Singular Traveling Wave Solutions Research Articles

Solitons and symmetry reduction solutions of a nonlocal two‐mode Korteweg–de Vries equation

The two‐mode Korteweg–de Vries (TMKdV) equation describes propagation of two different waves modes simultaneously. By using consistent correlated bang (CCB) method, a nonlocal form of the TMKdV equation is constructed and converted into two local equations with its dependent variables having definite parity. Multiple soliton solutions of the nonlocal TMKdV equation are obtained by using known multiple soliton solutions of the TMKdV equation, and some singular travelling wave solutions are also generated in a similar way. The classical Lie symmetry method is also carried on the nonlocal TMKdV equation to get its symmetry reduction solutions.

The Soliton Wave Solutions and Bifurcations of the (2 + 1)-Dimensional Dissipative Long Wave Equation

With the help of the bifurcation theory of dynamical differential system and maple software, we shall devote ourselves to research travelling wave solutions and bifurcations of the (2 + 1)-dimensional dissipative long wave equation. The study of travelling wave solutions for long wave equation derives a planar Hamiltonian system. Based on phase portraits, we obtain exact explicit expressions of some bounded traveling wave solutions and some important singular traveling wave solutions, under different parametric conditions.

Phase Portraits and Bounded and Singular Traveling Wave Solution of Stochastic Nonlinear Biswas–Arshed Equation

The main purpose of the current paper is to study the phase portraits and bounded and singular traveling wave solution of the stochastic nonlinear Biswas–Arshed equation by using the “three‐step method” of Professor Li’s method together with the phase orbit of planar dynamical system. Firstly, by employing the traveling wave transformation, the stochastic nonlinear Biswas–Arshed equation is simplified into deterministic nonlinear ordinary differential equation. Secondly, phase portraits of the stochastic nonlinear Biswas–Arshed equation are plotted by analyzing the planar dynamic system of the nonlinear ordinary differential equation. Finally, the bounded and singular traveling wave solutions of the stochastic nonlinear Biswas–Arshed equation are constructed.

Exact analytical solutions of the fractional biological population model, fractional EW and modified EW equations

In this paper, exact analytical solutions of the biological population model, the EW and the modified EW equations with a conformable derivative operator have been examined by means of the trial solution algorithm and the complete discrimination system. Dark, bright and singular traveling wave solutions of the equations have been obtained by algorithm. Also, revealed singular periodic solutions have been listed. All solutions were verified by substituting them into their corresponding equation via Mathematica package program.

Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

In this paper, we consider the traveling wave solutions of the one-dimensional Serre-Green-Naghdi (SGN) equations which are proposed to model dispersive nonlinear long water waves in a one-layer flow over flat bottom. We decouple the traveling wave system of SGN equations into two ordinary differential equations. By studying the bifurcations and phase portraits of each bifurcation set of one equation, we obtain the exact traveling wave solutions of SGN equations for the variable $ u(x, t) $ which represents average horizontal velocity of water wave. For the compacted orbits intersecting with the singular line in phase plane, we obtained two families of solutions: a family of smooth traveling wave solutions including periodic wave solutions and solitary wave solutions, and a family of compacted singular solutions which have continuous first-order derivative but discontinuous second-order derivative.

Singular traveling wave solutions for Boussinesq equation with power law nonlinearity and dual dispersion

In this paper we study the Boussinesq equation with power law nonlinearity and dual dispersion which arises in fluid dynamics. A particular kind of product of distributions is introduced and applied to solve non-smooth solutions of this equation. It is proved that, under certain conditions, a distribution solution as a singular Dirac delta function exists for this model. For the first time, this kind of product of distributions is used to deal with a fourth order nonlinear partial differential equation.

Some Interesting Traveling Waves in a Transversely Isotropic Incompressible Hyperelastic Semi-infinite Rod

In this paper, the bounded traveling wave solutions are examined for a system of partial differential equations with certain boundary conditions. Interestingly, this system can be used to describe the axially symmetric motion of a semi-infinite incompressible hyperelastic cylindrical rod. With the aid of traveling wave transformations, the partial differential equations can be reduced to a traveling wave equation. The implicit analytical expressions determining the traveling waves are derived. Significantly, the influences of material parameters on the qualitative properties are discussed in detail by using the phase portraits of the traveling wave system. Smooth solutions such as the periodic traveling wave solutions and the solitary wave solutions with the peak form are shown. In particular, some interesting singular traveling wave solutions including the solitary cusp wave solutions and the periodic cusp wave solutions with the peak form are obtained. Numerical examples for all these waves are given.

The effects of horizontal singular straight line in a generalized nonlinear Klein–Gordon model equation

In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein–Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution.

Different Kinds of Singular and Nonsingular Exact Traveling Wave Solutions of the Kudryashov-Sinelshchikov Equation in the Special Parametric Conditions

In this paper, by using the integral bifurcation method, we studied the Kudryashov-Sinelshchikov equation. In the special parametric conditions, some singular and nonsingular exact traveling wave solutions, such as periodic cusp-wave solutions, periodic loop-wave solutions, smooth loop-soliton solutions, smooth solitary wave solutions, periodic double wave solutions, periodic compacton solutions, and nonsmooth peakon solutions are obtained. Further more, the dynamic behaviors of these exact traveling wave solutions are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameters.

Exact Traveling Wave Solutions for a Nonlinear Evolution Equation of Generalized Tzitzéica-Dodd-Bullough-Mikhailov Type

By using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. Under different parameters, we investigated different kinds of exact traveling wave solutions of this generalized TDBM equation. Many singular traveling wave solutions with blow-up form and broken form, such as periodic blow-up wave solutions, solitary wave solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions, are obtained. In order to visually show dynamical behaviors of these exact solutions, we plot graphs of profiles for some exact solutions and discuss their dynamical properties.

Explicit and exact traveling wave solutions to the nonlinear LC circuit equation

Traveling wave in a nonlinear LC circuit with dissipation have been investigated theoretically. With the aid of the extended hyperbolic function method,developed by the authors in recent works to solve nonlinear partial differential equations exactly, the fourth order nonlinear wave equation with dissipation, which models shock wave propagation in a nonlinear LC circuit, have been analytically studied. Abundant explicit and exact traveling wave solutions to the fourth order nonlinear wave equation with dissipation are obtained. These solutions include exact shock wave solutions, singular traveling wave solutions, and periodic wave solutions in a rational form of trigonometric functions.

A Method for Constructing Traveling Wave Solutions to Nonlinear Evolution Equations

In this paper, an analytical method is proposed to construct explicitly exact and approximate solutions for nonlinear evolution equations. By using this method, some new traveling wave solutions of the Kuramoto-Sivashinsky equation and the Benny equation are obtained explicitly. These solutions include solitary wave solutions, singular traveling wave solutions and periodical wave solutions. These results indicate that in some cases our analytical approach is an effective method to obtain traveling solitary wave solutions of various nonlinear evolution equations. It can also be applied to some related nonlinear dynamical systems.

The First‐Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations

This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion‐acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell‐type for n and E, the solitary wave solutions of kink‐type for E and bell‐type for n, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.

The Bäcklund transformations and abundant explicit exact solutions for the AKNS–SWW equation

The Bäcklund transformations and abundant explicit exact solutions to the AKNS shallow water wave equation are obtained by combining the extended homogeneous balance method with the extended hyperbolic function method. The solutions obtained admit of multiple arbitrary parameters. These solutions include (a) a compound of the rational fractional function and a linear function, (b) a compound of solitary wave solution and a linear function, (c) a compound of the singular travelling wave solutions and a linear function, and (d) a compound of the periodic wave solutions of triangle function and a linear function. In special cases, we can obtain a series of soliton solutions, singular travelling wave solutions, periodic travelling wave solutions, and rational fractional function solution. In addition to re-deriving some known solutions in a systematic way, some brand-new exact solutions are also established.

New exact traveling wave solutions for the Klein–Gordon–Zakharov equations

Based on the extended hyperbolic functions method, we obtain the multiple exact explicit solutions of the Klein–Gordon–Zakharov equations. The solutions obtained in this paper include (a) the solitary wave solutions of bell-type for u and n , (b) the solitary wave solutions of kink-type for u and bell-type for n , (c) the solitary wave solutions of a compound of the bell-type and the kink-type for u and n , (d) the singular traveling wave solutions, (e) periodic traveling wave solutions of triangle function types, and solitary wave solutions of rational function types. We not only rederive all known solutions of the Klein–Gordon–Zakharov equations in a systematic way but also obtain several entirely new and more general solutions. The variety of structures of the exact solutions of the Klein–Gordon–Zakharov equations is illustrated.

Abundant exact and explicit solitary wave and periodic wave solutions to the Sharma–Tasso–Olver equation

This paper is concerned with a double nonlinear dispersive equation—the Sharma–Tasso–Olver equation. The extended hyperbolic function method is employed to investigate the solitary and periodic travelling waves in this equation. With the aid of Mathematica and Wu-elimination Method, the abundant exact explicit solutions of the nonlinear Sharma–Tasso–Olver equation are derived. The solutions obtained in this paper include (a) solitary wave solutions, (b) the singular travelling wave solutions, and (c) periodic travelling wave solutions of triangle function types. Several entirely new exact solutions to the equation are explicitly obtained, in addition to deriving all known solutions in a systematic way. This work can be regard as an extension to the recent work by Wazwaz.

The extended hyperbolic functions method and new exact solutions to the Zakharov equations

The multiple exact solutions for the nonlinear evolution equations describing the interaction of laser–plasma are developed. The extended hyperbolic function method are employed to reveal these new solutions. The solutions include that of the solitary wave solutions of bell-type for n and E, the solitary wave solutions of kink-type for E and bell-type for n, the solitary wave solutions of a compound of the bell-type and the kink-type for n and E, the singular traveling wave solutions, periodic traveling wave solutions of triangle function types, and solitary wave solutions of rational function types. In addition to re-deriving all known solutions in a systematic way, several new and more general solutions can be obtained by using our method.

The extended hyperbolic function method and exact solutions of the long–short wave resonance equations

The extended hyperbolic functions method for nonlinear wave equations is presented. Based on this method, we obtain a multiple exact explicit solutions for the nonlinear evolution equations which describe the resonance interaction between the long wave and the short wave. The solutions obtained in this paper include (a) the solitary wave solutions of bell-type for S and L, (b) the solitary wave solutions of kink-type for S and bell-type for L, (c) the solitary wave solutions of a compound of the bell-type and the kink-type for S and L, (d) the singular travelling wave solutions, (e) periodic travelling wave solutions of triangle function types, and solitary wave solutions of rational function types. The variety of structure to the exact solutions of the long–short wave equation is illustrated. The methods presented here can also be used to obtain exact solutions of nonlinear wave equations in n dimensions.

A unified trial function method in finding the explicit and exact solutions to three NPDEs

In this paper, by introducing a transformation and utilizing the trial function method, it is quite interesting that three nonlinear partial differential equations called the KdV equation, Boussinesq equation and CDGSK equation have exactly the same trial function form and that their explicit and exact solutions which include solitary wave solutions, singular travelling wave solutions, and triangle function-type periodic wave solutions can be easily derived in a unified and concise way. Among them, some are new travelling wave solutions of physical interest.

A new method for solving the KdV and KdV-Burgers equations

In this paper, by using the transformation introduced in Acta Phys. Sin., 53 (2004) 2828, a nonlinear partial differential equation can be reduced to a nonlinear ordinary differential equation, and then solve it by the ansatz technique and with the aid of Mathematica. As a result, abundant types of explicit and exact solutions to the KdV equation and the KdV-Burgers equation, which include the solitary wave solutions, the singular traveling wave solutions, the trigonometric function periodic wave solutions, are successfully obtained.