Soliton Equations Research Articles

Solitary Wave Solutions of Chafee-Infante Equation and (2+1)-Dimensional Breaking Soliton Equation by the Improved Kudryashov Method

In this paper, we apply the improved Kudryashov method for finding exact solution and then solitary wave solutions of the Chafee-Infante equation and (2+1)-dimensional breaking soliton equation, where mathematical software Maple-13 is used as an important mathematical tool for removing calculation complexity, justification of the solutions and its graphical representations.

Lump and interactional solutions of the (2+1)-dimensional generalized breaking soliton equation

In this paper, by using the long wave limit method, we study lump solution and interactional solution of the (2[Formula: see text]+[Formula: see text]1)-dimensional generalized breaking soliton equation without using bilinear form. The moving properties of the lump solution, and the interactional properties of a lump and a solitary wave, are analyzed theoretically and graphically with asymptotic analysis.

Matrix Integral Solutions to the q-Difference Two-Dimensional Toda Lattice Equation and Its Pfaffianized System

Matrix integrals used in random matrix theory have been revealed to provide τ-functions for several hierarchies of soliton equations. In this article, we extend this relation by showing that such i...

On a vector long wave‐short wave‐type model

AbstractA new vector long wave‐short wave‐type model is proposed by resorting to the zero‐curvature equation. Based on the resulting Riccati equations related to the Lax pair and the gauge transformations between the Lax pairs, multifold Darboux transformations are constructed for the vector long wave‐short wave‐type model. This method is general and is suitable for constructing the Darboux transformations of other soliton equations, especially in the absence of symmetric conditions for Lax pairs. As an illustrative example of the application of the Darboux transformations, exact solutions of the two‐component long wave‐short wave‐type model are obtained, including solitons, breathers, and rogue waves of the first, second, third, and fourth orders. All the solutions derived by the Darboux transformations involve square roots of functions, which is not observed in the investigation of other nonlinear integrable equations. This model describes new nonlinear phenomena.

Soliton surfaces induced by the Fokas-Lenells equation

In this paper, we study the application of the theory of solitons in differential geometry. The recently proposed soliton equation, which is Fokas-Lenells equation, has been investigated, and its two-dimensional soliton surface in the three-dimensional Euclidean space (R2 ! R3) has been constructed. Thus the connection between the Fokas-Lenells equation and the surface was established by using the Sym-Tafel formula. We find the first and the second quadratic forms, surface area, and Gaussian curvature. The obtained results have various applications in mathematical physics, the geometry of curves and the theory of surfaces.

Lump-type solutions, rogue wave type solutions and periodic lump-stripe interaction phenomena to a (3 + 1)-dimensional generalized breaking soliton equation

In this paper, we present a bilinear form for an extended (2+1)-dimensional generalized breaking soliton equation, namely, a (3+1)-dimensional generalized breaking soliton (GBS) equation, through using the Hirota direct method. Abundant lump-type solutions, rogue wave type solutions, breather lump wave solutions and interaction solutions are constructed, based on the Hirota bilinear form. Particularly, we generate a type of new interaction solutions in terms of a new combination of quadratic function, trigonometric function and exponential function, namely, a kind of periodic lump-stripe solitons, which is a sort of mixed type solutions of periodic lump-type solitons and stripe solitons. We show that the periodic lump-type solitons will be swallowed by the stripe solitons after they collide. Finally, dynamics characteristics and evolution behaviors are exhibited for the obtained solution waves through particular plots with proper choices of different values for the parameters.

Quantum theory of laser soliton

The Heisenberg–Langevin equation for a spatial laser soliton in a wide-aperture laser with saturable absorption is constructed within the framework of consistent quantum electrodynamics. We discuss in detail the canonical variables for the generation field and the material two-level medium, consisting of centres providing amplification and absorption. It is assumed that laser generation evolves in time much more slowly than an atomic media. This assumption makes it possible to apply the adiabatic approximation and construct a closed equation for the amplitude of a laser field. Much attention is paid to the formulation of Langevin sources when deriving the equation since they play a decisive role in the formation of solitons’ quantum statistical features. To provide an appropriate procedure for observing the quantum squeezing of a soliton, synchronization of laser generation by an external weak electromagnetic field is considered. Here we also present the results of the analysis of a classical laser soliton (neglecting the quantum fluctuations), which serves as the basis for further consideration of quantum effects.

N-soliton and rogue wave solutions of (2+1)-dimensional integrable system with Lax pair

A new generalized (2[Formula: see text]+[Formula: see text]1)-dimensional model, obtained from the Kadomtsev–Petviashvili (KP) equation considered as a system of nonlinear evolution partial differential equations (PDEs), is introduced. With the usage of the Hirota bilinear method and the KP-hierarchy reduction method, N-soliton solutions of the integrable system are constructed. Considering the case of s[Formula: see text]=[Formula: see text]−1 in the linear differential operators, L1 and L2, and a specific set of parameters, two dark solitons, mixed solutions consisting of soliton-type and periodic waves solution are obtained. Based on the particular definition of the matrix elements, one and two rogue waves solutions expressed in terms of rational functions are derived. It is shown that the fundamental rogue waves are line rogue waves, which is different from the property of the moving line solitons of the soliton equations.

Coupled Kähler–Ricci solitons on toric Fano manifolds

We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled K\"ahler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton type equations on toric Fano manifolds. Some of these solutions provide natural candidates for the large time limits of a certain geometric flow generalizing the K\"ahler-Ricci flow.

Resonant multiple wave solutions to some integrable soliton equations**Project supported by the Yue-Qi Scholar of the China University of Mining and Technology (Grant No. 102504180004) and the 333 Project of Jiangsu Province, China (Grant No. BRA2018320).

To transform the exponential traveling wave solutions to bilinear differential equations, a sufficient and necessary condition is proposed. Motivated by the condition, we extend the results to the (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation, the (3+1)-dimensional generalized Kadomtsev–Petviashvili (g-KP) equation, and the B-type Kadomtsev–Petviashvili (BKP) equation. Aa a result, we obtain some new resonant multiple wave solutions through the parameterization for wave numbers and frequencies via some linear combinations of exponential traveling waves. Finally, these new resonant type solutions can be displayed in graphs to illustrate the resonant behaviors of multiple wave solutions.

Anisotropic flow of convex hypersurfaces by the square root of the scalar curvature

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor ψ when the strictly convex initial hypersurface in Euclidean space is suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. For a general case in dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.

Ricci Solitons and Killing Fields on Generalized Cahen—Wallach Manifolds

We study Ricci solitons and Killing fields on generalized Cahen–Wallach manifolds. The Ricci soliton equation provides a generalization of the Einstein equation on (pseudo-)Riemannian manifolds which is closely connected with Ricci flows. We prove that the Ricci soliton equation is locally solvable with any constant in the Ricci soliton equation on generalized Cahen–Wallach manifolds. Using a Brinkmann coordinate system, we study the Killing fields on these manifolds and give constraints on the dimension of the space of Killing fields. Also, we obtain solutions to the Killing equations for 2-symmetric Lorentzian manifolds in small dimensions.

Periodic type and periodic cross-kink wave solutions to the (2 + 1)-dimensional breaking soliton equation arising in fluid dynamics

In this paper, we have acquired the periodic type and cross-kink wave solutions. In this paper, we use the Hirota bilinear method. With the help of the symbolic calculation and applying the used method, we solve the (2[Formula: see text]+[Formula: see text]1)-dimensional Breaking Soliton (BS) equation. We obtain some periodic wave and cross-kink wave that have greatly enriched the existing literature on the BS equation. All solutions have been verified back into its corresponding equation with the aid of the Maple package program via the three-dimensional images and density images with the help of Maple, the physical characteristics of these waves are described very well. These will be widely used to describe many interesting physical phenomena in the fields of gas, plasma, optics, acoustics, heat transfer, fluid dynamics, classical mechanics and so on.

Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

Radial basis function-based quasi-interpolation performs efficiently in high-dimensional approximation and its applications, which can attain the approximant and its derivatives directly without solving any large-scale linear system. In this paper, the bivariate multi-quadrics (MQ) quasi-interpolation is used to simulate two-dimensional (2-D) Burgers’ equation. Specifically, the spatial derivatives are approximated by using the quasi-interpolation, and the time derivatives are approximated by forward finite difference method. One advantage of the proposed scheme is its simplicity and easy implementation. More importantly, the proposed scheme opens the gate to meshless adaptive moving knots methods for the high-dimensional partial differential equations (PDEs) with shock or soliton waves. The scheme is also applicable to other non-linear high-dimensional PDEs. Two numerical examples of Burgers’ equation (shock wave equation) and one example of the Sine–Gordon equation (soliton wave equation) are presented to verify the high accuracy and efficiency of this method.

Lie symmetry reductions and dynamics of solitary wave solutions of breaking soliton equation

In this paper, some new invariant solutions of breaking soliton (BS) equation have been derived by using similarity transformations method. The system represents the interaction of Riemann waves propagating along [Formula: see text]-axis and long waves along [Formula: see text]-axis. The commutative relation and symmetry analysis of BS equation are derived using Lie group theory. Meanwhile, the method reduces the number of independent variables by one in each step. A repeated application of similarity transformations method reduces the BS equation into overdetermined equations, which provide invariant solutions. The derived results are more general than previous findings. The obtained solutions are supplemented by numerical simulation, which makes this research physically meaningful. Eventually, doubly soliton, multisoliton and asymptotic profiles of solutions are analyzed in the analysis and discussion section.

Some applications of the modified (G′/G2)-expansion method in mathematical physics

Abstract This study proposes a modified ( G ′ / G 2 ) -expansion method to seek new more general exact traveling wave solutions to nonlinear evolution equations (NLEEs). To demonstrate the validity of this method, two interesting NLEEs are considered: the (2 + 1)-dimensional typical breaking soliton equation and (1 + 1)-dimensional classical Boussinesq system of equations. Consequently, we construct some new exact solutions involving the free parameters of the considered equations, which are categorized into three unique forms including rational, periodic, and hyperbolic functions. Thus, this study illustrates the effectiveness and the simplicity of the proposed method with the aid of symbolically computational software such as MATHEMATICA.

Characteristics of integrability, bidirectional solitons and localized solutions for a ( $$3+1$$ 3 + 1 )-dimensional generalized breaking soliton equation

The characteristics of integrability, bidirectional solitons and localized solutions are investigated for a ( $$3+1$$ )-dimensional breaking soliton (GBS) equation with general forms. Firstly, starting from the GBS equation, we perform the singularity manifold analysis and obtain a new integrable model in the sense of Painleve property. Secondly, taking advantage of the Bell polynomial approach, we construct the Backlund transformation, Lax pair and an infinite sequence of conservation laws. Subsequently, this new equation is also found to allow bidirectional soliton solutions, and the head-on and overtaking collisions between solitons are illustrated by some illustrative graphs. Finally, some localized excitations, such as lump solution, multi-dromions, periodic solitary waves solution, are obtained.

Lump-type solutions and interaction phenomenon to the (2+1)-dimensional Breaking Soliton equation

In this article, we use the Hirota bilinear method. With the help of the symbolic calculation and applying the used method, we solve the (2+1)-dimensional Breaking Soliton (BS) equation. We obtain some interaction between lump soliton and solitary wave, the interaction between lump soliton and periodic wave, breather-type periodic soliton, periodic kink-wave, kink-soliton wave, and solitary wave solutions. All solutions have been verified back into its corresponding equation with the aid of Maple package program. The graphical representation of the solution is given by Maple and physically interpreted. The obtained results are useful in gaining the understanding of the various nonlinear scenarios in fluid dynamics and also the dynamics of higher dimensional nonlinear wave fields related to mathematical physics and engineering sciences.

Discrete and Ultradiscrete Periodic Phase Soliton Equations

We propose new type of discrete and ultradiscrete soliton equations, which admit extended soliton solution called periodic phase soliton solution. The discrete equation is derived from the discrete DKP equation and the ultradiscrete one is obtained by applying the ultradiscrete limit. The soliton solutions have internal freedom and change their shape periodically during propagation. In particular, the ultradiscrete solution reduces into the solution to the ultradiscrete hungry Lotka-Volterra equation in a special case.

Radially ingoing and outgoing dispersive and dissipative structures in electron‐ion plasmas in the presence of electron trapping

In this article, non‐linear propagation of ingoing and outgoing electrostatic waves on the ion time scale in an unmagnetized, non‐relativistic electron‐ion (ei) plasma in the presence of warm ions, ion kinematic viscosity, and trapped Maxwellian electrons was examined in a non‐planar geometry. In the weak non‐linearity limit, modified soliton and shock equations were derived with the inclusion of electron trapping in cylindrical and spherical geometries. The finite difference method was used to solve all these equations in the non‐planar geometries using the planar versions of these equations as an initial input. The results were compared with their counterparts with quadratic non‐linearity and the main differences were expounded. It was shown that the spatio‐temporal scales over which the shocks form for the non‐planar trapped Burgers equation are much shorter by comparison with the shocks admitted by the non‐planar trapped Korteweg de Vries Burgers equation. It was also found that unlike their non‐linear shock counterparts, the solitary structures admitted by the non‐planar trapped Korteweg de Vries equation exhibit a phase shift.