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Computer Algebra Recipes

Richard H Enns, George C McGuire

Chapter 7

The Hunt for Solitons - all with Video Answers

Educators

Chapter Questions

04:30

Problem 1

Explore how the sine-Gordon solitary waves vary in shape as the velocity $c$ is altered.

Shoukat Ali
Shoukat Ali
Other Schools
02:26

Problem 2

If the sine term is replaced with a cosine in the SGE, how would the solitarywave solutions be affected? Confirm your reasoning by running the text recipe with a cosine present, instead of the sine term. You will have to alter the initial conditions to obtain the new separatrixes.

Benjamin Chaback
Benjamin Chaback
Numerade Educator
02:34

Problem 3

Suppose that the nonlinear term in the SGE is replaced with $\sin ^2 \psi$. Using the phase-plane portrait approach, determine whether there is a solitary-wave solution to this modified SGE.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:56

Problem 4

Modify the text recipe to determine the solitary-wave solutions for the minussign case in the NLSE. Remembering that the physically observed intensity is proportional to $|\psi|^2$, confirm that these solutions are black solitary waves. These solitary waves are collisionally stable, so are black solitons.

Nick Johnson
Nick Johnson
Numerade Educator
01:18

Problem 5

The NLSE for a dielectric with a saturable refractive index takes the form
$$
i \frac{\partial \psi}{\partial x}+\frac{1}{2} \frac{\partial^2 \psi}{\partial t^2}+\frac{|\psi|^2}{1+a|\psi|^2} \psi=0,
$$
where $i=\sqrt{-1}$ and $a$ is a positive parameter. Taking $a=0.5$ and assuming a solution of the form $\psi(x, t)=U(t) e^{i b x}$ with $b=1$, use the phase-plane portrait to demonstrate graphically that a solitary-wave solution exists. An analytic form is not known for this solitary wave.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:58

Problem 6

Burger's equation
$$
\frac{\partial \psi}{\partial t}+\psi \frac{\partial \psi}{\partial x}=\sigma \frac{\partial^2 \psi}{\partial x^2},
$$
with $\sigma$ a positive parameter, is an example of a nonlinear diffusion equation. Graphically show that an antikink solitary-wave solution exists to Burgers' equation for a representative value of the diffusion coefficient $\sigma$.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
03:28

Problem 7

The Boussinesq wave equation, which was first derived in an attempt to describe shallow-water waves ( $\psi$ is the surface displacement) propagating in both directions, is
$$
\frac{\partial^2 \psi}{\partial x^2}-\frac{\partial^2 \psi}{\partial t^2}+6 \frac{\partial^2\left(\psi^2\right)}{\partial x^2}+\frac{\partial^4 \psi}{\partial x^4}=0 .
$$
Using the phase-portrait option, show that a bright solitary-wave solution exists for this equation.

Chai Santi
Chai Santi
Numerade Educator

Problem 8

Modify the recipe in the text to explicitly graph the solitary sound wave profile as a function of $z$.

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02:32

Problem 9

Discuss how the three solitary-wave profiles vary in shape as the values of $\gamma_1$, $\gamma_2$, and $\gamma_3$ are altered in the text recipe. For example, try $\gamma_1=1, \gamma_2=2$, and $\gamma_3=3$. Support your discussion with the profile plots in each case. Note whether any of the profiles is still a solitary wave or is a wave train.

Dading Chen
Dading Chen
Numerade Educator

Problem 10

Derive a solitary-wave solution of the modified KdV equation
$$
\frac{\partial \psi}{\partial t}+\alpha \psi^2 \frac{\partial \psi}{\partial x}+\frac{\partial^3 \psi}{\partial x^3}=0
$$
which appears in the theory of double layers in plasmas and as a model of ion acoustic solitons in a multicomponent plasma.

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05:03

Problem 11

The Boussinesq water wave equation is
$$
\frac{\partial^2 \psi}{\partial x^2}-\frac{\partial^2 \psi}{\partial t^2}+6 \frac{\partial^2}{\partial x^2}\left(\psi^2\right)+\frac{\partial^4 \psi}{\partial x^4}=0 .
$$
Derive the analytic form of the bright solitary-wave solution and animate it.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:09

Problem 12

Show that the choice of the negative square root yields an antikink solution.

Carson Merrill
Carson Merrill
Numerade Educator
01:04

Problem 13

Is there any relation between the maximum nonzero amplitude and the velocity? Is this the same sort of relationship as for the KdV solitary wave or is it different?

Mayukh Banik
Mayukh Banik
Numerade Educator
04:58

Problem 14

Burgers' nonlinear diffusion equation is of the form
$$
\frac{\partial U}{\partial t}+U \frac{\partial U}{\partial x}=\sigma \frac{\partial^2 U}{\partial x^2},
$$
where $\sigma$ is the positive diffusion coefficient. Analytically derive an antikink solitary-wave solution to Burgers' equation and animate it. How do the width of the antikink region and the velocity depend on amplitude?

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
10:30

Problem 15

The SGE permits a moving (velocity $v$ ) "breather"-mode solution, which is localized in space but oscillatory in time, of the form,
$$
\psi=4 \arctan \left(\sqrt{\frac{m}{1-m}} \frac{\sin (\gamma \sqrt{1-m}(t-v x))}{\cosh (\gamma \sqrt{m}(x-v t))}\right),
$$
with $\gamma=1 / \sqrt{1-v^2},-1<v<1$, and $0<m<1$. The factor $\gamma$ is the special Lorentz transformation of relativity with speed of light equal to one.
(a) Confirm that $\psi$ is a solution of the SGE.
(b) Animate $\psi$ for $m=\frac{1}{2}$ and (i) $v=0$, (ii) $v=0.5$, (iii) $v=-0.9$.

Gabriel Eduok
Gabriel Eduok
Numerade Educator
05:33

Problem 16

In the two-soliton kink-kink solution, replace the first $c$ by $1 / c, x$ by $c t$, and $c t$ by $x$. Animate the resulting solution and show that it represents a kinkantikink collision. Describe the observed behavior.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:15

Problem 17

Using the Taylor expansion, derive the approximation (7.5) to $\partial^3 \psi / \partial^3 x$.

Zachary Mitchell
Zachary Mitchell
Numerade Educator

Problem 18

In the Zabusky-Kruskal finite difference scheme for the KdV equation, $\psi$ in the nonlinear term $\psi(\partial \psi / \partial x)$ was approximated by the average of three $\psi$ terms at the grid points $(i+1, j),(i, j)$, and $(i-1, j)$. Compare the results obtained in the text recipe with those you would obtain if $U \equiv \psi$ were approximated by $U_{i, j}$ alone. Discuss your result.

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02:35

Problem 19

Multiply the smallest of the three solitary waves in the text recipe by a factor of 3 and interpret the resulting behavior when the worksheet is executed. Explore the effect of multiplying one or more pulses by numerical factors.

Mayukh Banik
Mayukh Banik
Numerade Educator
02:15

Problem 20

In the text recipe, change the sech ${ }^2$ terms in the input pulses to $\operatorname{sech}^4$ terms, then execute the modified recipe, and discuss the results.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
09:11

Problem 21

In the text recipe, double the amplitudes of the input kink and antikink solitarywave profiles. Remembering to also double the value $2 \pi$ in the initialization statement to avoid causing an end-effect problem, run the file with the amplified input and interpret the outcome.

Susan Hallstrom
Susan Hallstrom
Numerade Educator

Problem 22

Modify the recipe to simulate the collision of a kink solitary wave with another kink. Discuss the observed behavior.

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Problem 23

Modify the recipe to simulate the collision of an antikink solitary wave with another antikink. Discuss the observed behavior.

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Problem 24

Solve the problem of the text recipe using an explicit scheme based on a rectangular mesh.

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Problem 25

The interaction of two intense laser pulses of different frequencies as they pass through each other in opposite directions in a certain resonant absorbing fluid can be described [RE76] by the following normalized PDEs for the laser intensities $U$ and $V$,
$$
\frac{\partial U}{\partial x}+\frac{\partial U}{\partial y}=-g_1 U V-\alpha U, \quad \frac{\partial V}{\partial x}-\frac{\partial V}{\partial y}=-g_2 U V+\alpha V .
$$
Here $x$ is the normalized distance inside the fluid medium of length one unit, $y$ the normalized time, $g_1>0$ and $g_2>0$ are the "gain" coefficients, and $\alpha \geq 0$ the absorption coefficient. The $U$ pulse travels in the positive $x$ direction, while the $V$ pulse moves in the negative $x$ direction.
(a) Find the characteristic directions along which the PDEs reduce to ODEs.
(b) Devise an explicit numerical scheme that integrates the ODEs along the characteristic directions assuming that there are no pulses initially inside the fluid $(U(x, 0)=V(x, 0)=0$ for $0<x<1)$ and identical finiteduration $U$ and $V$ pulses are fed in at opposite ends $(U(0, y)=V(1, y)=$ $f(y)$ for $0 \leq y \leq Y=\frac{1}{2}$ and zero for $\left.y>Y\right)$.
(c) Numerically solve the equations and animate the results, assuming that $f(y)=1, g_1=0.4, g_2=20$, and (a) $\alpha=0$, (b) $\alpha=0.5$.
(d) Discuss the behavior of the two pulses as revealed in the animation.
(e) Repeat the calculation and animation for $f(y)=\sin (2 \pi y)$, the parameter values and all boundary and initial conditions remaining the same. Compare the results with those obtained for the rectangular pulses.

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