In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance.

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Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.

It is eminent that nonlinear partial differential equations NLPDEs are widely used as models to depict many important complex physical phenomena in a variety of fields of science and engineering, such as, nonlinear optics, solid state physics, plasma physics, chemical kinematics, fluid mechanics, chemistry, biology and many others. For instance, the Backlund transformation method Rogers and Shadwick , the inverse scattering method Ablowitz and Clarkson , the Darboux transformation method Rogers and Schief , the homogeneous balance method Wang ; Wang et al.

A further significant method was introduced by Wang et al. Thus, diverse group of scientists studied different kind of NLPDEs and constructed solitons, topological solitons, singular solitons and other kinds of general solitary wave solutions via this method. Applications of this method can be found in references, Akbar et al.

For example, Zhang et al. Li et al. Akbar et al. To depict the novelty, reliability and advantages, we compare the acquired solutions to those solutions attained by Zayed and Al-Joudi 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:. 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. Setting each coefficient to zero yields an over-determined system of algebraic equations.

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.

## An ansatz for solving nonlinear partial differential equations in mathematical physics

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. 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. This work was carried out in collaboration between the authors. Both authors have a good contribution to design the study, and to perform the analysis of this research work.

Both authors read and approved the final manuscript. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions.

The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems. It is eminent that nonlinear partial differential equations NLPDEs are widely used as models to depict many important complex physical phenomena in a variety of fields of science and engineering, such as, nonlinear optics, solid state physics, plasma physics, chemical kinematics, fluid mechanics, chemistry, biology and many others.

## [] Heun Functions and Some of Their Applications in Physics

For instance, the Backlund transformation method Rogers and Shadwick , the inverse scattering method Ablowitz and Clarkson , the Darboux transformation method Rogers and Schief , the homogeneous balance method Wang ; Wang et al. A further significant method was introduced by Wang et al. Thus, diverse group of scientists studied different kind of NLPDEs and constructed solitons, topological solitons, singular solitons and other kinds of general solitary wave solutions via this method.

Applications of this method can be found in references, Akbar et al. For example, Zhang et al. Li et al. Akbar et al.

To depict the novelty, reliability and advantages, we compare the acquired solutions to those solutions attained by Zayed and Al-Joudi 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:. 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. Setting each coefficient to zero yields an over-determined system of algebraic equations. 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. 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. 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.

### Bestselling Series

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. This work was carried out in collaboration between the authors.

Both authors have a good contribution to design the study, and to perform the analysis of this research work. Both authors read and approved the final manuscript. The authors also would like to express their gratitude to the anonymous referees for their valuable comments and suggestions.