• No results found

Introduction to the Hirota Direct Method

N/A
N/A
Protected

Academic year: 2021

Share "Introduction to the Hirota Direct Method"

Copied!
21
0
0

Loading.... (view fulltext now)

Full text

(1)

IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM SWEDEN 2021,

Introduction to the Hirota Direct Method

PASCAL CAPETILLO

JONATHAN HORNEWALL

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

(2)

Theoretical Physics

Introduction to the Hirota Direct Method

Pascal Capetillo and Jonathan Hornewall pascalca@kth.se

jonhor@kth.se

SA114X Degree Project in Engineering Physics, First Level Department of Theoretical Physics

Royal Institute of Technology (KTH) Supervisor: Edwin Langmann

June 8, 2021

(3)

Abstract

The primary subject matter of the report is the Hirota Direct Method, and the primary goal of the report is to describe and derive the method in detail, and then use it to produce analytic soliton solutions to the Boussinesq equation and the Korteweg-de Vries (KdV) equation. Our hope is that the report may also serve as an introduction to soliton theory at an undergraduate level.

The report follows the structure of first introducing Hirota’s bi-linear operator and giving an account of its relevant properties. The properties of the operator are then used to find soliton solutions for differential equations that can be expressed in a "bilinear"

form. Thereafter, a set of methods for finding the bilinear form of a more general non- linear differential equation are presented. Finally, we apply the tools to the Boussinesq and KdV equations respectively to derive their soliton solutions.

(4)

Sammanfattning

Rapportens huvudsakliga fokus är Hirotas direkta metod och dess huvudsakliga syfte är att redogöra för och härleda metoden grundligt för att sedan använda den för att lösa Korteweg-de Vries (KdV) ekvation och Boussinesq ekvation. Förhoppningsvis kan rapporten också fungera som en lämplig introduktion till soliton-teori på kandidatnivå.

Rapporten följer en struktur där först Hirotas bi-linjära operator introduceras och dess relevanta egenskaper beskrivs. Teorin för operatorn och dess egenskaper används sedan för att finna solitonlösningar till differentialekvationer som kan skrivas på en "bilinjär form". Sedan presenteras en uppsättning metoder för att hitta den bilinjära formen till mer generella icke-linjära differentialekvationer. Slutligen tillämpas samtliga av de verktyg som presenterats för att ta fram solitonlösningar till både Boussinesq-ekvationen och KdV-ekvationen.

(5)

Contents

1 Introduction 2

2 The Hirota Direct Method 3

2.1 Introduction . . . 3

2.2 The Hirota Bilinear Operator . . . 3

2.2.1 Definition . . . 3

2.2.2 Properties . . . 4

2.3 Finding soliton solutions for a bilinear equation . . . 5

2.4 How to find the Bilinear form . . . 8

2.4.1 Dependent variable transformations . . . 8

3 Solving the KdV and Boussinesq equations 11 3.1 Introduction . . . 11

3.2 Korteweg-de Vries equation . . . 11

3.2.1 Solution using the Hirota Direct Method . . . 11

3.3 The Boussinesq equation . . . 13

3.3.1 Solution using the Hirota Direct Method . . . 14

4 Summary and Conclusions 15

(6)

Chapter 1 Introduction

The purpose of the report is to give an account of the Hirota Direct Method, sometimes Hirota’s direct method, which is a mathematical method for finding a special class of solutions to non-linear differential equations [5]. In particular, we will use the method to derive analytical solution to the Korteweg-de Vries (KdV) equation and the Boussinesq equation, which appear in physics [1].

Hirota’s method is specifically used to find "soliton" solutions, which are a type of highly regular solutions with wave-like properties [2]. Solitons can intuitively be thought of as solitary, non-decaying, travelling waves. They occur frequently as analytic solutions to non-linear differential equations, and they are an active area of study in the theory of differential equations and mathematical physics today [2].

Regarding the equations: The Korteweg-De Vries equation and the Boussinesq equa- tions both appear in fluid mechanics, as they can be used to model the behavior of waves travelling over shallow water surfaces [1]. The KdV equation is also of historical impor- tance, since it was one of the first non-linear differential equations found to have analytic solutions [10]. It is usually studied in introductory texts on soliton theory, since it can be solved using various general solution methods of non-linear differential equations [4].

The report was written alongside the report "Numerical solutions to the Boussinesq equation and the Korteweg–de Vries equation"[8] in collaboration with its author Filip Sjölander.

(7)

Chapter 2

The Hirota Direct Method

2.1 Introduction

The following section, and indeed the rest of the report, will be focused on constructing Hirota’s direct method and exploring how it can be used to solve differential equations.

As we shall see, the fundament of the theory rests on the properties of a certain bilinear differential operator known as the Hirota bilinear operator (D-operator).

2.2 The Hirota Bilinear Operator

The D-operator introduced by Hirota is a binary form taking a pair of functions as input and outputting a new function. It exhibits many properties which make it useful in the analysis of differential equations. Specifically, it allows us to find analytic solutions to them in the form of soliton solutions. The operator itself has since its introduction become the object of much further study. The definition, and some important properties will now be listed and explained.

2.2.1 Definition

We define the operator acting on a pair of functions (f, g) of a real variable x, by Dx(f, g) ≡ ∂

∂x − ∂

∂x0



f (x)g(x0) x0=x

. (2.1)

For repeated application of the operator, and when applying the operator with regards to different variables, x and t in this definition but clearly generalizable to any number of real variables, we extend the definition to

DtmDxn(f, g) ≡ ∂

∂x − ∂

∂x0

m

 ∂

∂t − ∂

∂t0

n

f (x, t)g(x0, t0)

x0=x, t0=t

.

We will henceforth use the following notation for brevity and because it has become widely used when dealing with the D-operator.

Dx(f, g) ≡ Dxf · g.

(8)

In order to build some intuition for the D-operator some quick results are presented below.

Dxf · g = fxg − f gx,

Dx3f · g = fxxxg − 3fxxgx+ 3fxgxx− f gxxx, DxDtf · g = fxtg − fxgt− ftgx+ f gxt.

In fact we can note the following formula for the expansion of the n:th power of the operator, constructed by applying binomial expansion to powers of (2.1),

Dxnf · g =

n

X

k=0

(−1)kn k



xn−kxk0f (x)g(x0). (2.2)

This should be compared to the normal product rule for differentiation, differing only in the presence of the alternating sign before each term generated by the (−1)k.

The last defining property of the D-operator is that it is bilinear, as hinted at by its name. This means that it is linear in each argument of the input tuple, seperately. For example, if a, b, c, d are functions and δ is some scalar we have,

Dx(a + δb) · (c + δd) = Dx(a · c + a · δd + δb · c + δb · δd)

= Dx(a · c) + δDx(a · d + b · c) + δ2Dx(b · d).

2.2.2 Properties

Some special properties of the operator which we will make use of in solving bilinear equations will now be presented.

Let us look at a general polynomial P in D = (Dt, Dx, Dy, ...). We also introduce the notation ∂ = (∂t, ∂x, ∂y, ...). We now list a few useful properties,

P (D)f · 1 = P (∂)f, (2.3)

P (D)1 · f = P (−∂)f, (2.4)

Dnxf · g = (−1)nDxng · f, (2.5)

D2n+1x f · f = 0, n ∈ N, (2.6)

where (2.3)-(2.5) stem directly from the definition applied to the specified tuples of functions and constants, and (2.6) is a direct application of (2.5) on the tuple (f, f ).

If we now introduce the exponential function given by exp(ηi), with ηi = ωit + pix + qiy + ... + η0i and where η0i is some constant, we can present another important result,

P (D) exp(η1) · exp(η2) = P (ω1− ω2, p1− p2, q1− q2, ...) exp(η1+ η2). (2.7) In fact, this may be the key to why these types of equations have soliton solutions in the first place. For a more in-depth discussion of these properties, and the operator at large, see chapter 1.6 in Hirota’s book [4].

(9)

2.3 Finding soliton solutions for a bilinear equation

We proceed to use the operator in solving some differential equation written in bilinear form. Let’s assume that the bilinear equation we are to solve can be written as

P (D)f · f = 0, (2.8)

with P and D defined as in section 2.2.2 and with the additional constraint that

P (0) = 0. (2.9)

This last constraint is a requirement for finding soliton solutions [4], and the necessity of it is clearly seen in the following argument.

Now let us employ the standard perturbation method, expanding f as a formal power series in a small parameter ε giving the following expression,

f = 1 + εf1+ ε2f2+ ε3f3+ · · · . (2.10) Putting this expansion into (2.7) and using the bilinearity of the D-operator we get,

P (D)(1 · 1 + 1 · εf1+ εf1· 1 + εf1· εf1 + 1 · ε2f2+ ε2f2· 1 + ...) = 0,

Collecting the terms for each order of the exponent of the expansion parameter ε, we get the system of equations,

ε0 : P (D)(1 · 1) = 0, (2.11)

ε1 : P (D)(1 · f1+ f1· 1) = 0, (2.12)

ε2 : P (D)(1 · f2+ f1· f1 + f2· 1) = 0, (2.13) ε3 : P (D)(1 · f3+ f1· f2 + f2· f1+ f3· 1) = 0, (2.14)

. . .

First we note that (2.11) is automatically true by (2.9) since ∂x (1) = 0 regardless of what variable we differentiate with regards to. Next, we reduce (2.12) to

P (D)(1 · f1+ f1· 1) = 2P (D)(f1· 1) = 2P (∂)f1 = 0 ⇐⇒ P (∂)f1 = 0. (2.15) The first step is given by (2.5) and (2.6), whereas the second step is given by (2.3). What we see is that every term with an odd number of D-operators cancels and every term with an even number is doubled by the identities (2.5) and (2.6). What’s left in (2.15) is a linear partial differential equation with constant coefficients which is, as usual, solved by some exponential expression. Let us therefore assume the solution f1 = exp(η) with η = ωt + px + qy + ... + η0, and where η0 is some constant. We then get,

P (ω, p, q, ...) exp(η) = 0 ⇐⇒ P (ω, p, q, ...) = 0. (2.16) Thus, if the coefficients of the variables in the exponent, η, satisfy the polynomial the assumed solution will correspond to an actual solution. The equation P (ω, p, q, ...) = 0 is the dispersion relation of the solution. Compare this with the linear dispersion relation

(10)

from the simplified classical wave equation: ft+ cfx = 0 ⇒ ω + cp = 0, p is in this case usually called the wave number [9].

Let us note here that we are explicitly looking for exponentially decaying solutions as they are the ones forming solitons, as opposed to plane wave solutions, which are those usually analyzed in for example the above classical wave equation. Some more information about this can be found in Hirota’s book [4].

Now we look at (2.13). We rearrange it using bilinearity and then the exact same procedure as in (2.15) for the "mirrored" terms including f2 to get the expression,

2P (∂)f2 = −P (D)(f1· f1). (2.17) We see that our ansatz for f1 together with (3.6) and (3.8) sets the right hand side to zero. We are left with

P (∂)f2 = 0, (2.18)

from where it is clear that we can choose f2 = 0 and still have a valid solution.

For (2.14) this means that we can instantly reduce the expression into

P (∂)f3 = 0. (2.19)

Once again, choosing f3 = 0 is a valid choice.

The same procedure then repeats for all other powers of ε and we conclude that the perturbation series is truncated by fn= 0 for all n > 1. We have arrived at a solution,

f = 1 + εf1 = 1 + exp(η). (2.20)

In the last step we have absorbed the perturbation coefficient ε into the constant η0 in the exponent. This is the one-soliton solution!

To find the two-soliton solution, and indeed the more general N -soliton solution, we utilize the fact that (2.15) is a linear differential equation and therefore satisfies the superposition principle. For the two-soliton solution, we pick the following ansatz,

f1 = exp(η1) + exp(η2). (2.21)

Here η1 and η2 both separately satisfy the dispersion relation (2.16). Let’s see what happens to (2.12) with this assumption. As above we arrive, identically, at the expression (2.15). As opposed to last time for the one-soliton, the right hand side doesn’t evaluate to zero. Instead,

P (D)(f1 · f1)

= P (D)(exp(η1) + exp(η2)) · (exp(η1) + exp(η2))

= P (D)(exp(η1) · exp(η1) + exp(η1) · exp(η2) + exp(η2) · exp(η1) + exp(η2) · exp(η2))

= 2P (D)(exp(η1) · exp(η2)).

Where the second equals sign is simply a result of the bilinearity and the third one comes from (2.5)-(2.7): the argument about terms cancelling or doubling depending on if the derivative is of even or odd order gives us the coefficient 2, and (2.7) removes the pairs

(11)

of identical exponents of the form exp(ηi) · exp(ηi). Now we apply (2.7) again to get the following,

2P (D)(exp(η1) · exp(η2)) = 2P (ω1− ω2, p1− p2, q1− q2, ...) exp(η1+ η2). (2.22) We are as such guided to choose the solution

f2 = a12 exp(η1+ η2), (2.23)

where a12 is given by

a12= −P (ω1− ω2, p1− p2, q1 − q2, ...)

P (ω1+ ω2, p1+ p2, q1+ q2, ...). (2.24) Looking at (2.17) we see that the numerator of a12 cancels the coefficient in (2.22) and the denominator cancels the result of the dispersion relation of the solution f2 on the left hand side, in a similar manner to how we got (2.16)

We now proceed with equation (2.14). In a similar manner to how we dealt with (2.13), by splitting it into two expressions. We get

P (D)(1 · f3+ f3· 1) = P (D)(f1· f2+ f2· f1). (2.25) The left hand side is reduced in exactly the same manner as how we arrived at (2.15) so let us look at the right hand side of the equation. We insert our ansatz for f1 and the result we arrived at for f2 and get,

P (D)([exp(η1) + exp(η2)] · a12 exp(η1+ η2) + a12 exp(η1+ η2) · [exp(η1) + exp(η2)])

= 2a12P (D)([exp(η1) + exp(η2)] · a12 exp(η1+ η2))

= 2a12(P (D)(exp(η1) · exp(η1+ η2)) + P (D)(exp(η2) · exp(η1 + η2)).

We finally use the rule (3.6) to get,

2a12P (ω1, p1, q1, ...) exp(2η1+ η2) + 2a12P (ω2, p2, q2, ...) exp(η1+ 2η2). (2.26) Guided by our previous experiences, we would like this entire expression to equal zero:

then we can choose f3 to zero and truncate the expansion of the sum. In fact, it is already zero. We chose the function f1 so that each term, independently, satisfies the polynomial P , that is, by superimposing valid solutions of (2.15). Since every valid solution of (2.15) also, by (2.16), satisfies the dispersion relation we conclude that the right hand side of (2.25) evaluates to zero. Then we are in the exact same spot as previously and need f3

to satisfy

P (∂)f3 = 0,

whereupon we can clearly choose f3 = 0. By similar arguments as before, then the series has been successfully truncated and we have a solution,

f = 1 + exp(η1) + exp(η2) + a12exp(η1+ η2). (2.27)

(12)

As above the expansion coefficients have been absorbed into the constants of the expo- nents. This is the two-soliton solution!

This same argument can be repeated, and by applying the properties of the D-operator at appropriate times the general N -soliton solution is found. The key step is simply to choose a number of terms in f1. This in turn decides the N in the solution and is interpreted as the number of independent solitons that the solution will decay into over time. A closed expression for the N -soliton solution is given by Hirota and restated below for reference, though it is of little computational value because of its complexity.

f =X exp

" N X

i=1

µiηi+

(N )

X

i<j

Aijµiµj

#

, (2.28)

where the firstP means summation over all possible combinations of µi = 0, 1 for all 1 ≤ i ≤ N , andP(N )

i<j represents a summation over all possible pairs (i, j) such that 1 ≤ i, j ≤ N and i < j. For example if we let N = 3 we get the 3-soliton solution which then looks like,

f =1 + exp(η1) + exp(η2) + exp(η3)

+ exp(A12+ η1+ η2) + exp(A13 + η1+ η3)

+ exp(A23+ η2+ η3) + exp(A12+ A13+ A23+ η1+ η2+ η3).

Note also that the coefficients are related by aij = exp(Aij).

Now we have shown that, for any equation which takes the form P (D)f · f = 0, with the vacuum constraint given above, admits N -soliton solutions of exactly the form given above. The only difference between them, governed by the underlying equation P , will be given by the coefficients and the dispersion relations that they satisfy. Now you may be wondering if there is any value to this, seeing as you may never have seen an equation on the form above. So how do we find this elusive bilinear form?

2.4 How to find the Bilinear form

Differential equations encountered often by mathematicians in all fields, are seldom rep- resented in the form required to find the N -soliton solutions. And it appears that finding which equations can be rewritten in this way is more of an art than one may hope.

Nevertheless, in this section we give an account of some things to keep an eye out for.

The goal here is to rewrite an equation into being expressed with D-operators. The first step: a change of variables.

2.4.1 Dependent variable transformations

Here we go through two of the most common variable transformations used to find bilinear forms. There are of course many variations of this and some degree of sensitivity is required in order to choose the correct transformation. Some results that we will need to to see the validity of the transforms are,

(13)

exp(δ∂x)a(x) = a(x + δ), (2.29) exp(δDx)a(x) · b(x) = a(x + δ)b(x − δ), (2.30) 2 cosh(δ∂x)a(x) = a(x + δ) + a(x − δ), (2.31) cosh(δDx)a(x) · a(x) = a(x + δ)a(x − δ). (2.32) These are all motivated by looking at the Taylor expansion around the point x of a(x + δ) given by

a(x + δ) =

X

n=0

(∂nxa)(x)

n! ((x + δ) − x)n=

X

n=0

(δ∂x)n n! a(x)

Which directly gives (2.29). Changing the sign of the exponent will change the sign of every term with an odd number of derivatives in the Taylor expansion, and the same pattern is seen for the second function in the input of the D-operator. Both of these, by the same observation as above, give translation in the other (negative) direction, as seen for b in (2.30). The remaining results follow directly from these observations. Now let’s look at our two transformations.

The rational transform

We start with the rational transformation. For some solution u of a nonlinear partial equation we can transform it into some rational function by,

u = a

b. (2.33)

where u, a, and b are all functions of the same real variables. If the nonlinear terms of the equation are expressed as some polynomial of u and its derivatives, then this transformation will result in some homogeneous expression with respect to a, b which can be decoupled into a set of quadratic forms. The following identity motivates this transformation,

exp(δ∂x)a

b = exp(δDx)a · b

cosh(δDx)b · b. (2.34)

The proof follows from direct application of the results (2.29), (2.30), and (2.32). Ex- panding both sides of the equation (2.34), with respect to the parameter δ and collecting the terms for each order of δn, which admittedly requires some algebraic manipulation of the expansions, yields,

∂x a

b = Dxa · b

b2 , (2.35)

2

∂x2 a

b = Dx2a · b b2 − a

b

Dx2b · b

b2 , (2.36)

3

∂x3 a

b = Dxa · b

b3 − 3Dxa · b b2

Dx2b · b

b2 , (2.37)

and so on. These relations are then applied to any equation, which will then often be decoupled, introducing some arbitrary decoupling function λ into a system of bilinear equations [4, 3]. Let’s move on to the next transformation.

(14)

The logarithmic transformation

Perhaps more ubiquitous than the previous transform, is the logarithmic transformation given by

u = 2α (log f )xx, (2.38)

where u, and f are functions of real variables and α is a free parameter. For this transform another motivating identity exists, namely,

2α cosh

 δ ∂

∂x



log f (x) = α log[cosh(δDx)f (x) · f (x)]. (2.39) Proved by using (2.31) and (2.32), as well as standard results about the logarithm as follows,

LHS = α log f (x + δ) + log f (x − δ)

= α log[f (x + δ)f (z − δ)] = RHS.

As before, expanding both sides with respect to δ and collecting terms in powers of δ, gives the relations

2α ∂2

∂x2log f = αD2xf · f

f2 , (2.40)

2α ∂2

∂x∂tlog f = αDxDtf · f

f2 , (2.41)

2α ∂4

∂x4log f = α Dx4f · f

f2 − 3 D2xf · f f2

2

, (2.42)

and so on. (2.41) comes from the simple substitution of Dx to Dz = Dt + εDx and regarding f as a function of the independent variable z.

Now we are equipped with tools and knowledge about the D-operator strong enough to apply it to the KdV and Boussinesq equations.

(15)

Chapter 3

Solving the KdV and Boussinesq equations

3.1 Introduction

In this section we will show how to use the Hirota Direct Method to solve nonlinear differ- ential equations. Specifically, we will find the soliton solutions for both the Korteweg-de Vries (KdV) equation and the Boussinesq equation and present the closed form expres- sion for the first few of these.

3.2 Korteweg-de Vries equation

The KdV equation in (1 + 1) dimensions is given by,

ut+ uxxx+ 6uux = 0, (3.1)

where u(x, t) is a function of the real variables x and t > 0. The 6 can be exchanged with any arbitrary real constant via a simple change of variables, yet this form above is the most common way to present it.

It arises naturally in a range of physical applications pertaining to long, propagating, one-dimensional waves; for an example, it can be used to model waves on shallow water surfaces [6]. The equation is of profound historical importance; indeed, the advent of the modern soliton theory begins with the numerical discovery of its soliton solutions [10]. It is often treated in introductory texts, as its solution can serve as a demonstration of many fundamental techniques. Having given an account of Hirota’s method we will use it to find the soliton solutions of the KdV equation. As we shall see, the procedure will provide much insight into how the Hirota Direct Method is used to solve differential equations in practice.

3.2.1 Solution using the Hirota Direct Method

Our strategy will consist of three steps. First, we shall use one of the variable trans- formations presented in the previous section to recast the equation. Then, using the relations corresponding to that transformation we will rewrite the equation into bilinear form. Finally, the soliton solutions will be recovered explicitly using Hirota’s method.

(16)

Variable transformation

We begin by using the logarithmic transform as described in section 2.4.1 with α = 1.

That is, we apply the variable transformation given by,

u = 2(log f )xx, (3.2)

and insert this into (3.1). This yields,

2(log f )xxt+ 2(log f )xxxxx+ 24(log f )xx(log f )xxx = 0, (3.3) which can be integrated with respect to x once which, when setting the integration constant to zero gives the equation,

2(log f )xt+ 2(log f )xxxx+ 3(2 log f )2xx = 0. (3.4) To continue we must apply the relations between derivatives of the logarithm and the D-operator as discussed in section 2.4.1.

Finding the Bilinear form

Let us therefore apply relations (2.40)-(2.42), one for each term in equation (3.4). We get,

DxDtf · f

f2 + Dx4f · f

f2 − 3 Dx2f · f f2



+ 3 Dx2f · f f2

2

= DxDtf · f

f2 +D4xf · f f2 = 0.

Now multiplying by f2 on both sides, and factoring by Dx we arrive at,

Dx(Dt+ Dx3)f · f = 0, (3.5)

a bilinear form of the KdV equation. Now, let’s see how the soliton solutions of this equation look like.

Soliton solutions

As detailed, an equation written in the bilinear form (3.5) permits soliton solutions. Here we will present the first two. Let’s start with the one-soliton. It’s given by

f = 1 + exp(η), (3.6)

where η = ωt + P x + η0 and ω + P3 = 0. This function must of course be re-converted into the original function u. We note that the second derivative of the logarithm is given by

(log(f ))xx = f fxx− fx2

f2 (3.7)

(17)

which results in,

u = 2(1 + exp(η))P2exp(η) − (P exp(η))2 (1 + exp(η))2

= 2P2 exp(η)

(1 + exp(η))2 = 2P2 1

(exp(−η2) + exp(η2))2

= 2P2sech2

2), (3.8)

which is a nice closed form expression for the one-soliton equation for the KdV equation.

The two-soliton solution is in turn given by

f = 1 + exp(η1) + exp(η2) + a12exp(η1+ η2), (3.9) where ηi = ωit + Pix and ωi + Pi3 = 0 for i = 1, 2. As for the one-soliton we put this function into (3.2) and use (3.7) to get,

u = (P12exp(η1) + P22exp(η2) + (P1+ P2)2a12exp(η1+ η2)) (1 + exp(η1) + exp(η2) + a12exp(η1+ η2))

− (P1exp(η1) + P2exp(η2) + (P1+ P2)a12exp(η1+ η2)) (1 + exp(η1) + exp(η2) + a12exp(η1+ η2))

2

,

(3.10)

which is, after some relatively extensive algebraic manipulation, equivalent to, u = 2(P22− P12)P22cosech22) + P12sech21)

(P2coth(η2) − P1tanh(η1))2 . (3.11) As we start to realize, the procedure to express the N -soliton solution in a closed analytic expression gets increasingly cumbersome. However using some computer algebra software makes it quite dynamic to arrive at that point using simply the general form of the N - soliton given by (2.28) and applying the correct reverse transform.

Thus we have arrived at the analytic solutions of the KdV equation. As is often the case, there is much wisdom to be gained by looking at these equations and appreciating their properties, but the strength of these solutions will become even more evident as we look on our next equation, the Boussinesq equation.

3.3 The Boussinesq equation

The Boussinesq equation is given by

utt− uxx+ uxxxx− (u2)xx = 0, (3.12) where u(x, t) is a function of the real variables x and t > 0. Technically, the above equa- tion is known as the "good" Boussinesq equation; it is similar to the "bad" Boussinesq, with the latter differing only in the sign of the second term; the name comes from the former being numerically stable while the latter is not [7]. Much like the KdV equation, the Boussinesq equations can be used to successfully model very long, one-dimensional water waves [1] and, much like the KdV equation, it is solvable via the Hirota Direct Method. As we shall see, the processes for solving the two equations are very similar, demonstrating the generality of the tools we have developed this far.

(18)

3.3.1 Solution using the Hirota Direct Method

In solving the Boussinesq equation we will follow the exact same procedure as in the previous sections: we will begin by performing a suitable change of variables to transform the equation into a more manageable shape, we will then identify the bilinear form of the equation, before finally retrieving the soliton solutions.

Variable transformation

We, again, apply the logarithmic transformation. This time with different choice of α = −3. The transform is explicitly given by,

u = −6(log f )xx. (3.13)

This is inserted into (3.12) to get,

−6(log f )xxtt+ 6(log f )xxxx− 6(log f )xxxxxx− 36((log f )xx)2)xx = 0, (3.14) which is integrated twice, to give,

−6(log f )tt+ 6(log f )xx− 6(log f )xxxx− 36(log f )xx)2 = 0. (3.15) Now we divide both sides by −3 and factor the coefficient of the non-linear term to get the equation,

2(log f )tt− 2(log f )xx+ 2(log f )xxxx+ 3(2(log f )xx)2 = 0. (3.16) Now that we have a form with terms we can deal with we can move on to finding the bilinear form.

Finding the Bilinear form

We apply relations (2.40) and (2.42) to (3.16) and get the following equation, D2tf · f

f2 − Dx2f · f

f2 + Dx4f · f

f2 − 3 Dx2f · f f2



+ 3 Dx2f · f f2

2

= 0, which gives us the final bilinear form of the Boussinesq equation,

(Dt2− Dx2+ Dx4)f · f = 0. (3.17) Soliton solutions

Once again, we have arrived at an equation in bilinear form which, as we now know, permits soliton solutions. The procedure is exactly the same as in section 3.2, aside from the fact that the exponents, ηi = ωt + P x, must satisfy the new dispersion relation given by ωi2 − Pi2 + Pi4 = 0, which is derived directly from (3.17), as well as changing the multiple of the solutions. Thus we get, the one soliton:

u = −6P2sech2

2), (3.18)

and the two soliton:

u = −6(P22− P12)P22cosech22) + P12sech21)

(P2coth(η2) − P1tanh(η1))2 . (3.19) And just like that, we see the power of Hirota’s direct method in making sense of equations

(19)

Chapter 4

Summary and Conclusions

We have now gone through a brief history of the soliton and the class of equations associated with it. We have then, in detail, derived and motivated the Hirota Direct Method, introducing the D-operator, and showing some canonical ways of finding the bilinear form of a differential equation. Finally we have showcased the strength of this method in finding these soliton solutions for a class of functions exemplified by the KdV equation and the Boussinesq equation.

We finally arrived at seeing the strength of the Hirota Direct Method. Although there are other powerful, but cruder methods such as the inverse scattering transform [5], the direct method has proved to be an accessible and enlightening way of analyzing bilinearizable differential equations. In fact, because of the distinct, formulaic shape of the soliton solutions the entire problem reduces to finding a good transform for the considered equation.

(20)

Bibliography

[1] Joseph (1842-1929). Auteur du texte Boussinesq. Essai sur la théorie des eaux courantes / par J. Boussinesq. fre. Impr. nationale (Paris), 1877.

[2] P. G Drazin. Solitons : an introduction. eng. Cambridge texts in applied mathe- matics. Cambridge: Cambridge Univ. Press, 1989. isbn: 0-521-33389-x.

[3] P R Graves-Morris. “Pade approximants and their applications. Proceedings of a conference held at Canterbury, England, July 17–21, 1972”. In: (Jan. 1973). url:

https://www.osti.gov/biblio/4480428.

[4] R. Hirota. The Direct Method in Soliton Theory. Trans. Japanese by C. Gilson A. Nagai J. Nimmo. Reading, Massachusetts: Cambridge University Press, 2004.

[5] Ryogo Hirota. “Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons”. In: 27.18 (Nov. 1971), pp. 1192–1194. doi: 10 . 1103 / PhysRevLett.27.1192.

[6] D. J KORTEWEG and G DE VRIES. “XLI. On the Change of Form of Long Waves advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves”.

eng. In: Philosophical magazine (Abingdon, England) 91.4-6 (2011), pp. 1007–1028.

issn: 1478-6435.

[7] V. S MANORANJAN, A. R MITCHELL, and J. L MORRIS. “Numerical solutions of the good boussinesq equation”. eng. In: SIAM journal on scientific and statistical computing 5.4 (1984), pp. 946–957. issn: 0196-5204.

[8] F. Sjölander. “Numerical solutions to the Boussinesq equation and the Korteweg–de Vries equation”. 2021.

[9] Hugh D. Young. Sears and Zemansky’s university physics : with modern physics.

eng. 14. ed., global edition. 2015. isbn: 9781292100319.

[10] Yi Zang. “Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the”. eng. In: Encyclopedia of Complexity and Systems Science. New York, NY: Springer New York, pp. 5138–5143. isbn: 0387758887.

(21)

www.kth.se

References

Related documents

I dag uppgår denna del av befolkningen till knappt 4 200 personer och år 2030 beräknas det finnas drygt 4 800 personer i Gällivare kommun som är 65 år eller äldre i

Detta projekt utvecklar policymixen för strategin Smart industri (Näringsdepartementet, 2016a). En av anledningarna till en stark avgränsning är att analysen bygger på djupa

DIN representerar Tyskland i ISO och CEN, och har en permanent plats i ISO:s råd. Det ger dem en bra position för att påverka strategiska frågor inom den internationella

While firms that receive Almi loans often are extremely small, they have borrowed money with the intent to grow the firm, which should ensure that these firm have growth ambitions even

Effekter av statliga lån: en kunskapslucka Målet med studien som presenteras i Tillväxtanalys WP 2018:02 Take it to the (Public) Bank: The Efficiency of Public Bank Loans to

Indien, ett land med 1,2 miljarder invånare där 65 procent av befolkningen är under 30 år står inför stora utmaningar vad gäller kvaliteten på, och tillgången till,

Det finns många initiativ och aktiviteter för att främja och stärka internationellt samarbete bland forskare och studenter, de flesta på initiativ av och med budget från departementet

Den här utvecklingen, att både Kina och Indien satsar för att öka antalet kliniska pröv- ningar kan potentiellt sett bidra till att minska antalet kliniska prövningar i Sverige.. Men