Different Kinds Of Exact Solutions Research Articles

Exact solution of the generalized Riemann problem for the Eulerian droplet model with particle pressure

In this paper, we investigate the generalized Riemann problem for the Eulerian droplet model with particle pressure. By introducing two new state variables, the generalized Riemann problem for a system of balance law (nonhomogeneous system) is neatly transferred into the Riemann problem for a system of conservation law (homogeneous system). Four different kinds of exact solutions are presented in fully explicit forms, which are given in terms of two nonconstant states separated by a shock or a generalized rarefaction wave or by a combination of such wave profiles.

Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs.

Solitary wave solution, traveling wave solution and other solutions of variable-coefficients Chiral Schrödinger equation

This paper investigates different kinds of exact solutions of variable-coefficients Chiral Schrödinger equation [Formula: see text]. In this equation, the fractional quantum Hall effect edge states are described. By the unified method, we successfully get the soliton, solitary and elliptic wave solutions. Through the [Formula: see text]-expansion and the exponential expansion methods, we acquire different traveling wave solutions. Using the extended [Formula: see text]-expansion method, we attain the Jacobi elliptic function solutions. So as to get more physical information about the exact solutions of VCCNLSE, [Formula: see text], density and contour maps are described.

Soliton solutions for two kinds of fourth-order nonlinear nonlocal Schrödinger equations

In this paper, we will take two symmetric reduction conditions to obtain two kinds of fourth-order nonlinear nonlocal Schrödinger (NLS) equations, namely the fourth-order nonlocal reverse space and reverse space–time NLS equations. For the former, we will construct one- and N-fold Darboux Transformations (DTs) to obtain the symmetry preserving and broken solutions. In addition, we will study the different combinations of collision scenarios of dark and anti-dark soliton solutions. This process will be expressed in the form of quasi-determinant. Finally, we will get the symmetry preserving and broken solutions can exist at the same time. For the latter, the one- and N-fold DTs will be constructed. Taking the seed solutions as zero and continue wave solutions, we will obtain different kinds of exact solutions under nonlocal constraints, such as soliton, breather and rogue wave solutions, which is different from the classical case.

On a coupled nonlocal nonlinear Schrödinger system

Recently, Ablowitz and Musslimani have introduced a new integrable nonlocal nonlinear Schrödinger equation. In this paper, we investigate an integrable coupled nonlocal nonlinear Schrödinger equation which can be derived from the AKNS system. The Darboux transformation is constructed for this equation. Via this Darboux transformation, we obtain its different kinds of exact solutions including soliton, kink, periodic solutions and so on. Dynamics and interactions of different kinds of soliton solutions are discussed. Finally, we compare the obtained results with standard coupled NLS equation.

Separation variable method combined with integral bifurcation method for solving time-fractional reaction–diffusion models

In this paper, based on the previous works, we improve a computational method on solving time-fractional partial differential equation. Using the improved method, a series of time-fractional reaction–diffusion models with Fisher–KPP type are studied from mathematical point of view. Different kinds of exact solutions of four time-fractional reaction–diffusion models are obtained. The forms of these solutions include parametric form, explicit form, and implicit form. Most of them (solutions) have degenerate property according as time increase. The dynamical properties of some representative exact solutions are illustrated by graphs.

Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg-de Vries equation

Two nonlocal versions of the semi-discrete modified Korteweg-de Vries equation are derived by different nonlocal reductions from a coupled equation set in the Ablowitz–Ladik hierarchy. Different kinds of exact solutions in terms of double Casoratians to the reduced equations are obtained by imposing constraint conditions on the double Casorati determinant solutions of the coupled equation set. Dynamics of the soliton solutions for the real and complex nonlocal semi-discrete modified Korteweg-de Vries equations are analyzed and illustrated by asymptotic analysis.

Exact solutions of nonlocal Fokas–Lenells equation

In this paper we propose a nonlocal Fokas–Lenells (FL) equation which can be derived from the Kaup–Newell (KN) linear scattering problem. By constructing the Darboux transformation of nonlocal FL equation, we obtain its different kinds of exact solutions including bright/dark solitons, kink solutions, periodic solutions and several types of mixed soliton solutions. It is shown that the solutions of nonlocal FL equation possess different properties from the normal FL equation.

Idea of invariant subspace combined with elementary integral method for investigating exact solutions of time-fractional NPDEs

In this paper, inspired by the idea of invariant subspace method and combined with elementary integral method, we introduced a novel approach for investigating exact solutions of a time-fractional nonlinear partial differential equation (NPDE). Based on hypothetical structure of solution of separated variable, a time-fractional NPDE defined by time and space variables can be reduced to a nonlinear ordinary differential equation (NODE) or NODEs defined by space variable alone, and then using the elementary integral method to solve the NODE or NODEs, different kinds of exact solutions of a time-fractional NPDE are obtained finally. As examples, the time-fractional Hunter–Saxton equation and time-fractional Li–Olver equation were studied. Different kinds of exact solutions of these equations were obtained and their dynamical properties were illustrated.

Applications of integral bifurcation method together with homogeneous balanced principle on investigating exact solutions of time fractional nonlinear PDEs

In this paper, it is verified that the fractional chain rule appeared in some references does not hold under Riemann–Liouville definition and Caputo definition of fractional derivative. It implies that this chain rule is invalid on searching exact solutions of fractional nonlinear partial differential equations (PDEs). Avoiding this chain rule, a new way of investigating exact solutions of fractional nonlinear PDEs is introduced. As examples, a series of time fractional nonlinear PDEs with diffusion and convection terms are studied. Different kinds of exact solutions of these equations are obtained, their dynamical properties are discussed, and the profiles of several representative exact solutions are illustrated.

On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions

Very recently, Ablowitz and Musslimani introduced a new integrable nonlocal nonlinear Schrödinger equation. In this paper, we investigate an integrable nonlocal modified Korteweg-de Vries equation (mKdV) which can be derived from the well-known AKNS system. We construct the Darboux transformation for the nonlocal mKdV equation. Using the Darboux transformation, we obtain its different kinds of exact solutions including soliton, kink, antikink, complexiton, rogue-wave solution, and nonlocalized solution with singularities. It is shown that these solutions possess new properties which are different from the ones for mKdV equation.

Notes on “Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind”

The goal of this paper is to employ the approach of dynamical system to obtain the analytic expression for different kinds of periodic wave solutions of the two-component short pulse equations of the first kind: uxt=u+16(u3+3uv2)xx,vxt=v+16(v3+3u2v)xx, which are not obtained in Rui (2013) [20].

Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind

In this paper, by using the integral bifurcation method and the Sakovich’s transformations, we study the two-component short pulse equations of the first kind, different kinds of exact traveling wave solutions with two-loop character, such as two-loop soliton solutions, periodic loop-compacton wave solutions and different kinds of periodic two-loop wave solutions are obtained. Further, we discuss their dynamical behaviors of these exact traveling wave solutions and show their profiles of time evolution by illustrations. This is first time in our research area that we obtain two-soliton solutions of nonlinear partial differential equations under no help of Hirota’s method, inverse scattering method, Darboux transformation and Bächlund transformation.

A New Discrete Integrable System Derived from a Generalized Ablowitz‐Ladik Hierarchy and Its Darboux Transformation

We find an interesting phenomenon that the discrete system appearing in a reference can be reduced to the old integrable system given by Merola, Ragnisco, and Tu in another reference. Differing from the works appearing in the above two references, a new discrete integrable system is obtained by the generalized Ablowitz‐Ladik hierarchy; the Darboux transformation of this new discrete integrable system is established further. As applications of this Darboux transformation, different kinds of exact solutions of this new system are explicitly given. Investigatingthe properties of these exact solutions, we find that these exact solutions are not pure soliton solutions, but their dynamic characteristics are very interesting.