New interaction solutions and nonlocal symmetry of an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form

The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevéexpansion method. The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables. The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly. Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.     

Fusion,fission, and annihilation of complex waves for the(2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff system

With the help of the symbolic computation system, Maple and Riccati equation( ξ= a0+ a1ξ+ a22ξ), expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx + my + nt + Γ(x, y,t) for the(2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff system(GCBS) are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

New exact solutions of the generalized Zakharov–Kuznetsov modified equal-width equation

In this paper, new exact solutions, including soliton, rational and elliptic integral function solutions, for the generalized Zakharov–Kuznetsov modified equal-width equation are obtained using a new approach called the extended trial equation method. In this discussion, a new version of the trial equation method for the generalized nonlinear partial differential equations is offered.     

Lie point symmetries,conservation laws and exact solutions of ($$1+ n$$)-dimensional modified Zakharov–Kuznetsov equation describing the waves in plasma physics

An approximate solution for the nuclear Hulthén plus atomic Hulthén potentials is constructed by solving the associated Volterra integral equation by series substitution method. Within the framework of supersymmetry-inspired factorisation method, this solution is exploited to construct higher partial wave interactions. The merit of our approach is examined by computing elastic scattering phases of the \(\alpha {-}\alpha \) system by the judicious use of phase function method. Reasonable agreements in phase shifts are obtained with standard data.     

Painlev property of the modified C-KdV equation and its exact solutions

In this paper,the Painlev’e properties of the modified C-KdV equation are verified by using the W-K algorithm.Then some exact soliton solutions are obtained by applying the standard truncated expansion method and the nonstandard truncated expansion method with the help of Maple software,respectively.     

The exact solutions to (2 1)-dimensional nonlinear Schrdinger equation

1　IntroductionConsiderthe ( 2 1 ) dimensionalnonlinearSchr dingerequation (NLSE)iψt =ψxy γ2 vψ ( 1 )vx =2 ( ｜ψ｜2 ) y ( 2 )whereγ2 =± 1 .Eqs.( 1 )and ( 2 )playimportantroleinnonlinearopticalphysicalfield (see ,e .g .[1 ]) .InRef.[2 ],ZakharovappliedinversescatteringmethodtoderivethesolitonsolutionsforEqs.( 1 )and ( 2 ) ,andNsolitonsolutionscanbefoundinRef.[3].InRef.[4],RadhaandlakshmananhasinvestigatedthePainlev啨ｐｒｏｐｅｒｔｉｅｓｆｏｒＥｑｓ.( 1 )and ( 2 ) .Thehiddensymme…     

Resonance Y-type soliton solutions and some new types of hybrid solutions in the(2+1)-dimensional Sawada–Kotera equation

In this paper, based on N-soliton solutions, we introduce a new constraint among parameters to find the resonance Y-type soliton solutions in (2+1)-dimensional integrable systems. Then, we take the (2+1)-dimensional Sawada–Kotera equation as an example to illustrate how to generate these resonance Y-type soliton solutions with this new constraint. Next, by the long wave limit method, velocity resonance and module resonance, we can obtain some new types of hybrid solutions of resonance Y-type solitons with line waves, breather waves, high-order lump waves respectively. Finally, we also study the dynamics of these interaction solutions and indicate mathematically that these interactions are elastic.     

Positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations

Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations...     

Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation

This paper studies the analytical and semi-analytic solutions of the generalized Calogero–Bogoyavlenskii–Schiff(CBS) equation. This model describes the(2 + 1)–dimensional interaction between Riemann-wave propagation along the y-axis and the x-axis wave. The extended simplest equation(ESE) method is applied to the model, and a variety of novel solitarywave solutions is given. These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma. The accuracy of the obtained solution is verified using a variational iteration(VI) semi-analytical scheme. The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained. The adopted scheme's performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.     

High-order rational solutions and resonance solutions for a (3+1)-dimensional Kudryashov–Sinelshchikov equation

We mainly investigate the rational solutions and N-wave resonance solutions for the(3+1)-dimensional Kudryashov–Sinelshchikov equation, which could be used to describe the liquid containing gas bubbles. With appropriate transformations, two kinds of bilinear forms are derived. Employing the two bilinear equations, dynamical behaviors of nine district solutions for this equation are discussed in detail, including bright rogue wave-type solution, dark rogue wave-type solution, bright W-shaped solution, dark W-shaped rational solution, generalized rational solution and bright-fusion, darkfusion, bright-fission, and dark-fission resonance solutions. In addition, the generalized rational solutions, which depending on two arbitrary parameters, have an interesting structure: splitting from two peaks into three peaks.     

Chaotic solutions of (2+1)-dimensional Broek Kaup equation with variable coefficients

In this paper,an improved projective approach is used to obtain the variable separation solutions with two arbitrary functions of the (2+1)-dimensional Broek-Kaup equation with variable coefficients (VCBK). Based on the derived solitary wave solution and using a known chaotic system,some novel chaotic solutions are investigated.     

Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation

In this paper, we obtain the 1-soliton solutions of the (3?+?1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and the generalized Benjamin equation. By using two solitary wave ansatz in terms of sech ^p and $\tanh^{p}$ functions, we obtain exact analytical bright and dark soliton solutions for the considered model. These solutions may be useful and desirable for explaining some nonlinear physical phenomena in genuinely nonlinear dynamical systems.     

Painlevé analysis,infinite dimensional symmetry group and symmetry reductions for the(2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

The(2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani(KdVSKR) equation is studied by the singularity structure analysis. It is proven that it admits the Painlevé property. The Lie algebras which depend on three arbitrary functions of time t are obtained by the Lie point symmetry method. It is shown that the KdVSKR equation possesses an infinite-dimensional Kac–Moody–Virasoro symmetry algebra. By selecting first-order polynomials in t, a finitedimensional subalgebra of physical transformati...     

Abundant exact solutions for the (3+1)-dimensional generalized nonlinear Schrödinger equation with variable coefficients

The (3+1)-dimensional generalized nonlinear Schrodinger equation with variable coefficients (3D-VcgNLSE) and optical lattice is investigated. Bright and dark soliton solutions are presented by two direct ansätz. Two similar solutions are obtained in terms of the elliptic and the second type of Painlevé transcendent functions. Furthermore, hyperbolic and trigonometric solutions are studied via the G′/G-expansion method. The dynamical behaviors are demonstrated in some 3D- and contour plots.     

Self-similar cnoidal and solitary wave solutions of the (1+1)-dimensional generalized nonlinear Schrödinger equation

With the help of the similarity transformation and the solvable stationary nonlinear Schrödinger equation (NLSE),we obtain exact chirped and chirp-free self-similar cnoidal wave and solitary wave solutions of the generalized NLSE exhibitingspatial inhomogeneity, inhomogeneous nonlinearity and gain or loss at the same time. As an example, we investigate their propagationdynamics in a nonlinear optical system, and present a series of interesting properties of optical waves.     

Soliton solutions and symmetries of the 2+1 dimensional Kaup–Kupershmidt equation

The 2+1 dimensional Kaup–Kupershmidt (KK) equation is considered. A bilinear form for the equation is found and then 3-soliton solutions are obtained with the assistance of Mathematica. Six symmetries of the bilinear 2+1 dimensional KK equation are given and their symmetry algebra is identified.