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.

Dynamical Soliton Wave Structures of One-Dimensional Lie Subalgebras via Group-Invariant Solutions of a Higher-Dimensional Soliton Equation with Various Applications in Ocean Physics and Mechatronics Engineering

Having realized various significant roles that higher-dimensional nonlinear partial differ-ential equations(NLPDEs)play in engineering,we analytically investigate in this paper,a higher-dimensional soliton equation,with applications particularly in ocean physics and mechatronics(electrical electronics and mechanical)engineering.Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differen-tial equations.In addition,we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved.Further,a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation.This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations(ODEs).On solving the achieved nonlinear differential equations,we secure various solitonic solutions.In conse-quence,these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures,ranging from periodic,kink and kink-shaped nanopteron,soliton(bright and dark)to breather waves with extensive wave collisions depicted.We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions,two dimensions and density plots.Moreover,the gained group-invariant solutions involved several arbitrary functions,thus exhibiting rich physical structures.We also implore the power series technique to solve part of the complicated differential equa-tions and give valid comments on their results.Later,we outline some applications of our results in ocean physics and mechatronics engineering.

Fractional Breaking Soliton Equation Reduced from a Linear Spectral Problem Associated with Fractional Self-Dual Yang-Mills Equations

Fractional or fractal calculus is everywhere and very important.It is reported that the fractal approach is suitable for insight into the effect of porous structure on thermo-properties of cloth.A novel local fractional breaking soliton equation is derived from the reduction of the linear spectral problem associated with the local fractional non-isospectral self-dual Yang-Mills equations.More specifically,the employed linear spectral problem is first reduced to the(2+1)-dimensional local fractional zero-curvature equation through variable transformations.Based on the reduced local fractional zero-curvature equation,the fractional breaking soliton equation is then constructed by the method of undetermined coefficients.This paper shows that some other local fractional models can be obtained by generalizing the existing methods of generating nonlinear partial differential equations with integer orders.

On a Formation of Singularities of Solutions to Soliton Equations Represented by L,A,B-triples

We discuss the mechanism of formation of singularities of solutions to the Novikov-Veselov,modified Novikov-Veselov,and Davey-Stewartson II(DSII)equations obtained by the Moutard type transformations.These equations admit the L,A,B-triple presentation,the generalization of the L,Apairs for 2+1-soliton equations.We relate the blow-up of solutions to the non-conservation of the zero level of discrete spectrum of the L-operator.We also present a class of exact solutions,of the DSII system,which depend on two functional parameters,and show that all possible singularities of solutions to DSII equation obtained by the Moutard transformation are indeterminancies,i.e.,points when approaching which in different spatial directions the solution has different limits.

Singular and non-topological soliton solutions for nonlinear fractional differential equations

In this article, the fractional derivatives are described in the modified Riemann-Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations（FDEs） based on a fractional complex transform and apply it to solve nonlinear space-time fractional equations. As a result, the non-topological as well as the singular soliton solutions are obtained. This method can be suitable and more powerful for solving other kinds of nonlinear fractional FDEs arising in mathematical physics.

A deep learning method for solving high-order nonlinear soliton equations

We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.

Soliton solutions,travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics

Many physical systems can be successfully modelled using equations that admit the soliton solutions.In addition,equations with soliton solutions have a significant mathematical structure.In this paper,we study and analyze a three-dimensional soliton equation,which has applications in plasma physics and other nonlinear sciences such as fluid mechanics,atomic physics,biophysics,nonlinear optics,classical and quantum fields theories.Indeed,solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour.We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time.Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function,elliptic functions,elementary trigonometric and hyperbolic functions solutions of the equation.Besides,various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique.These solutions comprise dark soliton,doubly-periodic soliton,trigonometric soliton,explosive/blowup and singular solitons.We further exhibit the dynamics of the solutions with pictorial representations and discuss them.In conclusion,we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula.We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.

New lump, lump-kink, breather waves and other interaction solutions to the (3+1)-dimensional soliton equation

This study investigates the (3+1)-dimensional soliton equation via the Hirota bilinear approach and symbolic computations. We successfully construct some new lump, lump-kink, breather wave, lump periodic, and some other new interaction solutions. All the reported solutions are verified by inserting them into the original equation with the help of the Wolfram Mathematica package. The solution’s visual characteristics are graphically represented in order to shed more light on the results obtained. The findings obtained are useful in understanding the basic nonlinear fluid dynamic scenarios as well as the dynamics of computational physics and engineering sciences in the related nonlinear higher dimensional wave fields.

Stable-range Approach to the （2＋1）-dimensional Breaking Soliton and Kadomtsev-Petviashvili Equations

By using Xu's stable-range method,families of explicit exact solutions with multiple parameter functions for the(2+1)-dimensional breaking soliton and KadomtsevPetviashvili equations.These parameter functions make our solutions more applicable to related practical models and boundary value problems.

SITEM for the conformable space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations

In the present paper,new analytical solutions for the space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations are obtained by using the simplified tan(φ(ξ)2)-expansion method.Here,fractional derivatives are defined in the conformable sense.To show the correctness of the obtained traveling wave solutions,residual error function is defined.It is observed that the new solutions are very close to the exact solutions.The solutions obtained by the presented method have not been reported in former literature.

A HIERARCHY OF INTEGRABLE LATTICE SOLITON EQUATIONS AND ITS INTEGRABLE SYMPLECTIC MAP

A hierarchy of integrable lattice soliton equations and its Hamiitonian struc ture associated a 3×3 matrix spectral problem are got. An integrable symplectic map is obtained by nonlinearization of Lax pairs and ad joint Lax pairs of the hierarchy. Moreover, the solutions to the prototype system of lattice equations in the hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map.

THE INTEGRABILITY AND PARAMETRIC REPRESENTATION OF A HIERARCHY OF SOLITON EQUATIONS

The parametric representation for finite-band solutions of a stationary soliton equation is discussed. This parametric representation can be represented as a Hamiltonian system which is integrable in Liouville sense. The nonconfocal involutive integral representations {Fm} are obtained also The finite-band solutions of the soliton equation can be represented as the solutions of two set of ordinary differential equations.

Soliton Solutions to the Coupled Gerdjikov-Ivanov Equation with Rogue-Wave-Like Phenomena

Bilinear forms of the coupled Gerdjikov–Ivanov equation are derived. The $N$-soliton solutions to the equation are obtained by Hirota＇s method. It is interesting that the two-soliton solutions can generate the rogue-wave-like phenomena by selecting special parameters. The equation can be reduced to the Gerdjikov–Ivanov equation as well as its bilinear forms and its solutions.

EXISTENCE AND NONEXISTENCE FOR THE INITIAL BOUNDARY VALUE PROBLEM OF ONE CLASS OF SYSTEM OF MULTIDIMENSIONAL NONLINEAR SCHRDINGER EQUATIONS WITH OPERATOR AND THEIR SOLITON SOLUTIONS

The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type of equations, which are satisfied by transverse velocity of higher frequency electron, as we study soliton in plasma physics. In this paper, under some condition we study the existence and nonexistence for this equations in the cases possessing different signs in nonlinear term.

Hierarchy of Lax Integrable Lattice Equations, Darboux Transformation and Conservation Laws

A discrete isospectral problem and the associated hierarchy of Lax integrable lattice equations were investigated. A Darboux transformation for the discrete spectral problem was found. Finally, an infinite number of conservation laws were given for the corresponding hierarchy.

Exact Solutions of Non-isospectral sine-Gordon Equation with Self-consistent Sources

The non-isospectral sine-Gordon equation with self-consistent sources is derived.Its solutions are obtainedby means of Hirota method and Wronskian technique,respectively.Non-isospectral dynamics including one-solitoncharacteristics,two-soliton scattering,and ghost solitons,are investigated.

Propagation of an electromagnetic soliton in an anisotropic biquadratic ferromagnetic medium

Information storage technology based on anisotropic ferromagnets with sufficiently high magneto-optical effects has received much attention in recent years.Magneto-optical recording combines the merits of magnetic and optical techniques.We investigate the magneto-optical effects on a biquadratic ferromagnet and show that the dynamics of the system are governed by a perturbed nonlinear Schro¨dinger equation.The evolutions of amplitude and velocity of the soliton are found to be time independent,thereby admitting the lossless propagation of the electromagnetic soliton in the medium,which may have potential applications in soliton based optical communication systems.We also exploit the role of perturbation,which has a significant impact on the propagation of an electromagnetic soliton.

An infinity of conservation laws of fKdV equation

An infinity of conservation laws of fKdV equation is derived in terms of the Miura and Gardner's transform. The pseudo-mass and energy theorems are studied by the first two conservation laws. As a typical example, the theoretical mean wave resistance and the regional distribution of energy of the precursor soliton generation are determined by means of the first and the second conservation laws.

Three-dimensional Bose Einstein condensate vortex solitons under optical lattice and harmonic confinements

We predict three-dimensional vortex solitons in a Bose-Einstein condensate under a complex potential,which is the combination of a two-dimensional parabolic trap along the transverse radial direction and a one-dimensional optical-lattice potential along the z axis direction.The vortex solitons are built in the form of a layer-chain structure made of several fundamental vortices along the optical-lattice direction.This has not been reported before in the three-dimensional Bose-Einstein condensate.By using a combination of the energy density functional method with direct numerical simulation,we find three-dimensional vortex solitons with topological charges χ=1,χ=2,and χ=3.Moreover,the macroscopic quantum tunneling and chirp phenomena of the vortex solitons are shown in the evolution.Therein,the occurrence of macroscopic quantum tunneling provides the possibility for the experimental realization of quantum tunneling.Specifically,we successfully manipulate the vortex solitons along the optical lattice direction.The stability limits for dragging the vortex solitons from an initial fixed position to a prescribed location are further pursued.

Propagation of traveling wave solutions for nonlinear evolution equation through the implementation of the extended modi ed direct algebraic method

In this work,di erent kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional(3-D)nonlinear evolution equations(NEEs)through the implementation of the modi ed extended direct algebraic method.Bright-singular and dark-singular combo solitons,Jacobi's elliptic functions,Weierstrass elliptic functions,constant wave solutions and so on are attained beside their existing conditions.Physical interpretation of the solutions to the 3-D modi ed KdV-Zakharov-Kuznetsov equation are also given.