^{1}

This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.

Soliton was first discovered in 1834 by Russell [

In a highly informative as well as entertaining article [

His ideas did not earn attention until 1965 when Zabusky and Kruskal [

A soliton is a special traveling wave that after a collision with another soliton eventually emerges unscathed. Solitons are solutions of partial differential equations that model phenomena like water waves or waves along a weakly anharmonic mass-spring chain. A soliton is a bell-like solution as illustrated in Figure

Bell-like solitary wave.

The soliton can be written in a standard form, which is

It is obvious that

The soliton obeys a superposition-like principle: solitons passing through one another emerge unmodified, see Figure

Collision of two solitary waves.

A compacton is a special solitary traveling wave that, unlike a soliton, does not have exponential tails. A compacton-like solution is a special wave solution which can be expressed by the squares of sinusoidal or cosinoidal functions.

Compactons are special solitons with finite wavelength. It was Rosenau and Hyman [

Now consider a modified version of KdV equation in the form

We rewrite (

In case

We consider two common-order differential equations whose exact solutions are important for physical understanding:

The crucial difference between these two very simple equations is the sign of the coefficient of

Equation (

Compacton wave without tails.

Thermal conduction in fuel/Bi_{2}Te_{3}/Al_{2}O_{3} or fuel/Bi_{2}Te_{3}/terracotta systems always results in strong oscillation of the output signals. A criterion for oscillatory thermopower waves is much needed.

Recently, Walia et al. proposed a theory of thermopower wave oscillations to describe coupled thermal waves in fuel/Bi_{2}Te_{3}/Al_{2}O_{3} or fuel/Bi_{2}Te_{3}/terracotta systems [

Ignoring the nonlinear term in (_{2}Te_{3} films.

The system, (

By the transform, (

Equation (

In (

We search for a periodic solution of (

This is a criterion for oscillatory thermopower waves. When

Lin and Hildemann [

This paper aims at a wave solution of (

Equation (

Considering the boundary condition, (

Equation (

For a wave solution, the initial condition should be expressed in the form of (

There is plainly a tendency in the modern nonlinear science community to obtain exact solutions for nonlinear equations. There are many results on the exact solutions of nonlinear equations where the initial or boundary conditions are not considered. These solutions are called

An asymptotic approach is, however, to search for an asymptotic solution with physical understanding. If, for example, we feel interest in a solitary solution of (

For

For a two-wave solution, we can assume that

Some asymptotic methods are easy and accessible to all nonmathematicians using only pencil and paper. Consider a nonlinear differential equation for corneal shape [

Hereby we suggest a Taylor series method to find an asymptotic solution [

We rewrite (

The accuracy can be further improved if the solution procedure continues.

Comparing the Okrasiński and Płociniczak’s method with our pencil-and-paper method, we conclude that the solution process is accessible to nonmathematicians to solve any nonlinear two-point boundary problems.

The investigation of soliton solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. There are many analytical approaches to the search for solitary solutions, see [

We consider the following perturbed nonlinear evolution equation [

In case

Using multiple time scales (a slow time

Substituting (

In most cases, the nonlinear term

In order to overcome the shortcoming arising in the above solution process, hereby we applied the modified multitime expansions (see Section 2.9 of [

The equation for

We use the Duffing equation to illustrate the solution procedure [

Solving (

Substituting

Eliminating secular terms needs

If only the first-order approximate solution is searched for, from (

The obtained frequency-amplitude relationship, (

This section is an elementary introduction to the concepts of the calculus of variations and its applications to solitary solutions. Generally speaking, there exist two basic ways to describe a nonlinear problem: (1) by differential equations (DE) with initial/boundary conditions; (2) by variational principles (VP). The former is widely used, while the later is rarely used in solitary theory. The VP model has many advantages over its DE partner: simple and compact in form while comprehensive in content, encompassing implicitly almost all information characterizing the problem under consideration. Variational methods have been, and continue to be, popular tools for nonlinear problems. When contrasted with other approximate analytical methods, variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions.

The inverse problem of calculus of variations is to establish a variational formulation directly from governing equations and boundary/initial conditions. We will use the semi-inverse method [

Consider the well-known Korteweg-de Vries (KdV) equation

We rewrite it in a conserved form

Generally, the multiplier can be expressed in the form after identification

We, therefore, obtain the following needed variational principle:

Let us come back to (

Sometimes the Lagrange crisis can be eliminated by replacing some variables in the original variational principle using the constraint equation; a detailed discussion was systematically given in [

We replace

Now we introduce a new variable

It is easy to establish a variational formulation by introducing a potential function, and we can also establish a variational principle without auxiliary special function. To elucidate this, we consider the KdV equation in the form

Finally, we have the following needed Lagrangian in the form of velocity:

The research on traffic flow began at the beginning of the 20th century. Lighthill and Whitham first proposed the fluid-dynamical model for traffic flow [

In 1994, Zheng [

In order to establish a variational principle for the system, we rewrite (

The essence of the semi-inverse method [

There exist various approaches to the establishment of energy-like trial functionals for a physical problem, and illustrative examples can be found in [

The advantage of the above trial functional lies on the fact that the stationary condition with respective to

Calculating the functional (

The Euler equations of the above functional (

From the above three-field variational functional, we can easily obtain two-field or one-field variational function by substituting one- or two-field equations into the functional (

By a paralleling operation, we can also establish a variational functional with free fields

Constraining the functional (

Maupertuis-Lagrange’s principle of least kinetic potential action for a particle with mass

For arbitrary

In optics, Fermat’s principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light

The light trajectory that connects these two endpoints

For arbitrary

Consider a general nonlinear oscillator in the form

Assume that its solution can be expressed as

Substituting (

Explanation of (

Consider a nonlinear oscillator with fractional potential [

As an illustration, consider the following chemical reaction [

In order to obtain a variational model, we differentiate both sides of (

Assume that the solution can be expressed in the form

Substituting (

We can also choose the following trial function:

In the review article [

Considering the KdV equation, we seek its traveling wave solutions in the following frame:

Integrating (

The Hamiltonian invariant (

The semi-inverse method is a powerful mathematical tool to the search for variational formulae for real-life physical problems.

By Ritz method, we search for a solitary wave solution in the form

Substituting (

For the KdV equation expressed in (

Substituting (

We begin with the definition of the action functional as time integral over the Lagrangian

Newton’s motion equation can be obtained from the stationary condition of the functional (

We can introduce some constraints to the action functional (

Now we introduce a generalized velocity

From (

By the same manipulation, from (

A more generalized action can be obtained by linear combination of

Linearly combining

The Hamilton principle can be written in the form

Here

The Hamilton’s principle holds only for the conditions prescribed at the beginning and at the end of the motion and is therefore useless to deal with the usual initial condition problems, both as an analytical tool and as a basis for approximate solution methods. It is impossible for most real-life physical problems to prescribe terminal conditions.

In order to eliminate the unnecessary final condition at

Liu [

It would be a landmark in the history of calculus of variations after Hamilton if we can extend the principle to all initial-value problems without prescribing both initial and final conditions. In order to deal with the final condition, we consider the conserved Hamiltonian:

In order to convert the initial conditions

Making the above functional stationary, we obtain the following stationary conditions at

Accordingly, we can assume that

Making the obtained functional (

in the solution domain (

natural initial conditions

natural final condition

We have alternative approaches to identifying

To summarize, we can conclude from the above derivation and strict proof that the obtained modified Hamilton principles, which are first deduced in the history, and valid for all initial-value problems, are extremely important in both pure and applied sciences due to complete elimination of the long-existing shortcomings in Hamilton principle. The stationary conditions of the obtained variational principle satisfy the Newton’s motion equation, and all initial conditions, furthermore, the natural final condition

In this paper, we consider the following general oscillator:

It is easy to establish a variational principle for (

In the functional (

Assume that the solution can be expressed as

Submitting (

From (

Consider the Duffing equation

Its variational formulation can be written as

The Hamiltonian approach to nonlinear oscillators has been now widely used [

The Euler-Lagrange equation of (

It is obvious that the above Euler-Lagrange equation is the KdV equation.

We write the Lagrange function in the form

If a solitary solution is solved, considering tail property of a soliton, we assume that

The assumption, (

Substituting (

The residual equation is

Simplifying (

Consider the following generalized KdV equation:

Strong storms and cyclones, underwater earthquakes, high-speed ferries, and aerial and submarine landslides can cause giant surface waves approaching the coast and frequently cause extensive coastal flooding, destruction of coastal constructions, and loss of lives. A fast but reliable prediction of a tsunami pulse is of critical importance, and a simple equation like Bernoulli equation is, therefore, much needed.

To this end, we use a one-dimensional nonlinear shallow wave propagation for describing runup of irregular waves on a beach. The basic equations are [

We will establish a variational principle for the system of (

In order to establish a variational formulation for the system of (

Using the semi-inverse method [

It is obvious that the stationary condition of (

Making the functional, (

We can rewrite (

Equation (

We call (

Tsunami prediction, for example, after an earthquake, is of critical importance to save life and property. The extent of catastrophe depends mainly on the height of the Tsunami pulses, that is, the value of

It is well known that the Hamilton’s principle can be applied to a single fluid particle or a closed system by the involutory transformation. For an isoentropy rotational flow, a variational principle can be established using Lin’s constraints [

In 1950s, great progress had been made on the research of variational principle in solid mechanics, with Hu-Washizu variational principle [

Hamilton’s principle was so successfully and powerfully applied to particle mechanics that many attempts have been made to obtain the momentum equations from a variational principle patterned after Hamilton’s principle. These attempts have not all been successful except in the case of isoentropy irrotational flow. For a more general one, Lin’s constraints [

Considering a 3D unsteady inviscid compressible rotational flow, we have the following equations [

Momentum equation:

Equation of state:

Continuity equation:

Isoentropic equation:

Steady Bernoulli’s equation:

For unsteady flow, we should use the following momentum equation:

From the equation of state, we can deduce the following equation:

Applying the Lagrange multipliers to remove the constraint equations (

According to Clebsch [

Lin introduced three-additional constraint equations (Lin’s constraint equations) to the functional (

Introducing the Lagrange multipliers to remove Lin’s constraints, we have the following general functional:

Now considering a fluid particle, and applying Hamilton’s principle to construct the following functional:

The involutory transformation is to convert Lagrange space into Euler space:

Using the Lagrange multipliers to remove the constraints, (

Making the involutory transformation, we obtain the following functional via Lagrange multiplier method

In order to guarantee functional (

Rotational equation with Clebsch variables can be expressed as follows:

Making the functional (

Hereby we have successfully explained the phenomenon of Lin’s constraints via involutory transformation and deduced a generalized variational principle with only 6 independent variables via semi-inverse method, which makes it possible to use FEM to calculate the 3D unsteady compressible rotational flow. The corresponding generalized variational principle with Lin’s constraints is actually an approximate one (for Herrivel’s principle actually is a wrong principle), which, however, can find some application, especially in 2D problems, due to the fact that we can deduce all the Eulerian equations from the functional when making it stationary.

The variational iteration method [

The variational iteration method has been shown to solve a large class of nonlinear problems effectively, easily, and accurately with the approximations converging rapidly to accurate solutions.

To illustrate the basic idea of the technique, we consider the following general nonlinear system:

The basic concept of the method is to construct a correction functional for the system (

Consider the following nonlinear equation of

The main merit of this iteration formula lies in the fact that

For initial value problems, we can begin with

For boundary value problems, the initial guess can be expressed in the form

After identifying the Lagrange multiplier

Note:

From (

We can also apply Laplace transform to identify the Lagrange multiplier. By Laplace transform, we have

Here is an incomplete list for variational iteration formulas for various differential equations [

For discontinuous solitons, we can assume, for example, the following form:

For compacton-like solution, we assume that the solution has the form

As an illustrating example, we consider the following modified KdV equation

Substituting (

Setting

The initial solution (trial function) can be also constructed in a solitary form. Now we begin with

So we obtain the following needed solitary solution:

It is also interesting to note that the solitary solution can be converted into a compacton-like solution if we choose

Solving

Substituting (

Before proceeding the method, we give an interesting application of the homotopy technology to an asymptotic match of Rayleigh grain size distribution and Hillert grain size distribution. Grain growth is a well-known phenomenon in the evolution of crystalline microstructures. A stochastic continuity or Fokker-Planck continuity equation was proposed to accurately predict properties of grain growth [

An asymptotic match of (

Equation (

In this short paper we suggest an asymptotic match to bridge the two limited cases, and the obtained result is valid for the whole case. The asymptotic match is a simple and useful mathematical tool in engineering for reliable treatment of a nonlinear problem whose analytical solution can be easily obtained for two limited cases (e.g.,

Consider another example of the relativistic oscillator

It is easy to obtain the following approximate frequency [

In order to match both the cases

The two most important steps in application of the homotopy perturbation method [

Consider a general nonlinear equation

The first step for the method is to construct a homotopy equation in the form

For example, consider a nonlinear oscillator [

When

If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form

In this paper, we suggest an alternative approach for construction of homotopy equation, which is

To illustrate the solution procedure, we consider a nonlinear oscillator in the form

Equation (

We construct the following homotopy equation with an auxiliary term [

Substituting

Eliminating the secular term needs

The solution procedure continues by submitting

No secular term in

Solving (

In order to compare with the perturbation solution and the exact solution, we set

In case

If a higher-order approximate solution is needed, we rewrite the homotopy equation in the form

Generally, the homotopy equation can be constructed in the form

Homotopy perturbation method [

For traveling wave solutions, we can use the following transformation:

We begin with a soliton in the form

Using in the

The

Let

The [

Couple of the homotopy perturbation method with the Laplace transform makes the solution procedure much simpler.

Consider the following foam drainage equation:

Slab detachment or break-off is appreciated as an important geological process. Ananalytical solution was given in [

The power-law flow law for the layer can be written as

Schmalholz obtained the following detachment time [

The power-law fluid assumption, (

The governing equation for the dynamics of necking becomes

Rewrite (

We use the parameter-expansion method (see Section

The parameters