Analytical Investigation of Soliton Solutions to Three Quantum Zakharov-Kuznetsov Equations

In the present paper, the two-dimensional quantum Zakharov-Kuznetsov (QZK) equation, three-dimensional quantum Zakharov-Kuznetsov equation and the three-dimensional modified quantum Zakharov-Kuznetsov equation are analytically investigated for exact solutions using the modified extended tanh-expansion based method. A variety of new and important soliton solutions are obtained including the dark soliton solution, singular soliton solution, combined dark-singular soliton solution and many other trigonometric function solutions. The used method is implemented on the Mathematica software for the computations as well as the graphical illustrations.     

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions axe derived.     

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless (CID) equations is derived. It will be shown that this bilinear form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions are derived.     

New Exact Solutions of Fractional Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations Using Fractional Sub-Equation Method

In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetsov(m ZK) equations by using fractional sub-equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie's modified Riemann–Liouville sense.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

Zakharov-Shabat Equations for Dark Soliton Solutions to the NLS Equation

For dark soliton solutions of the NLS equation, an inverse scattering transform is redeveloped. Deductions are essentially simplified in terms of an auxiliary spectral parameter from the beginning. Equations of inverse scattering transform in the form of Zakharov-Shabat are found to be simpler than those in the form of Marchenko. An explicate expression for the dark N-soliton solution and its asymptotic behaviors in the limits as t →±∞ are simply derived.     

New Type Soliton Solutions to Korteweg-de Vries and Benjamin-Bona-Mahony Equations

We study the Korteweg-de Vries equation and the Benjamin-Bona-Mahony equation, and obtain three kinds of new type soliton solutions, i.e. peakon solutions, double-peak （peaked-point and peaked-compacton） soliton solutions. A double solitary wave with blow-up points is also contained.     

Multiple Soliton Solutions of the High Order Broer-Kaup Equations

Using the extended homogeneous balance method, which is very concise and primary, we find the multiple soliton solutions of the high order Broer-Kaup equations. The method can be generalized to dealing with high-dimensional Broer-Kaup equations and other class of nonlinear equations.     

A Constructive Approach to the Soliton Solutions of Integrable Quadrilateral Lattice Equations

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Bäcklund transformation, and the method of solving a Riccati map by exploiting two known particular solutions. This leads to an expression for the N-soliton-type solutions of a generic equation within this class. As a particular instance we give an explicit N-soliton solution for the primary model, which is Adler’s lattice equation (or Q4).     

New Weierstrass Semi-rational Expansion Method to Doubly Periodic Solutions of Soliton Equations

Based on the Weierstrass elliptic function equation, a new Weierstrass semi-rational expansion method and its algorithm are presented. The main idea of the method changes the problem solving soliton equations into another one solving the corresponding set of nonlinear algebraic equations. With the aid of Maple, we choose the modified KdV equation, (2+1)-dimensional KP equation, and (3+1)-dimensional Jimbo-Miwa equation to illustrate our algorithm. As a consequence, many types of new doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Moreover the corresponding new Jacobi elliptic function solutions and solitary wave solutions are also presented as simple limits of doubly periodic solutions.     

Relationship Between Soliton-like Solutions and Soliton Solutions to a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m→1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.     

Soliton Chain Solutions to the Two-Dimensional Higher Order Korteweg-de Vries Equations of Sawada-Kotera Hierarchy

A series of two-dimensional equations with the two-dimensional Korteweg-de Vries equation being their first one are constructed as Sawada-Kotera did in one-dimensional case. Their soliton chain solutions are obtained and investigated.     

New Exact Solutions of Zakharov-Kuznetsov Equation

The Zakharov-Kuznetsov equation is proved to be nonintegrable by standard Painleve approach and three new types of soliton solutions are obtained by means of the nonstandard truncation of the extended Painleve analysis approach.     

Solutions of Fractional Partial Differential Equations of Quantum Mechanics

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox's $H$-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177-190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.     

Universal Solutions of Quantum Dynamical Yang–Baxter Equations

We construct a universal trigonometric solution of the Gervais–Neveu–Felder equation in the case of finite-dimensional simple Lie algebras and finite-dimensional contragredient simple Lie superalgebras.     

Analytic Approximations for Soliton Solutions of Short-Wave Models for Camassa-Holm and Degasperis-Procesi Equations

In this paper, the short-wave model equations are investigated, which are associated with the Camassa- Holm （CH） and Degasperis Procesi （DP） shallow-water wave equations. Firstly, by means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. Secondly, the equation is solved by homotopy analysis method. Lastly, by the transformatioas back to the original independent variables, the solution of the original partial differential equation is obtained. The two types of solutions of the short-wave models are obtained in parametric form, one is one-cusp soliton for the CH equation while the other one is one-loop soliton for the DP equation. The approximate analytic solutions expressed by a series of exponential functions agree well with the exact solutions. It demonstrates the validity and great potential of homotopy analysis method for complicated nonlinear solitary wave problems.     

Soliton solutions and interactions of the Zakharov-Kuznetsov equation in the electron-positron-ion plasmas

Analytically investigated in this paper is the Zakharov-Kuznetsov equation which describes the propagation of the electrostatic excitations in the electron-positron-ion plasmas. By means of the Hirota method and symbolic computation, the bilinear form for the Zakharov-Kuznetsov equation is derived, and then the N-soliton solution is constructed. Parametric analysis is carried out in order to illustrate that the soliton amplitude and width are affected by the phase velocity, ion-to-electron density ratio, rotation frequency and cyclotron frequency. Propagation characteristics and interaction behaviors of the solitons are also discussed through the graphical analysis. The effects of the nonlinearity A, dispersion B and disturbed wave velocity C on the amplitude and velocity of the solitons are derived. First, the amplitude is proportional to the nonlinearity A and inversely proportional to dispersion B. Second, the velocity increases as the dispersion B increases. Third, the velocity increases as the disturbed wave velocity C (4B > C) increases; the velocity decreases as the disturbed wave velocity C (4B < C) increases.     

Approximate Analytical Solutions for a Class of Laminar Boundary-Layer Equations

A simple and efficient approximate analytical technique is presented to obtain solutions to a class of two-point boundary value similarity problems in fluid mechanics. This technique is based on the decomposition method which yields a genera/analytic solution in the form of a convergent infinite series with easily computable terms. Comparative study is carried out to show the accuracy and effectiveness of the technique.     

Quantum Theory of Vibron Soliton

A quantum mechanical treatment of Takeno model for energy transport in protein is presented and the cubic anharmonicity of the hydrogen-bonded interaction is also taken into account. Under the continuum approximation, two coupled nonlinear differential equations are obtained, and then exact and approximate solitary wave solutions are found. The ideal parameters for protein are used to show whether the soliton solutions can exist for real protein molecules.