Series New Exact Solutions to Nonlinear Nizhnik-Novikov-Veselov System Analytical Solution, Fixed Point Theory of Partially Ordered Space

ABSTRACT

One new solving expression is built for Nizhnik-Novikov-Veselov system in the paper. Through corresponding auxiliary equation arrangement, more than 150 analytical solutions of elementary and Jacobi elliptic functions are obtained so that the NNV system has a wider range of physical meaning. At the same time, the existence and uniqueness of this systematic solution are discussed by fixed point theory of partially ordered space. The expression of the unique solution could be gained if making use of the technique of computer.

One new solving expression is built for Nizhnik-Novikov-Veselov system in the paper. Through corresponding auxiliary equation arrangement, more than 150 analytical solutions of elementary and Jacobi elliptic functions are obtained so that the NNV system has a wider range of physical meaning. At the same time, the existence and uniqueness of this systematic solution are discussed by fixed point theory of partially ordered space. The expression of the unique solution could be gained if making use of the technique of computer.

Cite this paper

nullJ. Hao, W. Zhong and X. Yan, "Series New Exact Solutions to Nonlinear Nizhnik-Novikov-Veselov System Analytical Solution, Fixed Point Theory of Partially Ordered Space,"*Applied Mathematics*, Vol. 1 No. 5, 2010, pp. 406-410. doi: 10.4236/am.2010.15053.

nullJ. Hao, W. Zhong and X. Yan, "Series New Exact Solutions to Nonlinear Nizhnik-Novikov-Veselov System Analytical Solution, Fixed Point Theory of Partially Ordered Space,"

References

[1] Y. Chen and Q. Wang, “A Series of New Double Periodic Solutions to a (2+1)-Dimensional Asymmetric Nizhnik- Novikov-Veselov Equation,” Chinese Physics, Vol. 13, No. 11, 2004, pp. 1796-1800.

[2] C. Q. Dai and G. Q. Zhou, “Exotic Interactions between Solitons Of the (2+1)-Dimensional Asymmetric Nizhnik- Novikov-Veselov System,” Chinese Physics, Vol. 16, No. 5, 2007, pp. 1201-1208.

[3] S. H. Ma, J. P. Fang and Q. B. Ren, “New Mapping Solutions and Localized Structures for the (2+1)- Dimensional Asymmetric Nizhnik-Novikov-Veselov System,” Acta Physica Sinica, Vol. 56, No. 12, 2007, pp. 6784-6790. (in Chinese)

[4] B. Q. Li and Y. L. Ma, “(G′/G)-Expansion Method and New Exact Solutions for (2+1)-Dimensional Asymmetrical Nizhnik-Novikov-Veselov System,” Acta Physica Sinica, Vol. 58, No. 7, 2009, pp. 4373-4378. (in Chinese)

[5] D. C. Lu, B. J. Hong and L. X. Tian, “Explicit and Exact Solutions to the Variable Coefficient Combined KdV Equation with Forced Term,” Acta Physica Sinica, Vol. 55, No. 11, 2006, pp. 5617-5622. (in Chinese)

[6] H. M. Li, “Mapping Deformation Method and Its Application to Nonlinear Equations,” Chinese Physics, Vol. 11, No. 11, 2002, pp. 1111-1114.

[7] H. M. Li, “New Exact Solutions of Nonlinear Gross- Pitaevskii Equation with Weak Bias Magnetic and Time- Dependent Laser Fields,” Chinese Physics, Vol. 14, No. 2, 2005, pp. 251-256.

[8] Taogetusang, Sirendaoerji, “A Method for Constructing Exact Solutions of Nonlinear Evolution Equation with Variable Coefficient,” Acta Physica Sinica, Vol. 58, No. 7, 2009, pp. 2121-2126. (in Chinese)

[9] X. L. Yan, “Existence and Uniqueness Theorems of Solution for Symmetric Contracted Operator Equations and Their Applications,” Chinese Science Bulletin, Vol, 36, No. 10, 1991, pp. 800-805.

[10] X. L. Yan, “Exact Solution to Nonlinear Equation,” Economic Science Press, Beijing, 2004. (in Chinese)

[11] J. P. Hao and X. L. Yan, “Exact Solution of Large Deformation Basic Equations of Circular Membrane under Central Force,” Journal of Applied Mathematics and Mechanics, Vol. 27, No. 10, 2006, pp. 1333-1337.

[1] Y. Chen and Q. Wang, “A Series of New Double Periodic Solutions to a (2+1)-Dimensional Asymmetric Nizhnik- Novikov-Veselov Equation,” Chinese Physics, Vol. 13, No. 11, 2004, pp. 1796-1800.

[2] C. Q. Dai and G. Q. Zhou, “Exotic Interactions between Solitons Of the (2+1)-Dimensional Asymmetric Nizhnik- Novikov-Veselov System,” Chinese Physics, Vol. 16, No. 5, 2007, pp. 1201-1208.

[3] S. H. Ma, J. P. Fang and Q. B. Ren, “New Mapping Solutions and Localized Structures for the (2+1)- Dimensional Asymmetric Nizhnik-Novikov-Veselov System,” Acta Physica Sinica, Vol. 56, No. 12, 2007, pp. 6784-6790. (in Chinese)

[4] B. Q. Li and Y. L. Ma, “(G′/G)-Expansion Method and New Exact Solutions for (2+1)-Dimensional Asymmetrical Nizhnik-Novikov-Veselov System,” Acta Physica Sinica, Vol. 58, No. 7, 2009, pp. 4373-4378. (in Chinese)

[5] D. C. Lu, B. J. Hong and L. X. Tian, “Explicit and Exact Solutions to the Variable Coefficient Combined KdV Equation with Forced Term,” Acta Physica Sinica, Vol. 55, No. 11, 2006, pp. 5617-5622. (in Chinese)

[6] H. M. Li, “Mapping Deformation Method and Its Application to Nonlinear Equations,” Chinese Physics, Vol. 11, No. 11, 2002, pp. 1111-1114.

[7] H. M. Li, “New Exact Solutions of Nonlinear Gross- Pitaevskii Equation with Weak Bias Magnetic and Time- Dependent Laser Fields,” Chinese Physics, Vol. 14, No. 2, 2005, pp. 251-256.

[8] Taogetusang, Sirendaoerji, “A Method for Constructing Exact Solutions of Nonlinear Evolution Equation with Variable Coefficient,” Acta Physica Sinica, Vol. 58, No. 7, 2009, pp. 2121-2126. (in Chinese)

[9] X. L. Yan, “Existence and Uniqueness Theorems of Solution for Symmetric Contracted Operator Equations and Their Applications,” Chinese Science Bulletin, Vol, 36, No. 10, 1991, pp. 800-805.

[10] X. L. Yan, “Exact Solution to Nonlinear Equation,” Economic Science Press, Beijing, 2004. (in Chinese)

[11] J. P. Hao and X. L. Yan, “Exact Solution of Large Deformation Basic Equations of Circular Membrane under Central Force,” Journal of Applied Mathematics and Mechanics, Vol. 27, No. 10, 2006, pp. 1333-1337.