Consistent Riccati expansion solvable classification and soliton-cnoidal wave interaction solutions for an extended Korteweg-de Vries equation

In this paper, we consider an extended Korteweg-de Vries (KdV) equation. Using the consistent Riccati expansion (CRE) method of Lou, the extended KdV equation is proved to be CRE solvable in only two distinct cases. These two CRE solvable models are the KdV-Lax and KdV-Sawada-Kotera (KdV-SK) equations. In addition, applying the nonauto-Bäcklund transformations which are provided by the CRE method, we present the explicit construction for soliton-cnoidal wave interaction solutions which represent a soliton propagating on a cnoidal periodic wave background in the KdV-Lax and KdV-SK equations, respectively.     

Soliton molecules and the CRE method in the extended mKdV equation

The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition, which is called velocity resonance. One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters. The soliton molecules including the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton are exhibited in figures 1–3, respectively.Meanwhile, the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevé method. The consistent Riccati expansion(CRE) method is applied to the extended mKdV equation. It demonstrates that the extended mKdV equation is a CRE solvable system. A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.     

An integrable coupled Alice–Bob modified Korteweg de-Vries system: Lax pairs,Bäcklund transformations,residual symmetries and exact solutions

ABSTRACT

A coupled Alice–Bob modified Korteweg de-Vries (mKdV) system is established from the mKdV equation in this paper, which is nonlocal and suitable to model two-place entangled events. The Lax integrability of the coupled Alice–Bob mKdV system is proved by demonstrating three types of Lax pairs. By means of the truncated Painlevé expansion, auto-Bäcklund transformation of the coupled Alice–Bob mKdV system and Bäcklund transformation between the coupled Alice–Bob mKdV system and the Schwarzian mKdV equation are demonstrated. Nonlocal residual symmetries of the coupled Alice–Bob mKdV system are researched. To obtain localized Lie point symmetries of residual symmetries, the coupled Alice–Bob mKdV system is extended to a system consisting six equations. Calculation on the prolonged system shows that it is invariant under the scaling transformations, space-time translations, phase translations and Galilean translations. One-parameter group transformation and one-parameter subgroup invariant solutions are obtained. The consistent Riccati expansion (CRE) solvability of the coupled Alice–Bob mKdV system is proved and some interaction structures between soliton–cnoidal waves are obtained by CRE. Moreover, Jacobi periodic wave solutions, solitary wave solutions and singular solutions are obtained by elliptic function expansion and exponential function expansion.     

B?cklund transformations,consistent Riccati expansion solvability,and soliton–cnoidal interaction wave solutions of Kadomtsev–Petviashvili equation

The famous Kadomtsev-Petviashvili(KP)equation 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlev6 expansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE) solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.     

A New Integrable (2+1)-Dimensional Generalized Breaking Soliton Equation: N-Soliton Solutions and Traveling Wave Solutions

In this work, we study a new (2+1)-dimensional generalized breaking soliton equation which admits the Painleve property for one special set of parameters. We derive multiple soliton solutions, traveling wave solutions, and periodic solutions as well. We use the simplified Hirotas method and a variety of ansatze to achieve our goal.     

Nonlocal symmetry and consistent Riccati expansion integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system

Under investigation in this paper is nonlocal symmetry, consistent Riccati expansion (CRE) integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system, which can be used to describes a bidirectional soliton for wave propagation. We construct the Bäcklund transformation and consider the truncated Painlevé expansion of the system. It’s Schwarzian form is derived, whose nonlocal symmetry is localized to provide the corresponding nonlocal group. Furthermore, we verify that the system is solvable via the CRE. Based on the CRE, we further present its soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral.     

Nonlocal Symmetry and Interaction Solutions of a Generalized Kadomtsev-Petviashvili Equation

A generalized Kadomtsev-Petviashvili equation is studied by nonlocal symmetry method and consistent Riccati expansion (CRE) method in this paper. Applying the truncated Painlevé analysis to the generalized Kadomtsev-Petviashvili equation, some Bäcklund transformations (BTs) including auto-BT and non-auto-BT are obtained. The auto-BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion. Then the nonlocal symmetry is localized to the corresponding nonlocal group by introducing two new variables. Further, by applying the Lie point symmetry method to the prolonged system, a new type of finite symmetry transformation is derived. In addition, the generalized Kadomtsev-Petviashvili equation is proved consistent Riccati expansion (CRE) solvable. As a result, the soliton-cnoidal wave interaction solutions of the equation are explicitly given, which are difficult to be found by other traditional methods. Moreover, figures are given out to show the properties of the explicit analytic interaction solutions.     

Lump and lump-soliton interaction solutions for an integrable variable coefficient Kadomtsev-Petviashvili equation

For a variable coefficient Kadomtsev-Petviashvili (KP) equation the Lax pair as well as conjugate Lax pair are derived from the Painleve analysis.The N-fold binary Darboux transformation is presented in a compact form.As an application,the multi-lump,higher-order lump and general lump-soliton interaction solutions for the variable coefficient KP equation are obtained.Typical lump structures with amplitudes exponentially decaying to zero as the time tends to infinity and interactions between one lump and one soliton are shown.     

CTE Solvability,Nonlocal Symmetries and Exact Solutions of Dispersive Water Wave System

A consistent tanh expansion （CTE） method is developed for the dispersion water wave （DWW） system. For the CTE solvable DlVVC system, there are two branches related to tanh expansion, the main branch is consistent while the auxiliary branch is not consistent. From the consistent branch, we can obtain infinitely many exact significant solutions including the soliton-resonant solutions and soliton-periodic wave interactions. From the inconsistent branch, only one special solution can be found. The CTE related nonlocal symmetries are also proposed. The nonlocai symmetries can be localized to find finite Backlund transformations by prolonging the model to an enlarged one.     

Residual symmetry,interaction solutions,and conservation laws of the(2+1)-dimensional dispersive long-wave system

We explore the(2+1)-dimensional dispersive long-wave(DLW) system. From the standard truncated Painlev′e expansion, the B¨acklund transformation(BT) and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the(2+1)-dimensional DLW system is consistent Riccati expansion(CRE) solvable. If the special form of(CRE)-consistent tanh-function expansion(CTE) is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.     

New Exact Solutions of （1 ＋ 1）-Dimensional Coupled Integrable Dispersionless System

This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more generalvariable separation solutions with two arbitrary functions of (1+1)-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p(x) and q(t) inour solutions, the annihilation phenomena of the flat-basin soliton, arch-basin soliton, and flat-top soliton are discussed.     

Exact Solutions of a Coupled Burgers System

The exact solutions of a new coupled Burgers system are studied in three different ways. The first type of solutions are found thanks to the coupled Burgers system possessing a simple single Burgers reduction. The second type of multiple soliton solutions are revealed via the decouple procedure. The third type of exact solutions are found by means of a prior ansatz and solutions of the heat conduction equation. Two different kinds of soliton fission phenomena of the model are discovered and a special type of completely elastic soliton collision without phase shift of the model is also displayed.     

Chirped and chirp-free self-similar cnoidal and solitary wave solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients

By means of the similarity transformation connecting with the solvable stationary cubic-quintic nonlinear Schrödinger equation (CQNLSE), we construct explicit chirped and chirp-free self-similar cnoidal wave and solitary wave solutions of the generalized CQNLSE with spatially inhomogeneous group velocity dispersion (GVD) and amplification or attenuation. As an example, we investigate their propagation dynamics in a soliton control system.     

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq–Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq–Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.     

Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation

We investigated the soliton solution for N$N$ coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz–Kaup–Newell–Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management.     

Deriving the New Traveling Wave Solutions for the Nonlinear Dispersive Equation,KdV-ZK Equation and Complex Coupled KdV System Using Extended Simplest Equation Method

In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.     

Matter wave solitons in coupled system with external potentials

We present Lax-pair corresponding to the coupled Gross-Pitaevskii equation (CGPE) which governs the evolution of the macroscopic wave function of two components Bose-Einstein condensates trapped in time-dependent harmonic potential. Kinds of soliton solutions can be derived from the Lax-pair through Darboux transformation conveniently. Furthermore, soliton management in two-component Bose-Einstein condensate would be realized base on that the shape and motion of soliton in both components are investigated analytically. Moreover, it is found that there is a transformation existed between the nonautonomous coupled system and Manakov model.     

CRE Method for Solving m Kd V Equation and New Interactions Between Solitons and Cnoidal Periodic Waves

In nonlinear physics, the modified Korteweg de-Vries(m Kd V) as one of the important equation of nonlinear partial differential equations, its various solutions have been found by many methods. In this paper, the CRE method is presented for constructing new exact solutions. In addition to the new solutions of the m Kd V equation, the consistent Riccati expansion(CRE) method can unearth other equations.     

Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation

The real, nonsingular elliptic solutions of the Korteweg-de Vries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic-Quintic Nonlinearities

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.