Decompositions of the Kadomtsev-Petviashvili equation and their symmetry reductions

Starting with a decomposition conjecture,we carefully explain the basic decompositions for the Kadomtsev-Petviashvili(KP)equation as well as the necessary calculation procedures,and it is shown that the KP equation allows the Burgers-STO(BSTO)decomposition,two types of reducible coupled BSTO decompositions and the BSTO-KdV decomposition.Furthermore,we concentrate ourselves on pointing out the main idea and result of Bäcklund transformation of the KP equation based on a special superposition principle in the particular context of the BSTO decompositions.Using the framework of standard Lie point symmetry theory,these decompositions are studied and the problem of computing the corresponding symmetry constraints is treated.

Symmetry, Reductions and New Exact Solutions of ANNV Equation Through Lax Pair

In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the （2＋1）-dimensional Lax pair is reduced to （1＋1）-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.

BACKLUND TRANSFORMATION AND SIMILARITY REDUCTIONS OF BOUSSINESQ EQUATION

The homogeneous balance method, which is simple and straightforward, is extended to seek for Backlund transformation, exact bell shape soliton solutions and similarity reduction of the Boussinesq equation. The method can be used in general.

Chinese Public's Willingness to Pay for CO＿2 Emissions Reductions: A Case Study from Four Provinces/Cities

Contingent Valuation Method(CVM) was used to investigate the Chinese public's willingness to pay(WTP) for a policy to reduce CO2 emissions. Face to face interviews were conducted to collect 1,653 valid questionnaires from Beijing, Shanghai, Shandong province, and Fujian province. A model was constructed to understand the factors that influence WTP. The results indicate that the Chinese public is willing to pay CN 201.86 annually to support the policy of reducing greenhouse gas emissions. Participants from Beijing show the highest WTP, followed by participants from Fujian and Shandong, while those from Shanghai report the lowest WTP. The findings reveal that participants with higher income, higher satisfaction with their current life, and awareness of climate issues are willing to pay more for CO2 emissions reductions. In addition, those who are young, male and members of the Communist Party also indicate a higher WTP. The results imply that translating the public's willingness to protect climate into actions should be taken into account in China's low carbon policy. There is a need to consider the difference in degree of willingness, among different social groups, to pay for emissions reductions if the market-based mechanisms such as carbon tax were designed to facilitate emissions reductions.

Similarity Reductions and Painlevé Property of Coupled KdV Equations

Using Ablowitz-Ramani Segur algorithm, the coupled KdV systems are reclassified under the Painlevé integrable sense while the similarity reductions of the model are obtained by using the Clarkson and Kruskal＇s direct method. Some new types of Painlevé integrable models including a model with different dispersion relations for two layer fluids are found.

Symmetries and Similarity Reductions of a New (2+1)-Dimensional Shallow Water Wave System

The symmetries of a （2＋1）-dimensional shallow water wave system, which is newly constructed through applying variation principle of analytic mechanics, are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. The （1＋1） dimensional displacement shallow water wave equation can be derived from the reductions when special symmetry parameters are chosen.

Lie Symmetries,1-Dimensional Optimal System and Optimal Reductions of(1+2)-Dimensional Nonlinear Schrodinger Equation

For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.

A Note on Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

Applying the classical Lie symmetry approach to the barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a β-plane channel, we find a new symmetry, which is not presented in a previous work [F. Huang, Commun. Theor. Phys. （Beijing, China） 42 （2004） 903]. A general finite transformation group is obtained based on the full Lie point symmetry, Furthermore, two new types of similarity reduction solutions are obtained.

Similarity Reductions of Nearly Concentric KdV Equation

Basing on the direct method developed by Clarkson and Kruskal, the nearly concentric Korteweg-de Vries （ncKdV） equation can be reduced to three types of （1＋1）-dimensional variable coefficients partial differential equations （PDEs） and three types of variable coefficients ordinary differential equation. Furthermore, three types of （1＋1）-dimensional variable coefficients PDEs are all reduced to constant coefficients PDEs by some transformations.

Similarity Reductions of Nonisospectral BKP Equation

Basing on the direct method developed by Clarkson and Kruskal,the nonisospectral BKP equation can bereduced to three types of(1+1)-dimensional variable coefficients partial differential equations(PDEs).Furthermore,onthe basis of the idea of the symmetry group direct method by Lou et al.,three types of reduction PDEs are all reducedto the related constant coefficients PDEs by some transformations.

Symmetry Reductions of a 1.5-Layer Ocean Circulation Model

The （2＋1）-dimensional nonlinear 1.5-layer ocean circulation model without external wind stress forcing is analyzed by using the classical Lie group approach. Some Lie point symmetries and their corresponding two-dimensional reduction equations are obtained.

Painlevé property, local and nonlocal symmetries, and symmetry reductions for a (2+1)-dimensional integrable KdV equation

The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.

What Do Tariff Reductions Mean?

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Reductions in non-point source pollution through different management practices for an agricultural watershed in the Three Gorges Reservoir Area

Non-point source water pollution generated by agricultural production is considered a major environmental issue in the Three Gorges Reservoir Area （TGRA） of China. The Annualised Agricultural Non-Point Source Pollution （AnnAGNPS） model was selected to assess the impact of the application of various management treats, including seven crops, five fertilizer levels and three-group management practice scenarios, on water quality from Heigou River Watershed in TGRA. The scenario subsets include conservation tillage practice （CTP）, conservation reserve program （CRP） and conversion of cropland into forestland program （CCFP）. Results indicated that tea can not be replaced by other crops because comparatively tea resulted in a higher sediment yield. CTP with no-tillage was more effective to reduce sediment yield, but could increased nutrient loss. CRP reduced sediment yield significantly, but slightly benefited on nutrient loss. CCFP reduced not only sediment yield but also the nutrient loss significantly. The conversion of cropland with a slope greater than 10° into forestland was found to be the best scenario as the sediment yield export is less than 5 tons/ha and nutrient loss is within the permissible limit.

Conditional Similarity Reductions of Jimbo—Miwa Equation via the Classical Lie Group Approach

Recently, the Clarkson and Kruskal direct method has been modified to find new similarity reductions (conditional similarity reductions) of nonlinear systems and the results obtained by the modified direct method cannot be obtained by the current classical and/or non-classical Lie group approach. In this paper, we show that the conditional similarity reductions of the Jimbo-Miwa equation can be reobtained by adding an additional constraint equation to the original model to form a conditional equation system first and then solving the model system by means of the classical Lie group approach.

Regio-and Stereo-selective Reductions of Steroidal 4-en-3,6-dione

The different regioselective and stereoselective products were obtained by the reduction of stigmast-4,22-dien-3,6-dione with NaBH_4-CH_3OH when different kinds of metal ions was added to the reaction.

Crystallographic texture and earing behavior analysis for different second cold reductions of double-reduction tinplate

Since the production of tinplate with non-earing properties is difficult, especially when it is produced via the double-reduction process, the optimal degree of second cold reduction is particularly important for achieving desirable drawing properties. The evolution of texture and the earing propensity of double-reduction tinplate with different extents of second reduction were investigated in this study. Optical microscopy and scanning electron microscopy were used to observe the changes in the microstructure at various extents of reduction. Two common testing methods, X-ray diffraction（XRD） and electron backscatter diffraction, were used to investigate the texture of the specimens, which revealed the effects of deformation percentage on the final texture development and the change in the grain boundary. The earing rate was determined via earing tests involving measurement of the height of any ear. The results obtained from both XRD analyses and earing tests revealed the same ideal value for the second cold reduction on the basis of the relationship between crystallographic texture and the degree of earing.

Multiple Soliton-Like Solutions and Similarity Reductions of a Spherical Kadomtsev-Petviashvili Equation from Plasma Physics

A spherical Kadomtsev-Petviashvili （SKP） equation for dust acoustic or ion-acoustic waves is studied. Similarity reductions of the SKP equation are obtained with the one-parameter （ε） Lie group of infinitesimal transformations and Clarkson-Kruskal direct method, The SKP equation is also solved with a generalized tanh function method.

Symmetry Reductions, Integrability and Solitary Wave Solutions to High-Order Modified Boussinesq Equations with Damping Term

Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.

New Similarity Reductions and Compacton Solutions for Boussinesq-Like Equations with Fully Nonlinear Dispersion

In this paper, similarity rcductions of Boussinesq-like equations with nonlinear dispersion (simply called B(m, n) equations) utt = (un)xx + (um) which is a generalized model of Boussinesq equation uts = (u2)xx + u and modified Bousinesq equation utt = (u3)xx + uxxxx, are considered by using the direct reduction method. As a result,several new types of similarity reductions are found. Based on the reduction equations and some simple transformations,we obtain the solitary wave solutions and compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they re-emerge with the same coherent shape) of B(1, n) equations and B(m, m)equations, respectively.