The concave or convex peaked and smooth soliton solutions of Camassa-Holm equation

ＩｎｔｒｏｄｕｃｔｉｏｎＣａｍａｓｓａ ,Ｈｏｌｍ[1]ｏｂｔａｉｎｅｄａｃｌａｓｓｏｆｎｅｗｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｓｈａｌｌｏｗｗａｔｅｒｅｑｕａｔｉｏｎ ,ｉ.ｅ.,Ｃａｍａｓｓａ＿Ｈｏｌｍｅｑｕａｔｉｏｎ2ｕｔ+ 2ｋｕｘ-12 ｕｘｘｔ+ 6ｕｕｘ =ｕｘｕｘｘｘ+ 12 ｕｕｘｘｘ. ( 1 )Ｆｏｒｅｖｅｒｙｋ,ｔｈｅＥｑ .( 1 )ｉｓａｃｌａｓｓｏｆｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｓｙｓｔｅｍ .Ｔｈｉｓｃｌａｓｓｏｆｅｑｕａｔｉｏｎｉｓａｃｌａｓｓｏｆｎｏｔｏｎｌｙｓｔｒａｎｇｅｂｕｔａｌｓｏ…     

Soliton solutions and Bäcklund transformations of a (2 + 1)-dimensional nonlinear evolution equation via the Jaulent–Miodek hierarchy

In this paper, a (2 + 1)-dimensional nonlinear evolution equation generated via the Jaulent–Miodek hierarchy is investigated. Based on the Bell polynomials and Hirota method, bilinear forms and Bäcklund transformations are derived. One- and two-soliton solutions are constructed via symbolic computation. Soliton solutions are obtained through the Bäcklund transformations. We can get three types by choosing different parameters: the kink, bell-shape, and anti-bell-shape solitons. Propagation of the one soliton and elastic interactions between the two solitons are discussed graphically. After the interaction of the two bell-shape or anti-bell-shape solitons, solitonic shapes and amplitudes keep invariant except for some phase shifts, while after the interaction of the kink soliton and anti-bell-shape soliton, the anti-bell-shape soliton turns into a bell-shape one, and the kink soliton keeps its shape, with their amplitudes unchanged.     

Hybrid solutions to Mel’nikov system

Based on the KP hierarchy reduction technique, explicit two kinds of breather solutions to Mel’nikov system are constructed, one breather is localized in the x-direction and period in the y-direction, the other is the opposite, that is localized in the y-direction and period in the x-direction. Moreover, these two kinds of breather solutions are reduced to the homoclinic orbits and dark soliton or anti-dark soliton solution under suitable parameters constraint respectively. It is interesting that the interaction between the dark soliton and anti-dark soliton is similar to a resonance soliton. In addition, with the long-wave limit, some rational solutions are derived, which possess two different behaviors: lump solution and line rogue wave. Then the dynamics properties of interactions among the obtained solutions are shown through some figures, especially, we not only get the parallel breather but also the intersectional breather during the discussion of the interaction to the two-breather solution. Furthermore, a new three-state interaction composed of dark soliton, rogue wave and breather is generated, this novel pattern is a fantastic phenomenon for the Mel’nikov system.

     

Spatiotemporal solitons on cnoidal wave backgrounds in three media with different distributed transverse diffraction and dispersion

The (3+1)-dimensional nonlinear Schrödinger equation with different distributed transverse diffraction and dispersion is studied based on the similarity transformation, and exact bright soliton solution on cnoidal wave backgrounds is derived. Moreover, three kinds of dynamical behaviors of these soliton solutions in three different dispersion/diffraction decreasing media with the Gaussian, hyperbolic, and Logarithmic profiles are discussed. Solitons interact with cnws and/or the change of characteristics of solitons by an addition of cnws are studied. Result of comparison with three media indicates that for the same parameters, the bright soliton in the Gaussian profile is compressed to the utmost degree. These results are potentially useful for future experiments in the optical communications, long-haul telecommunication networks, and Bose–Einstein condensations.     

Propagation properties of optical soliton in an erbium-doped tapered parabolic index nonlinear fiber: soliton control

In this paper, with the aid of symbolic computation, we investigate the generalized nonlinear Schrödinger Maxwell–Bloch equation, which describes the propagation of the optical soliton through an inhomogeneous two-level dielectric tapered fiber medium. By virtue of the Darboux transformation method, two-soliton solutions are generated based on the constructed Lax pair and figures are plotted to illustrate the properties of the obtained solutions. Moreover, through manipulating the dispersion and nonlinearity profiles, various soliton control systems are investigated which is promising for potential applications in the design of soliton compressor, soliton amplification and high-speed optical devices in ultralarge capacity transmission systems. This means that we are able to control the soliton types with suitably selected values of the parameters. Additionally more soliton control techniques are proposed and investigated. We expect that the above analysis could be observed in future experiments.     

Numerical simulation of standing solitons and their interaction

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔａｎｄｉｎｇｓｏｌｉｔｏｎｗａｓｄｉｓｃｏｖｅｒｅｄｉｎ 1 984ｂｙＤｒ.ＷＵ[1],ａＣｈｉｎｅｓｅｖｉｓｉｔｉｎｇｓｃｈｏｌａｒｉｎＵ .Ｓ .Ａ .ｆｒｏｍＮａｎｊｉｎｇＵｎｉｖｅｒｓｉｔｙ .Ｈｅｐｏｕｒｅｄｔｈｅｗａｔｅｒｉｎｔｏａｎａｒｒｏｗａｎｄｌｏｎｇｒｅｃｔａｎｇｕｌａｒｃｈａｎｎｅｌ,ｔｈｅｎｐｕｔｔｈｅｃｈａｎｎｅｌｏｎａｌｏｕｄｓｐｅａｋｅｒａｎｄｖｉｂｒａｔｅｄｉｔｖｅｒｔｉｃａｌｌｙｏｒｈｏｒｉｚｏｎｔａｌｌｙ .Ｃｏｎｔｒｏｌｌｉｎｇｔｈｅｖｉｂｒａ…     

Bright hump solitons for the higher-order nonlinear Schrödinger equation in optical fibers

Under investigation is the higher-order nonlinear Schrödinger equation with the third-order dispersion (TOD), self-steepening (SS) and self-frequency shift, which can be used to describe the propagation and interaction of ultrashort pulses in the subpicosecond or femtosecond regime. Through the introduction of an auxiliary function, bilinear form is derived. Bright one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. From the one-soliton solutions, we present the parametric regions for the existence of single- and double-hump solitons, and find that they are affected by the coefficients of the group velocity dispersion (GVD) and TOD. Besides, propagation of the one single- or double-hump soliton is observed. We analytically obtain the amplitudes for the single- and double-hump solitons, and calculate the interval between the two peaks for the double-hump soliton. Moreover, soliton amplitudes are related to the coefficients of the GVD, TOD and SS, while the interval between the two peaks for the double-hump soliton is dependent on the coefficients of the GVD and TOD. Interactions are seen between the (i) two single-hump solitons, (ii) two double-hump solitons, and (iii) single- and double-hump solitons. Those interactions are proved to be elastic via the asymptotic analysis.     

Nonautonomous bright soliton solutions on continuous wave and cnoidal wave backgrounds in blood vessels

We discuss the dissipative nonlinear Schrödinger equation (NLSE) with a variable coefficient in blood vessels via a NLSE-based constructive method and obtain exact nonautonomous soliton solutions including bright soliton solutions on continuous wave (cw) and cnoidal wave (cnw) backgrounds. Moreover, the dynamical behaviors of these soliton solutions are studied. The impact of the cw background on the separating and interactive soliton behaviors is investigated. These behaviors of the soliton can be modulated by adjusting the amplitude of background wave. The propagation behaviors of solitons on the cnw background in different dispersion systems are also studied. These results are potentially useful for future experiments in various blood vessels.     

Direct determination of bulk strain soliton parameters in solid polymeric waveguides

Bulk strain solitons in solids attract considerable attention in applications due to their very small decay, permanent bell shape and the wave parameter dependence on waveguide elasticity and geometry. One of the reasons of an evident gap between theory and numerical simulation of strain solitons propagation in various waveguides and comparatively rare experimental verification of rigorous results lies in extensive variability of physical constants data available for polymers. We show how a dramatic improvement of experimental setup provides new opportunities in solitary wave observation and its parameters measurements.Digital holography based on high-speed registration cameras allowed us to refine the accuracy of measurements, and precise pulse synchronization provided direct measurements of bulk strain soliton velocity with proper accuracy. It confirms the fact that the soliton parameters depend rather on the waveguide geometry and material, not on an initial pulse, which power provides either a single soliton or a soliton train.     

Darboux transformation of generalized coupled KdV soliton equation and its odd-soliton solutions

Based on the resulting Lax pairs of the generalized coupled KdV soliton equation, a new Darboux transformation with multi-parameters for the generalized coupled KdV soliton equation is derived with the help of a gauge transformation of the spectral problem. By using Darboux transformation, the generalized odd-soliton solutions of the generalized coupled KdV soliton equation are given and presented in determinant form. As an application, the first two cases are given.     

Dark and gray solitons of ($$$$)-dimensional nonlocal nonlinear media with periodic response function

By using the standard symmetry reduction method, some exact analytical solutions including gray solitons and gray soliton lattice solutions are derived for the (\(2+1\))-dimensional nonlinear optical media with periodic nonlocal response. Furthermore, dark/gray soliton solutions and dark soliton lattice solutions are found by means of hyperbolic function expansion method and elliptic function expansion method for the nonlocal nonlinear system, respectively. It is found that two critical points exist for soliton solutions, and the switching dynamics of solitons may be described by the critical points.     

The expression of soliton solution for Sine-Gordon equation

In this paper, we study the inverse scattering solution for Sine-Grodon equation A particularly concise expression of the soliton solution is obtained, and the single soliton and double soliton solutions are discussed.     

Dynamics of diverse data-driven solitons for the three-component coupled nonlinear Schrödinger model by the MPS-PINN method

Nonlinear Dynamics - We improve the physical information neural network by adding multiple parallel subnets to predict seven types of soliton dynamics, such as one soliton, two solitons and soliton...     

TYPHOON,SOLITON AND HELICAL STRUCTURE-AN APPROACH TO COMBINE DIFFERENT DISCPLINES

台风和孤立子都是小尺度不稳定流动受扰动的激励,吸收基本流场的能量而形成的. 当它们各自所受的耗散、弥散、平流或对流作用分别达到平衡时,其运动形态在一段较长时间内保持稳定. 通过研究螺旋结构,考察它们在局域能量维持方面相似的动力学特性. 通过理论分析和大型地球流体模拟转盘的双台风实验、以及拟开展的数值实验,将探索和检验“能否将台风看作是大尺度三维螺旋孤立子”,这既是研究台风和孤立子的形成机制及其动力学特性的一种新的思路,也是将二者融合起来研究的方法论的一种新途径.     

Perturbation theory for nonlinear klein-gordon equation

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅｎｏｎｌｉｎｅａｒＫｌｅｉｎ＿Ｇｏｒｄｏｎｅｑｕａｔｉｏｎｐｌａｙｓａｖｅｒｙｉｍｐｏｒｔａｎｔｒｏｌｅｉｎｎｏｎｌｉｎｅａｒｐｈｙｓｉｃｓｂｅｃａｕｓｅｏｆｉｔｓｗｉｄｅｓｐｒｅａｄａｐｐｌｉｃａｔｉｏｎｉｎｍａｎｙｆｉｅｌｄｓｏｆｐｈｙｓｉｃｓ[1,2 ].ＩｔｓｇｅｎｅｒａｌｆｏｒｍｉｓＵＴＴ-ＵＸＸ+ｍ2 Ｕ+λＵ3=0 ,( 1 )ｗｈｅｒｅｔｈｅｓｕｂｓｃｒｉｐｔｓｓｔａｎｄｆｏｒｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｔｉｏｎｗｉｔｈｒｅｓｐ…     

Motion of strings, embedding problem and soliton equations

The motion of a flexible string of constant length in E ³ in interaction, corresponding to a variety of physical situations, is considered. It is pointed out that such a system could be studied in terms of the embedding problem in differential geometry, either as a moving helical space curve in E ³ or by the embedding equations of two dimensional surfaces in E ³. The resulting integrability equations are identifiable with standard soliton equations such as the non-linear Schrödinger, modified K-dV, sine-Gordon, Lund-Regge equations, etc. On appropriate reductions the embedding equations in conjunction with suitable local space-time and/or gauge symmetries reproduce the AKNS-type eigenvalue equations and Riccati equations associated with soliton equations. The group theoretical properties follow naturally from these studies. Thus the above procedure gives a simple geometric interpretation to a large class of the soliton possessing nonlinear evolution equations and at the same time solves the underlying string equations.     

Two-dimensional anisotropic vortex–bright soliton and its dynamics in dipolar Bose–Einstein condensates in optical lattice

We study construction and dynamics of two-dimensional (2D) anisotropic vortex–bright (VB) soliton in spinor dipolar Bose–Einstein condensates confined in a 2D optical lattice (OL), with two localized components linearly mixed by the spin–orbit coupling and long-range dipole–dipole interaction (DDI). It is found that the OL and DDI can support stable anisotropic VB soliton in the present setting for arbitrarily small value of norm N. We then present a new method via examining the mean square error of norm share of bright component to implement stability analysis. It is revealed that one can control the stability of anisotropic VB soliton only by adjusting OL depth for a fixed DDI. In addition, the dynamics of the anisotropic VB soliton was studied by applying the kick to them. The mobility of the single kicked VB soliton is Rabbi-like oscillation. However, for the collision dynamics of two kicked anisotropic VB solitons, their properties mainly depend on their initial distance and OL, and they can realize the transition from the bright component to the vortex component. Our work may provide a convenient way to prepare and manipulate anisotropic VB soliton in high-dimensional space.

     

Nonlocal symmetry and similarity reductions for a $$\varvec{}$$-dimensional Korteweg–de Vries equation

Based on the Lax pair, the nonlocal symmetries to \((2+1)\)-dimensional Korteweg–de Vries equation are investigated, which are also constructed by the truncated Painlevé expansion method. Through introducing some internal spectrum parameters, infinitely many nonlocal symmetries are given. By choosing four suitable auxiliary variables, nonlocal symmetries are localized to a closed prolonged system. Via solving the initial-value problems, the finite symmetry transformations are obtained to generate new solutions. Moreover, rich explicit interaction solutions are presented by similarity reductions. In particular, bright soliton, dark soliton, bell-typed soliton and soliton interacting with elliptic solutions are found. Through computer numerical simulation, the dynamical phenomena of these interaction solutions are displayed in graphical way, which show meaningful structures.     

Soliton-like solutions of a derivative nonlinear Schrödinger equation with variable coefficients in inhomogeneous optical fibers

Under investigation in this paper is a derivative nonlinear Schrödinger equation with variable coefficients, which governs the propagation of the subpicosecond soliton pulses in inhomogeneous optical fibers. Through the nonisospectral Kaup–Newell scheme, the Lax pair is constructed with some constraints on the variable coefficients. Under the integrable conditions, bright one- and multi-soliton-like solutions are derived via the Hirota method. By suitably choosing the dispersion coefficient function, several types of inhomogeneous solitons are obtained in, respectively: (1) exponentially decreasing dispersion profile, (2) linearly decreasing dispersion profile, (3) exponentially increasing dispersion profile, and (4) periodically fluctuating dispersion profile. The intensity of the inhomogeneous soliton can be controlled by means of modifying the loss/gain term. Asymptotic analysis of the two-soliton-like solution is performed, which shows that the changes of the widths, amplitudes, and energies before and after the collision are completely caused by the variable coefficients, but have nothing to do with the collision between two soliton-like envelopes. Through suitable choices of variable coefficients, figures are plotted to illustrate the collision behavior between two inhomogeneous solitons, which has some potential applications in the real optical communication systems.     

Generation and evolution of oblique solitary waves in supercritical flows

It is considered that a thin strut sits in a supercritical shallow water flow sheet over a homogeneous or very mildly varying topography. This stationary 3-D problem can be reduced from a Boussinesq-type equation into a KdV equation with a forcing term due to uneven topography, in which the transverse coordinate Y plays a same role as the time in original KdV equation. As the first example a multi-soliton wave pattern is shown by means of N-soliton solution. The second example deals with the generation of solitary wave-train by a wedge-shaped strut on an even bottom. Whitham's average method is applied to show that the shock wave jump at the wedge vertex develops to a cnoidal wave train and eventually to a solitary wavetrain. The third example is the evolution of a single oblique soliton over a periodically varying topography. The adiabatic perturbation result due to Karpman & Maslov (1978) is applied. Two coupled ordinary differential equations with periodic disturbance are obtained for the soliton amplitude and phase. Numerical solutions of these equations show chaotic patterns of this perturbed soliton.