Infinitely-many Conservation Laws Research Articles

Asymptotic analysis on bright solitons and breather solutions of a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide

Abstract We study a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide. We obtain the Lax pair and N-fold Darboux transformation (DT) with N being a positive integer. Based on Lax pair obtained by us, we derive the infinitely-many conservation laws. We give the bright one-, two-, and N-soliton solutions, and the first-, second-, and Nth-order breather solutions based on the N-fold DT. We conclude that the velocities of the bright solitons are influenced by the distributed gain function, g(z), and variable coefficients in equation, h 1(z), p 1(z), r 1(z), and s 1(z) via the asymptotic analysis, where z represents the propagation variable or spatial coordinate. We also graphically observe that: the velocities of the first- and second-order breathers will be affected by h 1(z), p 1(z), r 1(z), and s 1(z), and the background wave depends on g(z).

Conservation laws, bound-state solitons and modulation instability for a variable-coefficient higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a variable-coefficient sixth-order nonlinear Schrödinger equation, which describes the propagation of attosecond pulses in an optical fiber. Based on Lax pair, infinitely-many conservation laws are constructed. With the aid of auxiliary functions, bilinear forms are derived. In addition, the one- and two-soliton solutions are obtained via the Hirota method. The influences of variable coefficients for soliton velocity and profile are discussed. Particularly, the interaction periods and soliton separation factor of bound-state solitons are analyzed. Finally, modulation instability is investigated. The reported results could be used to understand related soliton molecule and optical instability phenomena in nonlinear optics.

Multi-solitons and integrability for a (2+1)-dimensional variable coefficients Date–Jimbo–Kashiwara–Miwa equation

Under investigation in this paper is a (2+1)-dimensional generalized variable coefficients Date–Jimbo–Kashiwara–Miwa equation, which describes the nonlinear dispersive wave in inhomogeneous media. Via the generalized Laurent series truncated at the constant-level term, an auto-Bäcklund transformation is derived. Bilinear forms, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota method.

Conservation laws, Darboux transformation and localized waves for the [formula omitted]-coupled nonautonomous Gross–Pitaevskii equations in the Bose–Einstein condensates

Under investigation in this paper is the N-coupled nonautonomous Gross–Pitaevskii equations, which describe the dynamics of the Bose–Einstein condensates. Based on the Lax pair, infinitely-many conservation laws and Mth-fold Darboux transformation are constructed. Three types of the nonautonomous localized waves are obtained via the Darboux transformation. The nonautonomous bound-state soliton is observed. The profile and energy distribution of the nonautonomous breather and rogue wave are shown. The influences of coefficients for the shape and position of background wave are discussed. In addition, the interactions between three types of the nonautonomous localized waves are analyzed graphically.

Conservation laws, [formula omitted]-fold Darboux transformation, [formula omitted]-dark-bright solitons and the [formula omitted]th-order breathers of a variable-coefficient fourth-order nonlinear Schrödinger system in an inhomogeneous optical fiber

Optical fiber communication system is one of the core supporting systems of the modern internet age. For the simultaneous propagation of nonlinear waves in an inhomogeneous optical fiber, in this work, a coupled variable-coefficient fourth-order nonlinear Schrödinger system is studied. Infinitely-many conservation laws and N-fold Darboux transformation (DT) based on the existing Lax pair under certain constraints are derived, where N is a positive integer. Via the N-fold DT, the N-dark-bright soliton and Nth-order breather solutions under certain constraints are given. Interactions between the two dark-bright solitons are depicted graphically when the dispersion coefficient in that system, γ1(t), and the group velocity dispersion coefficient in that system, σ(t), are the constants and periodic functions, where t means the normalized retarded time. It is found that the velocities of the two dark-bright solitons increase as γ1(t) and σ(t) increase when γ1(t) and σ(t) are the constants. Interactions between the two breathers are presented graphically.

Darboux transformation, localized waves and conservation laws for an [formula omitted]-coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber

Optical fiber communication system is one of the supporting systems in the modern internet age. We investigate an M-coupled variable-coefficient nonlinear Schrödinger system, which describes the simultaneous pulse propagation of the M-field components in an inhomogeneous optical fiber, where M is a positive integer. With respect to the complex amplitude of the jth-field (j=1,…,M) component in the optical fiber, we construct an n-fold Darboux transformation, where n is a positive integer. Based on the n-fold Darboux transformation, we obtain some one- and two-fold localized wave solutions for the above system with the mixed defocusing-focusing-type nonlinearity and M=2. We acquire the infinitely-many conservation laws. Via such solutions, we obtain some vector gray solitons, interactions between the two vector parabolic/cubic gray solitons, and interactions between the vector parabolic/cubic breathers and gray solitons with different β(z), γ(z) and δ(z), the coefficients of the group velocity dispersion, nonlinearity and amplification/absorption. It can be found that δ(z) affects the backgrounds of the breathers and gray solitons.

Conservation laws and breather-to-soliton transition for a variable-coefficient modified Hirota equation in an inhomogeneous optical fiber

Optical fibers are used in the communications, biological sensors and chemical sensors. We investigate a variable-coefficient modified Hirota equation for the amplification or absorption of pulses propagating in an inhomogeneous optical fiber. With respect to the complex envelope of the optical field, we construct the infinitely-many conservation laws based on the existing Lax pair. According to the existing Darboux transformation, we derive the three-soliton solutions, the higher-order breather solutions and breather-to-soliton transition condition. Amplitudes of the two solitons change after the interaction, while velocities of them are unchanged via asymptotic analysis. When P ( z ) = 0 , interactions among the three parabolic or wavy solitons, interaction between the two parabolic or wavy or crooked breathers, and interactions among the three parabolic and wavy breathers are presented, where P ( z ) is related to the nonlinear focus length. Velocities of three solitons or two crooked breathers with P ( z ) ≠ 0 are different from those with P ( z ) = 0 . Based on the breather-to-soliton transition condition, when P ( z ) = 0 , parabolic or wavy multi-peak and M-shaped solitons are presented; when P ( z ) ≠ 0 , the crooked periodic wave and anti-dark soliton are shown.

Darboux-dressing transformation, conservation laws and bound-state solutions of the vector Lakshmanan–Porsezian–Daniel equation

The vector Lakshmanan–Porsezian–Daniel (vLPD) equations are mainly studied in this paper. The Darboux-dressing transformation (DDT) and infinitely-many conservation laws of the vLPD equations are constructed by the Lax pair. Furthermore, one soliton and bound-state solitons solutions of the vLPD equations are obtained by the DDT method. The obtained graphs can directly reflect the dynamic behavior of the aforementioned solutions.

Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber

Optical fiber communication plays an important role in the modern communication. In this paper, we investigate a variable-coefficient coupled Hirota system which describes the vector optical pulses in an inhomogeneous optical fiber . With respect to the complex wave envelopes, we construct a Lax pair and a Darboux transformation, both different from the existing ones. Infinitely-many conservation laws are derived based on our Lax pair. We obtain the one/two-fold bright-bright soliton solutions, one/two-fold bright-dark soliton solutions and one/two-fold breather solutions via our Darboux transformation. When α ( t ) , β ( t ) and δ ( t ) are the trigonometric functions, we present the bright-bright soliton, bright-dark soliton and breather which are all periodic along the propagation direction, where α ( t ) , β ( t ) and δ ( t ) represent the group velocity dispersion, third-order dispersion and nonlinear terms of the self-phase modulation and cross-phase modulation. Interactions between the two bright-bright soliton, two bright-dark solitons and two breathers are presented. Bound state of the two bright-bright solitons is formed. Widths and velocities of the two bright-bright solitons do not change but their amplitudes change after their interaction via the asymptotic analysis . Periods of the bright-dark solitons decrease when the periods of the trigonometric α ( t ) , β ( t ) and δ ( t ) decrease.

Modulation instability, rogue waves and conservation laws in higher-order nonlinear Schrödinger equation

In this paper, the modulation instability (MI), rogue waves (RWs) and conservation laws of the coupled higher-order nonlinear Schrödinger equation are investigated. According to MI and the 2 × 2 Lax pair, Darboux-dressing transformation with an asymptotic expansion method, the existence and properties of the one-, second-, and third-order RWs for the higher-order nonlinear Schrödinger equation are constructed. In addition, the main characteristics of these solutions are discussed through some graphics, which are draw widespread attention in a variety of complex systems such as optics, Bose–Einstein condensates, capillary flow, superfluidity, fluid dynamics, and finance. In addition, infinitely-many conservation laws are established.

Lax pair, conservation laws, Darboux transformation, breathers and rogue waves for the coupled nonautonomous nonlinear Schrödinger system in an inhomogeneous plasma

Abstract Plasmas are believed to be possibly “the most abundant form of ordinary matter in the Universe”. In this paper, a coupled nonautonomous nonlinear Schrodinger system is investigated, which describes the propagation of two envelope solitons in a weakly inhomogeneous plasma with the t-dependent linear and parabolic density profiles and nonconstant collisional damping. Lax pair with the nonisospectral parameter and infinitely-many conservation laws are derived. Based on the Lax pair, the Nth-step Darboux transformation is constructed. Utilizing the Nth-step Darboux transformation, we obtain the breather and rogue wave solutions, and find that the amplitude of the nonzero background is nonconstant and dependent on the inhomogeneous coefficients in the system under investigation. Characteristics of the breathers and rogue waves are discussed, and effects of the inhomogeneous coefficients on the breathers and rogue waves are analyzed. Breathers and rogue waves with the dark or bright soliton together are also constructed and their characteristics are discussed. We find that the dark and bright solitons can coexist and generate the breather-like waves.

Integrability and dark soliton solutions for a high-order variable coefficients nonlinear Schrödinger equation

Under investigation in this paper is the integrability and dark soliton solutions for a fifth-order variable coefficients nonlinear Schrödinger equation, which is used in an inhomogeneous optical fiber. Bilinear forms, Lax pair and infinitely-many conservation laws are obtained under an integrable constraint. Dark one-, two- and N-soliton solutions are constructed via the Hirota bilinear method.

Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system

Optical fiber communication system is one of the core supporting systems of the modern internet age. In this paper, under investigation is a higher-order nonlinear Schrödinger system for the simultaneous propagation of two ultrashort optical pulses in an optical fiber. Based on the Lax pair, with respect to the two-component electric fields, infinitely-many conservation laws and one/N-fold binary Darboux transformations are derived, where N=2,3,…. Solitons are discussed: (1) Nondegenerate one dark–dark soliton, which is black or gray in each component, is obtained. ϵ has no relation with the soliton amplitude in each component, but has a linear correlation with the soliton velocities, where ϵ, σ1 and σ2 are the constant coefficients in the system; (2) Overtaking and head-on interactions between the two dark–dark solitons as well as the bound state are depicted. With the decreasing value of σ1, the gray solitons’ amplitudes increase, but the black solitons’ amplitudes decrease. With the decreasing value of σ2, the gray solitons’ amplitudes increase, but the black solitons’ amplitudes do not change; (3) Interaction among the three overtaking solitons and interaction between a bound state and one soliton are displayed. We find that the interactions between the two solitons and among the three solitons are elastic.

Dynamics of multi-soliton and breather solutions for a new semi-discrete coupled system related to coupled NLS and coupled complex mKdV equations

In this paper, a new semi-discrete coupled system which was firstly proposed by Bronsard and Pelinovsky is under investigation. Based on its known Lax pair, the infinitely-many conservation laws and discrete N-fold DT for this system are constructed. As applications, bell-shaped multi-soliton and breather solutions in terms of determinants for this system are firstly derived by means of the discrete N-fold DT. Propagation and elastic interaction structures of such soliton solutions are shown graphically: (1) Propagation characteristics of one-, two-, three- and four-soliton solutions are discussed from vanishing background. (2) Propagation characteristics of one- and two-breather solutions are analyzed from the plane wave background. The details of the dynamical evolutions for such soliton and breather solutions are studied via numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of these solitons with or without a noise in a relatively short period of time, while the evolutions exhibit obviously larger oscillations and strong instability with the increase in time. These results may be useful for understanding the propagation of orthogonally polarized optical waves in an isotropic medium and circularly polarized few-cycle pulses in Kerr media described by the coupled NLS and coupled complex mKdV equations, respectively.

Conservation laws, modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber

Under investigation in this paper is a sextic nonlinear Schrodinger equation, which describes the pulses propagating along an optical fiber. Based on the symbolic computation, Lax pair and infinitely-many conservation laws are derived. Via the modiied Hirota method, bilinear forms and multi-soliton solutions are obtained. Propagation and interactions of the solitons are illustrated graphically: Initial position and velocity of the soliton are related to the coefficient of the sixth-order dispersion, while the amplitude of the soliton is not affected by it. Head-on, overtaking and oscillating interactions between the two solitons are displayed. Through the asymptotic analysis, interaction between the two solitons is proved to be elastic. Based on the linear stability analysis, the modulation instability condition for the soliton solutions is obtained.

Conservation laws, solitons, breather and rogue waves for the (2+1)-dimensional variable-coefficient Nizhnik–Novikov–Veselov system in an inhomogeneous medium

In this paper, we consider the (2+1)-dimensional variable-coefficient Nizhnik–Novikov–Veselov system in an inhomogeneous medium, which is the isotropic Lax integrable extension of the Korteweg-de Vries equation. Infinitely-many conservation laws are constructed. N-soliton solutions in terms of the Wronskian are obtained via the Hirota method. Velocity and shape of the soliton can be influenced by those variable coefficients, while the amplitude of the soliton can not be affected. Collision between the two solitons is shown to be elastic. Breather wave solutions are constructed via the trilinear method. Such phenomena as the deceleration and compression of the breather waves are studied graphically. Rogue wave solutions are derived when the periods of the breather wave solutions go to the infinity. Separated and united composite rogue waves are discussed graphically.

Lax pair, infinitely-many conservation laws and soliton solutions for a set of the time-dependent Whitham-Broer-Kaup equations for the shallow water

Under investigation in this paper is a set of the time-dependent Whitham–Broer–Kaup equations, which is used for the shallow water under the Boussinesq approximation. The equations can be transformed into generalized time-dependent coefficient Ablowitz–Kaup–Newell–Segur system via the variable transformation. Lax pair, infinitely-many conservation laws and bilinear forms of the Ablowitz–Kaup–Newell–Segur system are obtained. One-, two- and three-soliton solutions are derived via the Hirota bilinear method. The solitons are physically related to the horizontal velocity field and height that deviates from equilibrium position of the water. Features of the solitons are studied: Soliton amplitude is related to the wave number parameters, while the soliton velocity is related to the wave number parameters and variable coefficient. Interactions between/among the solitons could be elastic or inelastic, determined by the wave number parameters. Interaction property could not be affected by the variable coefficient. Soliton stability is studied via the numerical calculation, which indicates that the solitons could only propagate steadily in a limited time.

Darboux transformation of the coupled nonisospectral Gross–Pitaevskii system and its multi-component generalization

In this paper, we extend the one-component Gross–Pitaevskii (GP) equation to the two-component coupled GP system including damping term, linear and parabolic density profiles. The Lax pair with nonisospectral parameter and infinitely-many conservation laws of this coupled GP system are presented. Actually, the Darboux transformation (DT) for this kind of nonautonomous system is essentially different from the autonomous case. Consequently, we construct the DT of the coupled GP equations, besides, nonautonomous multi-solitons, one-breather and the first-order rogue wave are also obtained. Various kinds of one-soliton solution are constructed, which include stationary one-soliton and nonautonomous one-soliton propagating along the negative (positive) direction of x-axis. The interaction of two solitons and two-soliton bound state are demonstrated respectively. We get the nonautonomous one-breather on a curved background and this background is completely controlled by the parameter β. Using a limiting process, the nonautonomous first-order rogue wave can be obtained. Furthermore, some dynamic structures of these analytical solutions are discussed in detail. In addition, the multi-component generalization of GP equations are given, then the corresponding Lax pair and DT are also constructed.

Bäcklund transformation, infinitely-many conservation laws, solitary and periodic waves of an extended (3 + 1)-dimensional Jimbo–Miwa equation with time-dependent coefficients

In this paper, an extended (3+1)-dimensional Jimbo–Miwa equation with time-dependent coefficients is investigated, which comes from the second member of the Kadomtsev–Petviashvili hierarchy and is shown to be conditionally integrable. Bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived via the binary Bell polynomials and symbolic computation. With the help of the bilinear form, one-, two- and three-soliton solutions are obtained via the Hirota method, one-periodic wave solutions are constructed via the Riemann theta function. Additionally, propagation and interaction of the solitons are investigated analytically and graphically, from which we find that the interaction between the solitons is elastic and the time-dependent coefficients can affect the soliton velocities, but the soliton amplitudes remain unchanged. One-periodic waves approach the one-solitary waves with the amplitudes vanishing and can be viewed as a superposition of the overlapping solitary waves, placed one period apart.

Integrability, solitons, periodic and travelling waves of a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

Under investigation in this paper is a generalized (3+1)-dimensional varible-coefficient nonlinear-wave equation, which has been presented for nonlinear waves in liquid with gas bubbles. The bilinear form, Backlund transformation, Lax pair and infinitely-many conservation laws are obtained via the binary Bell polynomials. One-, two- and three-soliton solutions are generated by virtue of the Hirota method. Travelling-wave solutions are derived with the aid of the polynomial expansion method. The one-periodic wave solutions are constructed by the Hirota-Riemann method. Discussions among the soliton, periodic- and travelling-wave solutions are presented: I) the soliton velocities are related to the variable coefficients, while the soliton amplitudes are unaffected; II) the interaction between the solitons is elastic; III) there are three cases of the travelling-wave solutions, i.e., the triangle-type periodical, bell-type and soliton-type travelling-wave solutions, while we notice that bell-type travelling-wave solutions can be converted into one-soliton solutions via taking suitable parameters; IV) the one-periodic waves approach to the solitary waves under some conditions and can be viewed as a superposition of overlapping solitary waves, placed one period apart.