Manual Nonlinear Waves in Solid State Physics

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References Publications referenced by this paper. G -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Wang , X.

New exact travelling wave solutions using modified extended tanh-function method S. El-Wakil , M. The tanh method for traveling wave solutions of nonlinear equations Abdul-Majid Wazwaz. Partial differential equations : methods and applications Abdul-Majid Wazwaz. These solutions include soliton, topological soliton, singular soliton and generalized solitary wave solutions which might be helpful to analyze complex phenomena.

In this section, we present the description of the method. The most important steps of the new ansatz are as follows:.

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Step 1: In order to find traveling wave solutions of 1 , we introduce the wave variable. Step 2: Integrate 3 term by term one or more times according to possibility that yields constant s of integration. The integration constant s may be zero for simplicity. Step 3: We assume the traveling wave solution of 3 can be expressed in the following form:. Step 4: To fix the positive integer N , we consider the homogeneous balance between the highest-order linear terms with the nonlinear terms of the highest order in 3.

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Setting each coefficient to zero yields an over-determined system of algebraic equations. Step 6: The general solutions of 5 yield:. Substituting these constants and the solutions given in 6 into 4 , we obtain wave solutions of Eq. This equation was derived to describe an approximation for surface long waves in nonlinear dispersive media. It can also characterize the hydromagnetic waves in cold plasma, acoustic waves in inharmonic crystals and acoustic gravity waves in compressible fluids.

We now examine solutions to 7 by using the proposed ansatz scheme. By means of the traveling wave transformation 2 and integrating once, we obtain. Therefore, the suggested ansatz scheme allows us to set the solution of 8 as follows:.

Solution of this over-determined system of algebraic equations with the help of computer algebra yields:. Using 6 into the solution 9 and substituting Cases 1—9, we obtain abundant traveling wave solutions including soliton, singular soliton, periodic solution, etc.

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If we set specific values of A and B , various known solutions can be rediscovered. For example, soliton, periodic and complex solutions can be derived from the traveling wave solutions 10 :. These are exact bell-type soliton solutions. Solution 12 is the singular soliton solution.

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Singular solitons are another kind of solitary waves that appear with a singularity, usually infinite discontinuity Wazwaz Singular solitons can be connected to solitary waves when the center position of the solitary wave is imaginary Drazin and Johnson Therefore it is not irrelevant to address the issue of singular solitons. This solution has spike and therefore it can probably provide an explanation to the formation of Rogue waves. When A and B receive particular values, different known solutions will be rediscovered. For example:.

Periodic traveling waves play an important role in numerous physical phenomena, including reaction—diffusion—advection systems, impulsive systems, self-reinforcing systems, etc.

Mathematical modelling of many intricate physical events, for instance biology, chemistry, physics, mathematical physics and many more phenomena resemble periodic traveling wave solutions. Setting particular values of the free parameters involved in solutions 10 — 38 abundant soliton, singular solitons, periodic solutions and general solitary wave solutions can be found. It is noteworthy to refer that some of our obtained solutions are identical to the solutions achieved by Zayed and Al-Joudi which validate our solutions and some are new.

These solutions might be much important for the explanation of some special physical phenomena. The solitary wave ansatz method can be applied to high-dimensional or coupled nonlinear PDEs in mathematical physics. We expect the attained solutions may be useful for further numerical analysis and may help the researchers to explain complex physical phenomena.

The solutions obtained in this article are in more general forms and many known solutions to this equation are only special cases. This study shows that the proposed ansatz is reliable, effective and computerized which permit us to carry out complicated and tiresome algebraic calculation and giving new solutions to the applied equation. This ansatz can be applied to both single equation and coupled equations to establish further new solutions for other kinds of nonlinear partial differential equations.

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Cambridge University Press, Cambridge. Adomian G Solving frontier problems of physics: the decomposition method. Kluwer Academic, Boston. Prog Appl Math 1 2 — Commun Theor Phys 57 2 — Math Prob Engr World Appl Sci J 16 11 — Phys Scr 85 3 Appl Math Comput —