A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves

A new algebraic method is devised to uniformly construct a series of new travelling wave solutions for two variant Boussinesq equations. The solutions obtained in this paper include soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters.     

An algebraic method exactly solving two high-dimensional nonlinear evolution equations

An algebraic method is applied to construct soliton solutions, doubly periodic solutions and a range of other solutions of physical interest for two high-dimensional nonlinear evolution equations. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solutions according to some parameters.     

Some new and general solutions to the compound KdV-Burgers system with nonlinear terms of any order

In this letter, a new Riccati equation expansion method is presented for constructing exact travelling-wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the Riccati equation to construct exact travelling-wave solutions of nonlinear partial differential equations. As a result, some more generalized solutions, which contain triangular periodic solutions, exp function solutions and the soliton-like solutions, are obtained.     

Several classes of exact solutions to the 1 + 1 Born–Infeld equation

We obtain closed-form exact solutions to the 1 + 1 Born–Infeld equation arising in nonlinear electrodynamics. In particular, we obtain general traveling wave solutions of one wave variable, solutions of two wave variables, similarity solutions, multiplicatively separable solutions, and additively separable solutions. Then, putting the Born–Infeld model into correspondence with the minimal surface equation using a Wick rotation, we are able to construct complex helicoid solutions, transformed catenoid solutions, and complex analogues of Scherk’s first and second surfaces. Some of the obtained solutions are new, whereas others are generalizations of solutions in the literature. These exact solutions demonstrate the fact that solutions to the Born–Infeld model can exhibit a variety of behaviors. Exploiting the integrability of the Born–Infeld equation, the solutions are constructed elegantly, without the need for complicated analytical algorithms.     

A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method

In this paper, a series of abundant exact travelling wave solutions is established for a modified generalized Vakhnenko equation by using auxiliary equation method. These solutions can be expressed by Jacobi elliptic function. When Jacobi elliptic functions modulus m→1 or 0, the travelling wave solutions degenerate to four types of solutions, namely, the soliton solutions, the hyperbolic function solutions, the trigonometric function solutions, constant solutions.     

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.     

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.     

A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation

A new algebraic method is devised to obtain a series of exact solutions for general nonlinear equations. Compared with the most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters. For illustration, we apply the method to solve a new two-dimensional perturbed KdV equation and successfully construct the various kind of exact solutions including line soliton solutions, rational solutions, triangular periodic solutions, Jacobi, and Weierstrass doubly periodic solutions.     

Irregular Partially Invariant Solutions of Rank 2 and Defect 1 to Equations of Gas Dynamics

We consider three-dimensional subalgebras admitted by the equations of gas dynamics having time as an invariant and containing no rotation operator. For such subalgebras we seek for irregular partially invariant solutions of rank 2 and defect 1. The representation for solutions has the form which generalizes motion of a gas with a linear velocity field. We show that partially invariant solutions exist for each subalgebra. We describe the set of these solutions. We find solutions with the indicated representation that are not partially invariant. The solutions reducible to invariant solutions are generalized to new submodels.     

Rational Solutions to the KdV Equation in Terms of Particular Polynomials

Here, we construct rational solutions to the KdV equation by particular polynomials. We get the solutions in terms of determinants of the order $n$ for any positive integer $n$, and we call these solutions, solutions of the order $n$. Therefore, we obtain a very efficient method to get rational solutions to the KdV equation, and we can construct explicit solutions very easily. In the following, we present some solutions until order $10$.     

Existence of kink and unbounded traveling wave solutions of the Casimir equation for the Ito system

This paper study the traveling wave solutions of the Casimir equation for the Ito system. Since the derivative function of the wave function is a solution of a planar dynamical system, from which the exact parametric representations of solutions and bifurcations of phase portraits can be obtained. Thus, we show that corresponding to the compacton solutions of the derivative function system, there exist uncountably infinite kink wave solutions of the wave equation. Corresponding to the positive or negative periodic solutions and homoclinic solutions of the derivative function system, there exist unbounded wave solutions of the wave function equation.     

Symbolic computation of exact solutions for a nonlinear evolution equation

In this paper, by means of the Jacobi elliptic function method, exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are presented. It can be shown that not only the obtained solitary wave solutions have the property of loop-shaped, cusp-shaped and hump-shaped for different values of parameters, but also different types of double periodic wave solutions are possible, namely periodic loop-shaped wave solutions, periodic hump-shaped wave solutions or periodic cusp-shaped wave solutions. Furthermore, periodic loop-shaped wave solutions will be degenerated to loop-shaped solitary wave solutions for the same values of parameters. So do cusp-shaped solutions and hump-shaped solutions. All these solutions are new and first reported here.     

Solutions to the system of operator equations AXB=C=BXA

In this paper, we present some necessary and sufficient conditions for the existence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB=C=BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.     

An algebraic method with computerized symbolic computation for the one-dimensional generalized BBM equation of any order

In this paper, an extended algebraic method with symbolic computation is applied to construct a series of travelling wave solutions of the one-dimensional generalized BBM equation of any order with positive and negative exponents. As a result, the proposed method gives many explicit exact solutions such as solitary wave solutions, periodic solutions, solitary patterns solutions and compacton solutions.     

Variable-coefficient Jacobi elliptic function expansion method for (2+1)-dimensional Nizhnik-Novikov-Vesselov equations

In this paper, a variable-coefficient Jacobi elliptic function expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations. Being concise and straightforward, this method is applied to the (2+1)-dimensional Nizhnik-Novikov-Vesselov equations. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions. To give more physical insights to the obtained solutions, we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions.     

Lump solutions to a generalized Hietarinta-type equation via symbolic computation

Lump solutions are one of important solutions to partial differential equations, both linear and nonlinear. This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms. The key is a Hirota bilinear form. Lump solutions are constructed via symbolic computations with Maple, and specific reductions of the resulting lump solutions are made. Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented, together with three-dimensional plots and density plots of the lump solutions.     

A second Wronskian formulation of the Boussinesq equation

A Wronskian formulation leading to rational solutions is presented for the Boussinesq equation. It involves third-order linear partial differential equations, whose representative systems are systematically solved. The resulting solutions formulas provide a direct but powerful approach for constructing rational solutions, positon solutions and complexiton solutions to the Boussinesq equation. Various examples of exact solutions of those three kinds are computed. The newly presented Wronskian formulation is different from the one previously presented by Li et al., which does not yield rational solutions.     

Bäcklund transformations and abundant exact explicit solutions of the Sharma-Tasso-Olver equation

By using an extension of the homogeneous balance method and Maple, the Bäcklund transformations for the Sharma-Tasso-Olver equation are derived. The connections between the Sharma-Tasso-Olver equation and some linear partial differential equations are found. With the aid of the transformations given here and the computer program Maple 12, abundant exact explicit special solutions to the Sharma-Tasso-Olver equation are constructed. In addition to all known solutions re-deriving in a systematic way, several entirely new and more general exact explicit solitary wave solutions can also be obtained. These solutions include (a) the algebraic solitary wave solution of rational function, (b) single-soliton solutions, (c) double-soliton solutions, (d) N-soliton solutions, (e) singular traveling solutions, (f) the periodic wave solutions of trigonometric function type, and (g) many non-traveling solutions. By using the Airy’s function and the Bäcklund transformations obtained here, the exact explicit solution of the initial value problem for the STO equation is presented. The variety of the structure of the solutions for the Sharma-Tasso-Olver equation is illustrated.     

Approximation to a parabolic system modeling the thermoelastic contacts of two rods

In this article, we study a sequence of finite difference approximate solutions to a parabolic system, which models two dissimilar rods that may come into contact as a result of thermoelastic expansion. We construct the approximate solutions based on a set of finite difference schemes to the system, and we will prove that the approximate solutions converge strongly to the exact solutions. Moreover, we obtain and prove rigorously the error bound, which measures the difference between the exact solutions and approximate solutions in a reasonable norm. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 1–25, 1998     

Existence of multiple solutions to second-order discrete Neumann boundary value problems

By using the invariant set of descending flow and variational method, we establish the existence of multiple solutions to a class of second-order discrete Neumann boundary value problems. The solutions include sign-changing solutions, positive solutions, and negative solutions. An example is given to illustrate our results.