Dear Amitha,

Travelling waves arise naturally in many physically systems, usually described by partial differential equations. Solitary waves, also known as 'solitons', are are a particular class of travelling wave that have special properties.

Solitons can usually propagate over large distances without dissipation due to certain nonlinear effects cancelling out dispersive effects. They also have the additional property that they can interact with other solitons such that they emerge following a collision without changing shape, apart for a small phase change.

John Scott-Russel was the first person to report, in 1844, the sight of a solitary wave on the Edinburgh-Glasgow canal. It travelled approximately 2 miles at around 9 miles/hour.

There is a very comprehensive literature on the subject, and a famous equation that sustains solitons is the Korteweg-deVries equation - see Wikipedia entry. Many additional resources will be found on the Internet.

I hope this helps a little.

Kind regards,

Graham W Griffiths

Thank you very much dear Prof.

But how to make out whether the Fig. Represents travelling wave or solitary wave?

What is the nature of solution wave for mKdV-Burgers’ equation for very small diffusive parameter?

It becomes nonlinear dispersive PDE with nonconvex flux function

Thank you once again sir for your answer

Dear Amitha,

A solitary wave, as its name suggests, usually consists of a single disturbance that propagates over time. It is also possible for a train of solitary waves to exist in a medium at the same time and in close proximity. These may consist of solitary waves of different shapes and sizes.

A common solitary wave travelling with velocity 'c', typically has the form:

u(x,t) = (1/2)*sech^2( (c/2)*sqrt(c*(x-c*t))/2 )

The general term "travelling waves" applies to all forms/types of wave that move through a medium.

Travelling waves that are of the soliton type usually have 'invariant' properties such that certain integrals over all space are constant.

For a general overview refer to:

Griffths, G. W., and W. E. Schiesser, Linear and nonlinear waves, Scholarpedia, 4, no. 7, 2009, p 4308. Available online at:

http://www.scholarpedia.org/article/Linear and nonlinear waves.

Also books at: www.pdecomp.net

To see how travelling wave solutions can be obtained study the tanh() method introduced by the following papers:

Malfliet, W., Solitary wave solutions of nonlinear wave equation, American Journal of Physics, 60, no. 7, 1992, pp 650-654

Malfliet, W., and W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54, 1996, pp 563-568

Malfliet, W. and W. Hereman, The tanh method: II. Perturbation technique for conservative systems, Physica Scripta, 54, 1996, pp 569-575

Additional methods such as: exp, Riccati, coth, etc. are also useful.

The Computer algebra system 'Maple' also has a general method 'TWSolutions' built into the program that uses all the above approaches, and more.

Kind regards,

Graham W Griffiths

Thank you so much sir.

Infact, I was going through all the types of nonlinear waves you have mentioned in

Griffths, G. W., and W. E. Schiesser, Linear and nonlinear waves, Scholarpedia, 4, no. 7, 2009, p 4308.

My doubts are almost clear now.

Thank you ,

Regards

Dear Amitha,

To follow on from my previous answer, it appears that as the diffusive parameter 'nu' of the mKdV_Burgers equation becomes small, the wave amplitude will reduce and the velocity of propagation will slow. This is based on a tanh() solution of the form:

u(x,t) = (1/18)*sqrt(18)*sqrt(1/mu)*nu-(1/18)* sqrt(18)*sqrt(1/mu)*nu*tanh((1/54)*nu*(9*x*mu+9*x0*mu-2*nu^2*t)/mu^2)

Not sure if that helps.

Kind regards,

Graham W Griffiths

Yes Prof. Graham W Griffiths. Its of great help.

I am using Laplace Decomposition Method to solve mKdV-Burgers' equation and studying the dissipative effects.

Thanks a lot

Dear sir,

In continuation of the question mentioned regarding the solitary wave solution and travelling solution, what type of wave is observed in the attachment Fig.

The Fig. depicts 1-term, 3-term and 5-term LDM of modified KdVB equation with quadratic source term and hyperbolic initial condition ?

Can one make out what type of wave just by looking at the FIg.?

Thanks in advance

I am not sure from your plots exactly what type of TW you have shown. Generally, solitons travel without distortion, but they come in different types, each with their own distinct shape, see:

http://www.scholarpedia.org/article/Soliton

Not sure that this helps.

Graham W Griffiths

Thank you very much sir

My doubt was

just by looking at any type of graph, is it possible to identify what type of wave it is?

Say for example, a kink wave is of some particular shape, Peakons of different kind, etc.

The Fig. in the attachment is the successive approximations of

u_t+〖〖αu〗^2 u〗_x-u+u^2=〖ν u〗_xx+〖δu〗_(xxx ) for u(x,0)=〖-2sech〗^2 (x) and α=1, ν=1,δ=1

Laplace Decomposition method has been used to obtain the successive approximations.

I once again thank you sir for your kind reply. I am just going through the nature solitary waves from all possible resources available.

Thank you for your resource material and it is definitely useful.

Regards

Amitha Rao

Dear Amitha,

The initial condition you are using does not seem to be a solution to the PDE at t=0.

The numerical solution of a PDE that has soliton solutions 'must' be initialised to an exact soliton TW solution. Otherwise the solution is likely to become unstable or develop a train of solitons. It is this behaviour that started Zabasky and Kruskal on their investigations in the early 1960's, see:

- Zabusky, N. J., and M. Kruskal, (1965). Interaction of Solitons in a Collisionless Plasma and the Recurrence of Initial States. Physical Review Letters, 15, vol. 6, 240-243.

- Zabusky, N. J., and M. A. Porter (2010). Solitons. Scholarpedia, 5(8):2068. Available on-line at: http://www.scholarpedia.org/article/Solitons.

Kind regards,

Graham W Griffiths

Great sir

Thank you very much

Then you mean to say the initial condition should satisfy the given PDE ?

Or is it ok if it is approximately equal to zero?

Or

Is it ok if PDE is satisfied for some values of flow parameters?

Thank you

Dear Amitha,

If the initial condition does not satisfy the PDE, then it is unlikely that it will generate a soliton that propagates through the medium. This is primarily due to the initial condition not satisfy the invariants.

Kind regards,

Graham W Griffiths

Dear sir

Thank you very much fof your reply.

Thank you very much for sparing your time to clear the doubts

Your answers are always with clarity and understandable.

Thank you for your kind support

Solitary wave is a special kind of travelling wave for a suitable nonlinear PDE. Solitary wave can form due to delicate balance between non-linearity and dispersion.

Thank you very much Prof. Asit Saha and Prof. GIles Bokolo for your input

Thanks Prof Graham W Griffiths for your comments on Solitary waves and solutions.

This will of great help to me also.

Regards

Thank you all for the discussions.

Thanks for all the above discussions.

Sir,

Continuing the above, I would like to know the following.

1) Is it possible to identify the solitary wave solution just by looking at the graphs of the solution ? It is something similar to the identification of the shapes of Sine wave, Exponential wave, Cosine wave, etc.

2) As I understood these solitary waves are due to the balance between nonlinearity and dispersion, how come the Burgers' equation has solitary wave solution?

In the Burgers' equation given by:

u_t+uu_x+u=mu_xx, there is no dispersion term(u_xxx).

Thanks in advance

In fact solitary wave or solitons are identified by graphs only. In solitary wave solution something is moving forward without changing its shape. If there are two solitons moving with their different speeds then they may collide to form a new shape temporarily and thereafter they move with same shapes.

Burgers' equation does not possess solitary solution. However, it does satisfy some conservation equations.

Thanks a lot sir for your answer

I read many publications where soliton wave solutions are obtained for Burgers' equation

I send you the details

Please see the attachment where soliton of the Burgers equation is obtained.

There are many publications which talk about soliton solutions of Burgers' equation.

Thank you sir

Dear Amitha, this is from what book. Thank you .

Very fruitful discussion indeed.

Thanks to all.

Nice discussion. I have a doubt, what is the difference between solitary waves and bores? Can we consider solitary wave as a synonym of the bore?