data of Klaus‐Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava, and Klein, we found that in a neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painlevé I tritronquée solution. A linear map can be explicitly made from the tritronquée solution to this neighborhood. Under this map: away from the tritronquée poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquée solution; localized defects appear at locations mapped from the poles of the tritronquée solution; the defects are proved universally to be a two‐parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann‐Hilbert problems, substantially generalizing [5] to establish universality beyond the context of solutions of a single equation. © 2021 Wiley Periodicals LLC.

Bounded states for breathers–soliton and breathers of sine–Gordon equation

The Wronskian solutions to the sine–Gordon (sG) equation that can provide interaction of different kinds of solutions are revisited. And a novel expression of N-soliton solution with a nonzero background to the sG equation is presented. The kinks (solitons), the breathers and the interactions among solitons and breathers are also derived directly from the novel expression. Due to the existences of abundant structures of the solitons and breathers, it is possible to search for the coherent structures, or bounded states of solitons and breathers. By introducing the velocity resonant conditions, the sG equation is proved to possess the bounded state for breather–soliton molecules (BSMs) and the bounded state for breather molecules (BMs). An approximately bounded state for kinks (solitons) is given for the wavenumber being nearly the same. In addition, it is demonstrated that the interactions among the BSMs, BMs, solitons and breathers may be inelastic by the particular meaning the sizes of the BSMs and BMs change.

INVESTIGATION OF FRACTIONAL ORDER SINE-GORDON EQUATION USING LAPLACE ADOMIAN DECOMPOSITION METHOD

We analytically investigate a nonlinear fractional-order sine-Gordon (sG) equation. The derivatives considered herein, are taken in Caputo’s sense. The Laplace transform together with the Adomian decomposition method (LADM) is applied to attain analytical approximation of the aforesaid equation. The sG equation having Caputo derivative is solved in the order of series solutions and the results are confirmed by considering two examples with appropriate initial conditions. The numerical simulations are accomplished to compare with the analytical approximations, where qualitatively better agreements are achieved.

Soliton and breather molecules in few-cycle-pulse optical model

The modified Korteweg–de Vries–sine-Gordon (mKdV–sG) equation is one of the models describing few-cycle-pulse optical solitons beyond the slowly varying envelope approximation. By introducing the velocity resonance mechanism to multiple soliton/breather solutions, it is found that the mKdV–sG model possesses abundant soliton molecule (SM), breather molecule (BM) and breather–soliton molecule (BSM) structures. Every SM includes only two solitons, a BM can be constructed by arbitrary number of breathers and a BSM can be formed from one or two solitons and arbitrary number of breathers. Though the interactions among the separated solitons and breathers are elastic, the size (the distances among solitons in a same molecule) will be changed after the interaction. Prospects of the studies overviewed in this work are given in the conclusions.

Half-cycle soliton and periodic waves of arbitrarily high amplitude in a two-level system

We show that the electromagnetic wave propagation problem based on a two-level atomic model admits exact traveling wave solutions, of both soliton and cnoidal wave types. These solutions extend the already known solutions of the same type of both the mKdV and sG equations, which have been derived from the same initial model, respectively, in a long-wave and in a short-wave approximation. The continuation is such that no modification of the wave profile is required, but that the wave velocity only has to be corrected.

Complex simplified Hirota’s forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV–Sine-Gordon equation

The present work consists of detailed exploration of modified KdV–Sine-Gordon equation in integrable form, owning to two-component nonlinear channel for modeling laser light propagation. For validating the behavior of this equation in the sense of integrability, we use the Painleve test. The simplified Hirota’s technique with new complex forms is developed suitably to construct multiple-soliton solutions with complex structure for considered equation. Moreover, Lie symmetry analysis has been implemented for perceiving symmetries of MKdV–SG equation and then culminating the invariant solitary wave solutions. The new findings obviously reveal that simplified Hirota’s technique with complex structure would be highly proficient for fabricating new multiple complex soliton solutions to other nonlinear equations with integrable properties from mathematical physics and dynamical systems community.

The Sylvester Equation and Integrable Equations: The Ablowitz—Kaup—Newell—Segur System

In this paper, we seek connections between the Sylvester equation and the Ablowitz—Kaup—Newell—Segur (AKNS) system. By the Sylvester equation KM — MK = r sT, we introduce master function S(i, j) = sT Kj (I + M)−1Kir. This function satisfies some recurrence relations. By imposing dispersion relations on r and s, we study the constructions of the AKNS system, where some AKNS type equations are investigated emphatically, including second-AKNS equation, second-modified AKNS (mAKNS) equation, third-AKNS equation, third-mAKNS equation and (—1)st-AKNS equation. The reductions of these equations to complex Korteweg—de Vries (KdV) equation, real and complex modified Korteweg—de Vries (mKdV) type equations, nonlinear Schrodinger (NLS) type equations and sine-Gordon (sG) equation are discussed.

Perturbed soliton and director deformation in a ferronematic liquid crystal

The nonlinear molecular deformation of the ferronematic liquid crystal in the presence of external applied magnetic field intensity is investigated in view of solitons for the director axis. The Frank’s free energy density of the nematic liquid crystal comprising the basic elastic deformations, molecular deformation associated with the nematic molecules and the suspended ferromagnetic particles and their interactions with magnetic field intensity is deduced to a sine-Gordon like equation using the classical Euler–Lagrange’s equation. Using the small angle approximation we establish the Ginzburg–Landau (GL) equation and a class of solutions are obtained. In the normal condition of large angle oscillation of the director axis, we constructed a damped sine-Gordon (sG) equation with the additional perturbation appears in the form of cosine function. The sG equation is solved using numerical simulation and kink excitations were obtained as the molecular deformation for the case of constant damping and distorted kink to a planar configuration transition as we increase the damping.

A systematic construction of parity-time ()-symmetric and non--symmetric complex potentials from the solutions of various real nonlinear evolution equations

We systematically construct a distinct class of complex potentials including parity-time () symmetric potentials for the stationary Schrödinger equation by using the soliton and periodic solutions of the four integrable real nonlinear evolution equations (NLEEs), namely the sine-Gordon (sG) equation, the modified Korteweg–de Vries (mKdV) equation, combined mKdV–sG equation and the Gardner equation. These potentials comprise of kink, breather, bion, elliptic bion, periodic and soliton potentials which are explicitly constructed from the various respective solutions of the NLEEs. We demonstrate the relevance between the identified complex potentials and the potential of the graphene model from an application point of view.

Solitons in antiferroelectric liquid crystal and Josephson junction: The double sine-Gordon model

In this paper we solve the double sine-Gordon (sG) equation with constant coefficients using the Hirota bilinearization method and generalize the result with the director dynamics of antiferroelectric liquid crystal and fluxon excitation in the Josephson junction. The obtained one and two soliton solutions are consistently described for the above two systems with real physical parameters.

Few-cycle solitons in supercontinuum generation dynamics

We review several propagation models that do not rely on the slowly-varying-envelope approximation (SVEA), and can thus be considered as fundamental models addressing the formation and propagation of few-cycle pulsed field structures and solitary waves arising in the course of intense ultrashort optical pulse evolution in nonlinear media and beyond octave-bandwidth optical spectrum broadening. These generic models are: the modified-Korteweg-de Vries (mKdV), the sine-Gordon (sG), and the mixed mKdV-sG equations. To include wave polarization dynamics, the vector extensions of both mKdV and sG equations are introduced. Multi-octave-spanning supercontinuum generation and few-cycle soliton structures are highlighted from numerical simulations.

Nonautonomous soliton solutions of the modified Korteweg–de Vries–sine-Gordon equation

Multisoliton solutions of the modified Korteweg–de Vries–sine-Gordon (mKdV–SG) equation with time-dependent coefficients are considered. Cases describing changes in the shape of soliton solutions (kinks and breathers) observed in gradual transitions between the mKdV, SG, and mKdV–SG equations are numerically studied.

Electron-Acoustic Solitary Waves in a Two-Temperature Plasma Having Electrons With Kappa Distribution

The excitation of nonlinear electron-acoustic solitary waves (SWs) is studied theoretically in an unmagnetized, uniform, and collisionless plasma. The plasma is composed of two types of electrons referred to as inertial cold electrons and inertialess Kappa (κ) distributed superthermal hot electrons along with stationary ions. Two parameters κ and β are introduced in the governing equations, where the density related parameter is defined by β = n co /n ho (n co and n ho are the cold and hot electron number densities, respectively). Using a reductive perturbation technique, nonlinear Korteweg-de Vries (KdV), modified KdV (mKdV), and standard Gardner (sG) equations have been derived, which admit SW solutions. The KdV equation allows to analyze both compressive (negative) and rarefactive (positive) solitons. But a significant solitary structure vanishes around a critical region corresponding to critical κ c and β c . To look a solitary structure in the critical region, it is proceeded to the next approach of mKdV solution with more higher order calculation of perturbation. But only rarefactive soliton is found in this approach. Furthermore, having higher order perturbation, the next approach of sG equation allows to rescue both compressive and rarefactive solitons even in the critical region. It is observed that the excitation of soliton pulses with various shapes depends on both parameters κ and β.

Nonlocal topological solitons of the sine-Gordon equation

A special nonlocal symmetry related to the Bäcklund transformation is obtained for the (1+1)-dimensional sine-Gordon (sG) equation in its polynomial form, which can further generate infinitely many nonlocal symmetries of the sG equation due to the inclusion of an arbitrary spectrum parameter. This nonlocal symmetry can be localized, and hence turn into a point symmetry for the augmented system through the introduction of an auxiliary function. Consequently, finite transformation, similarity reductions, and explicit invariant solutions related to the nonlocal symmetry are obtained. It is known that the sG equation has topological solitons or kinks (or antikinks). Based on the group invariant solutions, a different kind of topological soliton, which is a topological soliton situated on a cnoidal wave background, is obtained analytically and graphically displayed.

Detailed comparison of numerical methods for the perturbed sine-Gordon equation with impulsive forcing

The properties of various numerical methods for the study of the perturbed sine-Gordon (sG) equation with impulsive forcing are investigated. In particular, finite difference and pseudo-spectral methods for discretizing the equation are considered. Different methods of discretizing the Dirac delta are discussed. Various combinations of these methods are then used to model the soliton–defect interaction. A comprehensive study of convergence of all these combinations is presented. Detailed explanations are provided of various numerical issues that should be carefully considered when the sG equation with impulsive forcing is solved numerically. The properties of each method depend heavily on the specific representation chosen for the Dirac delta—and vice versa. Useful comparisons are provided that can be used for the design of the numerical scheme to study the singularly perturbed sG equation. Some interesting results are found. For example, the Gaussian approximation yields the worst results, while the domain decomposition method yields the best results, for both finite difference and spectral methods. These findings are corroborated by extensive numerical simulations.

A generalized exp-function method for multiwave solutions of sine-Gordon equation

In this paper, the exp-function method is generalized to sine-Gordon (sG) equation and single-, double- and three-wave solutions are obtained. It is shown that the generalized exp-function method combined with appropriate anstaz may provide a straightforward, effective and alternative method for constructing multiwave solutions of some nonlinear partial differential equations.

STOCHASTIC DYNAMICS OF DC AND AC DRIVEN DISLOCATION KINKS

Dynamics of a pinned dislocation kink controlled by the acting DC and AC forces is studied analytically. The motion of the kink, described by sine-Gordon (sG) equation, is explored within the framework of McLaughlin–Scott perturbation theory. Assuming weakness of the acting AC force, the equation of motion of the dislocation kink in the pinning potential is linearized. Based on the equations derived, we study stochastic behavior of the kink, and determine the probability of its depinning. The dependencies of the depinning probability on DC and AC forces are analyzed in detail.