New soliton solutions for Kaup-Boussinesq system

Abstract

In this Letter, we study Kaup-Boussinesq system by using the wellknown He's variational approach. In fact, the He's variational method is a promising method to various systems of linear and nonlinear equations.

Review of the Legal Principles of Qisas in Imamieh Jurisprudence: An Emphasis on Ayatollah Khoyi's Points of View

“…Ayatullah Khoyi says: "If the murderer is insane, he will not be retaliated, no difference the victim is wise or insane, but Agheleh has to pay for it. The criterion for the unreliability of retaliation is being insane at the time of committing the crime (Najafi, 2013).…”

“…Generally, maturity is one of the conditions for the punishment, so the immature child is not killed for killing someone. The late Sheikh Mohammad Hassan Najafi, the authour of the "Javaher al-Kalam", writes: the intentional and wrongful acts of the kids are the same; in other words, the intentional act of the kid is his fault (Najafi, 2013).…”

“…Allameh Helli in the "Mokhtasar" (Mohaqiq Heli, 1997) and in "Sharayee" (Mohaqiq Heli, 1988), Mohaqiq Heli in "Tabsareh" ( tabserat-al motaalemin fi ahkame al-din,), Najafi in "Javaher-el-Kalam" (Najafi, 2013), And Imam Khomeini, in "Tahrir al-Wasilah" (Khomeini, 1989), they all believe that the child, as long as he is not adult, is not sentenced to death, and "Shahid Thani" in "al-Ruzat al-Bahyah" (Shahid Thani, 1990) says: the discerning child, is not killed for murder, but Agheleh has to pay for it." Sheikh Tousi" in his book called "Al-khalaf" has claimed a consensus for non-retribution of the child (Tusi, 1987).…”

Variational approach, soliton solutions and singular solitons for new coupled ZK system

“…So we should search for a mathematical algorithm to discover the exact solutions of nonlinear partial differential equations. In recent years, powerful and efficient methods explored to find analytic solutions of nonlinear equations have drawn a lot of interest by a variety of scientists, such as Adomian decomposition method [2], the homotopy perturbation method [3,4], some new asymptotic methods searching for solitary solutions of nonlinear differential equations, nonlinear differential-difference equations and nonlinear fractional differential equations using the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform and ancient Chinese mathematics [4], the variational iteration method [5,6] which is used to introduce the definition of fractional derivatives [7,4], the He's variational approach [8], the extended homoclinic test approach [9,10], homogeneous balance method [11][12][13][14], Jacobi elliptic function method [15][16][17][18], Băclund transformation [19,20], G ′ /G expansion method for nonlinear partial differential equation [21,22], and fractional differential-difference equations of rational type [23][24][25] It is important to point out that a new constrained variational principle for heat conduction is obtained recently by the semi-inverse method combined with separation of variables [26], which is exactly the same with He-Lee's variational principle [27]. A short remark on the history of the semi-inverse method for establishment of a generalized variational principle is given in [28].…”

Using He's variational method to seek the traveling wave solution of PHI-four equation

“…Also, the explicit formulas may provide physical information and help us to understand the mechanism of related physical models. Recently, many effective methods to construct traveling wave solutions of NLPDEs have been established, such as Adomian decomposition method [1],the homotopy perturbation method [2],the variational iteration method [3,4] , the He's variational approach [5], the F −expansion method [6], three-wave method [7], extended homoclinic test approach [8,9], the ( [6] and the exp-function method [10].…”