VARIATIONAL APPROACH TO SOLITARY SOLUTIONS USING JACOBI-ELLIPTIC FUNCTIONS. Yue Wu

Σχετικά έγγραφα
Envelope Periodic Solutions to Coupled Nonlinear Equations

ECON 381 SC ASSIGNMENT 2

Diversity soliton solutions for the (2+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation

Approximation of distance between locations on earth given by latitude and longitude

New Soliton and Periodic Solutions for Nonlinear Wave Equation in Finite Deformation Elastic Rod. 1 Introduction

4.6 Autoregressive Moving Average Model ARMA(1,1)

D Alembert s Solution to the Wave Equation

Concrete Mathematics Exercises from 30 September 2016

Homework 3 Solutions

2 Composition. Invertible Mappings

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Other Test Constructions: Likelihood Ratio & Bayes Tests

Commutative Monoids in Intuitionistic Fuzzy Sets

Second Order RLC Filters

Second Order Partial Differential Equations

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

A summation formula ramified with hypergeometric function and involving recurrence relation

Example Sheet 3 Solutions

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

Lanczos and biorthogonalization methods for eigenvalues and eigenvectors of matrices

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

EE512: Error Control Coding

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

The kinetic and potential energies as T = 1 2. (m i η2 i k(η i+1 η i ) 2 ). (3) The Hooke s law F = Y ξ, (6) with a discrete analog

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

On a four-dimensional hyperbolic manifold with finite volume

The Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

( y) Partial Differential Equations

Matrices and Determinants

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

Όλνκα πνπδάζηξηαο: Γξεγνξία αββίδνπ Α.Δ.Μ:7859. Δπηβιέπνλ Καζεγεηήο: Παζραιίδεο Αζαλάζηνο ΑΝΩΣΑΣΟ ΣΔΥΝΟΛΟΓΗΚΟ ΔΚΠΑΗΓΔΤΣΗΚΟ ΗΓΡΤΜΑ ΚΑΒΑΛΑ

Numerical Analysis FMN011

derivation of the Laplacian from rectangular to spherical coordinates

Math221: HW# 1 solutions

The Simply Typed Lambda Calculus

Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

Section 8.3 Trigonometric Equations

Congruence Classes of Invertible Matrices of Order 3 over F 2

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Homomorphism in Intuitionistic Fuzzy Automata

Dr. D. Dinev, Department of Structural Mechanics, UACEG

Statistical Inference I Locally most powerful tests

High order interpolation function for surface contact problem

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

6.3 Forecasting ARMA processes

Solution Series 9. i=1 x i and i=1 x i.

Cyclic or elementary abelian Covers of K 4

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω

ON NEGATIVE MOMENTS OF CERTAIN DISCRETE DISTRIBUTIONS

ST5224: Advanced Statistical Theory II

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee

14 Lesson 2: The Omega Verb - Present Tense

: Monte Carlo EM 313, Louis (1982) EM, EM Newton-Raphson, /. EM, 2 Monte Carlo EM Newton-Raphson, Monte Carlo EM, Monte Carlo EM, /. 3, Monte Carlo EM

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

Solutions to Exercise Sheet 5

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation

CRASH COURSE IN PRECALCULUS

Laplace Expansion. Peter McCullagh. WHOA-PSI, St Louis August, Department of Statistics University of Chicago

J. of Math. (PRC) 6 n (nt ) + n V = 0, (1.1) n t + div. div(n T ) = n τ (T L(x) T ), (1.2) n)xx (nt ) x + nv x = J 0, (1.4) n. 6 n

PARTIAL NOTES for 6.1 Trigonometric Identities

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

Test Data Management in Practice

ADVANCED STRUCTURAL MECHANICS

Tridiagonal matrices. Gérard MEURANT. October, 2008

ΓΕΩΜΕΣΡΙΚΗ ΣΕΚΜΗΡΙΩΗ ΣΟΤ ΙΕΡΟΤ ΝΑΟΤ ΣΟΤ ΣΙΜΙΟΤ ΣΑΤΡΟΤ ΣΟ ΠΕΛΕΝΔΡΙ ΣΗ ΚΤΠΡΟΤ ΜΕ ΕΦΑΡΜΟΓΗ ΑΤΣΟΜΑΣΟΠΟΙΗΜΕΝΟΤ ΤΣΗΜΑΣΟ ΨΗΦΙΑΚΗ ΦΩΣΟΓΡΑΜΜΕΣΡΙΑ

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

The k-α-exponential Function

Συντακτικές λειτουργίες

Strain gauge and rosettes

Part III - Pricing A Down-And-Out Call Option

( ) 2 and compare to M.

Solutions to the Schrodinger equation atomic orbitals. Ψ 1 s Ψ 2 s Ψ 2 px Ψ 2 py Ψ 2 pz

Reminders: linear functions

Abstract Storage Devices

C.S. 430 Assignment 6, Sample Solutions

Partial Trace and Partial Transpose

Differential equations

HW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1)

Homomorphism of Intuitionistic Fuzzy Groups

Instruction Execution Times

SOME PROPERTIES OF FUZZY REAL NUMBERS

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

Every set of first-order formulas is equivalent to an independent set

J. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS

Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ "ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ ΥΓΕΙΑΣ "

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Lecture 26: Circular domains

Distances in Sierpiński Triangle Graphs

w o = R 1 p. (1) R = p =. = 1

Transcript:

Mathematical and Comptational Applications, Vol., No., pp. 9-93,. Association for Scientific Research VARIATIONAL APPROACH TO SOLITARY SOLUTIONS USING JACOBI-ELLIPTIC FUNCTIONS Ye W Economical Mathematics Office, Department of Economics and Management Shanghai Uniersity of Political Science and Law, Shanghai 7, China Email: noodlew@mail.dh.ed.cn Abstract- Partial differential eqations are transformed into ordinary differential eqations, and a ariational formlation is then established. The trial fnction is chosen sing Jacobi-elliptic fnction with some nknown parameters similar to the exp-fnction method. Varios approximate solitary soltions are obtained when making the obtained ariational principle stationary with respect to each nknown parameter in the trial-fnction. The copled Zakharo-Kznetso eqations are sed as an example to elcidate the soltion procedre. Keywords- Variational theory, semi-inerse method, solitary soltion, exp-fnction method, Jacobi-elliptic fnction. INTRODUCTION Solitary soltion is a hot topics in nonlinear science. Recently many new analytical methods were appeared to find solitary soltions of arios nonlinear eqations, sch as the exp-fnction method[-], the mlti-wae method[6], the G /G -expansion method[7], the tanh-method[8], the ariational iteration method[9-], ariational methods[-]and others[-]. Most analytical methods aim to searching for exact soltions withot considering the bondary/initial conditions, sch soltions might hae no physical nderstandings as pointed ot by Ji-Han He in his reiew article[9]. In this paper we trn or attention to approximate solitary soltions sing a ariational method. Generally speaking, there exist two basic ways to describe a physical problem[3] : ( by differential eqations (DE with bondary or initial conditions; ( by ariational principles (VP. The VP model has many adantages oer its DE partner: simple and compact in form while comprehensie in content, encompassing implicitly almost all information characterizing the problem nder consideration. In this paper, we will apply the semi-inerse method[3] to the search for a ariational principle for the discssed problem, then the trial-solitary soltions are expressed sing Jacobi-elliptic Fnctions.. VARIATIONAL FORMULATION In this paper, we will consider the copled Zakharo-Kznetso(ZK eqations []: t xxx yyx 6x x ( δ λ η 6μ α ( t xxx yyx x x x

Y. W 9 to illstrate the soltion procedre To seek the traeling solitary soltion of eqations(-(, we se the following transformation ( x, y, t, ( x, y, t, ξ kx ly wt Eqs.(-( become: k l w6k k ( ( ( ( 6 3 ''' ' ' δk λkl w kη kμ αk ( Integrating the aboe eqations, we obtain k l '' 3k wk d ( ( ( 3 '' δk λkl w kη 3kμ αk d (6 where dand d are constants. We apply the semi-inerse method to search for its ariational principle. According to the semi-inerse method[3], we write a Lagrangian in the form: ( ( ' 3 L k l k w ( k d F( (7 where F is an nknown fnction of. The Eler-Lagrange eqation with respect to is Eq.(. The Eler-Lagrange eqation with respect to reads: δ F k (8 δ δ F where is called He s ariational deriatie with respect to [3], defined as: δ δ F F F F (9 δ t t x x In iew of Eqation (6, we set: δ F k ( δ k 3 λ kl '' ( w k η μ d ( δ α From (, the nknown F can be identified as follows 3 ' 3 d F( ( δk λkl ( ( w kη kμ ( α α α α Finally we obtain the needed Lagrangian ( ( ' 3 L k l k w ( k d ( 3 ' 3 d ( δk λkl ( ( w kη kμ α α α α and the ariational formlation (3 (

9 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions (, J Ldξ ( ( ' 3 ( k l k w k d 3 ' 3 d ( δ k λkl ( ( w kη kμ dξ α α α α (3 3. TRIAL FUNCTIONS USING JACOBI ELLIPTIC FUNCTIONS In this section, we set d d a soltion in a general form d m for simplicity. The exp-fnction method[] admits nc q p a exp( nξ b n m exp( mξ Similarly in this paper we assme the soltion can be expressed in Jacobi elliptic fnctions: d nc q m p D NC Q MP asn n n bsn m m Asn n N B sn In order to make the soltion procedre mch simpler, we consider a simple case ξ A Aisn( ξ A i sn( ξ (7 where ( ( M M ( ( (6 ξ B B isn( ξ B i sn( ξ (8 A, A, A, B, B, B are nknown constants to be frther determined. sn is sn( ξ sn ξ / m, m (<m<, is the modls of the the Jacobi elliptic fnction, ( Jacobi elliptic fnction. Sbstitting (7 and (8 into Eq.(, we obtain J A, A, A, B, B, B L( A, A, A, B, B, B dξ (9 ( J J Setting and Ai Bi solitary soltions. 3. Stationary with respect to A Making J in Eq.(9 stationary with respect to A, J A respectiely, we can obtain arios approximate ( i i and setting each coefficients of cn( ξ dn( ξ sn, (, ce am ξ sn i to zero, we obtain a set of eqations for A, A, A, B, B, B : sn ξ ( (

Y. W 93 3 3-3αkmA α m ( k l A 3 αk (8 3 m m A αm ( ( k l ( 3 m m w( m A 3 αkm ( m A A 3 αkm ( m B α k m A 9α A A k m α B k m 9α A A k m 3 3 α A k m α B k m 3 αm ( ( k l ( m w A α A m w 8 α A k αa A km αa km αb km 6αA k m 6αA l m 9αA km αaakm 8αAkm 8αAAkm 3αBkm αbkm 6αAkm 6αAlm 3 3 3 3 3 9αAkm 6αAkm αakm 6αAlm αalm αamw 3α A m w α Am w 9 α A k m α A k m α A l m 8 α A k m α A A k m 8 α A k m α B k m 6 α A k m 6 α A l m 3 3 3 3 3 3 9 α A k m 6 α A A k m α A k m 8 α A A k m 3 α B k m α B k m 3 3 α A k m α A l m 8 α A A 9 α 3 α 8 α α A k m 8 α A l m α A l m α A k m α A l m α A m w 3 3 3 α A m w α A m w 3 α A m w km A km B km A k m 3 3 3 3 6 α A k m 6 α A l m 9 α A k m 8 α A A k m 3 α B k m 6 α A k m 3 6 α A l m 9 α A k m 6 α A k m α A k m 6 α A l m α A l m 3 3 3 3 3 3 3 α A k m 6 α A A k m α A k m α B k m 8 α A k m 8 α A l m 8 α A k m 8 α A l m α A m w 8 α A k m α A A k m α A k m α B k m α A A k m 8 α A k m α B k m α A m w3 α A m w α A m w 3 3 3 3 α A k m 6 α A A k m α A k m α B k m α A k m 3 3 α A l m 8 α A k m 8 α A l m α A m w 3 3 8 α A k m α A k m α A l m α A k m α A l m 3 8 α A k m α A k m α A l m Soling the aboe eqations simltaneosly with the help of some a mathematical software, we obtain the following soltion A, A, (k l A, ( m 6 m(k l ( k l w( k l B,, B B ( Sbstitting ( into (7 and (8, the traeling wae soltion is obtained which

9 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions reads ( ξ (k l isn ξ ( ( 6 m(k l ( m( k l w( k l i sn (3 sn tanh ξ, we hae the following special solitary wae soltion When m, ( ( ξ (k l tanh( i ξ ( 6 (k l 8 ( k l w( k l ξ i coth ( ( 3. Stationary condition with respect to A Similarly setting J A ( i i and setting each coefficients of cn( ξ dn( ξ sn, (, ce am ξ sn i to zero, we obtain a set of eqations for A, A, A, B, B, B : sn ξ ( 3 3 3 A k m A k m A l m 3 3 3 3 3 A A k m A k m B k m A k m A l m 8 α α α 8 α A k m α A k m α A l m 6 α α α 8 α 8 α 3 α A k m 8 α A k m 8 α A l m α A m w 6 α A A km α A km α B km 8 α A k m 8 α A l m 3 3 α A k m 8 α A k m 8 α A l m α A m w α A m w 3 3 3 3 3 9 α A km α A A km 8 α A A km 8 α A km 3 α B km 3 3 3 3 3 α B k m α A k m 6 α A k m α A l m 6 α A l m α A A km α A km α B km 6 α A k m 6 α A l m 3 3 8 α A k m 6 α A k m 6 α A l m α A m w α A m w α A m w α A m w 3 3 3 3 9 α A km 9 α A km 8 α A A km α A A km 8 α A A km 3 3 3 3 3 8 α A km 3 α B km 3 α B km α B km α A k m 3 3 3 6 α A k m α A l m 6 α A l m 6 α A A k m 8 α A km α B k m α A k m α A k m α A l m α A l m 3 3 3 α A k m 8 α A k m 8 α A l m 3 α A m w3 α A m w α A m w α A m w (6

Y. W 9 3 3 3 9 α A km 9 α A km α A A km 8 α A A km 3 3 3 3 3 8 α A k m 3 α B k m α B k m α A k m 6 α A k m 3 3 α A l m 6 α A l m α A A km α A km α B km 6 α A k 6 α 8 α 6 α 6 α m A l m A k m A k m A l m 3 3 A m w A m w A m w 3 α α α 9 α A k m α A A k m 8 α A A k m 8 α A k m 3 α B k m α B k m α A k m 6 α A k m α A l m 6 α A l m α A A km α A km α B km 6 α A k m 6 α A l 6 6 6 8 α A k m 6 α A k m 6 α A l m α A m w α A m w α A m w 3 9 α A km 9 α A km 8 α A A km α A A km 8 α A A km 8 α A km 3 α B km 3 α B km α B km α A k m 6 α A k m α A l m 6 α A l m 6 α A A k m 8 α A α B k m α A k m α A k m α A l m α A l m 6 6 6 α A km 8 α A k m 8 α A l m 3 α A m w 3 α A m w α A m w α A m w 3 9 α A km 9 α A km α A A km 8 α A A km 8 α A km m km 3 α B k m α B k m α A k m 6 α A k m α A l m 6 α A l m α A A k m α A k m α B k m 6 α A k 6 6 6 6 α A l m 8 α A k m 6 α A k m 6 α A l m 3 α A m w α α A m w A m w Soling the aboe ten eqations, we obtain the following soltions Case 8 ( m ( k l A,, A ( k l A 6 6 l ( k l m m k l B 6 km( k l k 8 ( k l w 8 ( k l m w ( ( B, m ( k l 3 w k l B ( ( m (

96 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions Case 8 ( m ( k l m(k l (k l A, A, A 8 8( 7mm ( k l 3w( m( k l B 7 k 8m( m( k l 3mw( k l B ( m( k l 3w( k l B ( Case 3 A, m ( k l A, A m( m 6m ( k l ( k l mw( k l B, B, ( B The aboe cases lead to the following soltions 8 ( m ( k l ( k l 3 ( ξ isn ( ξ (3 ( 6 l ( k l m 6 m ( k l 3 6 km( k l k ( m k l w( k l 8 ( k l w 8 ( k l m w ( 3 ( ξ ( m k l 8 ( m(k l (k l 88( 7mm ( k l 3w( m( k l ξ 7 k 8m( m( k l 3mw( k l isn ( m( k l 3w( k l isn ( isn( ξ ( isn( ξ isn ( (6

Y. W 97 m ( k l i sn (7 6m ( k l m( m( k l mw( k l sn tanh ξ, the aboe soltions trn ot to be familiar solitary i sn (8 When m, ( soltions, which read 6 6 7 6( k l ( k l coth( ξ i (9 ( ξ ( ( 6 6 6 ( k l k l wk l k ( 8( k l 3 w k l icoth (3 6( k l ( k l ( tanh( k i ξ l i coth( ξ (3 ( k l w( k l ( k l w( k l 8 6 6 3 7 k 7 8 ( k l w( k l itanh 8 3 icoth (3 ( k l tanh( ξ i ξ (33 ( 8 ( k l w( k l 6 ( k l 8 itanh (3 3.3. Stationary condition with respect to B J By setting, and doing the same as illstrated before, we obtain a set of B eqations for A, A, A, B, B, B : 3 3 3 3 δ μ B k m 9B k m B k l m λ

98 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions 3 3 3 aa km 6 B ek m 3 aa km aa km B ek m 3 aa km 3 3 3 3 3 δ δ μ μ 8 B e k m B k m B k m 8 B k B B k m 8 B k m μ μ μ μ 9 B k m 6 B B k m B k m 8 B B 3 μ 8 B B km 3 3 3 9 B km μ 6 B kl mλ B kl m λ8 B kl m λ B kl m λ 3 λ B k l m B m w3 B m w B m w3 B m w B k m η 3 3 B k m η B k m η 3 B k m η km μ 3 3 3 3 α A k m 6 B δ k m3 α A k m α A k m 6 B δ k m 6 B δ k m 3 3 δ μ μ μ μ B k m 8 B k B B k m B k m 9 B k m B B k m 3 8 B k m μ8 B B k m μ 9 B k m μ 6B k l m λ 6 B k l m λ 3 3 6 B k l m λ B k l m λ B m w 3 B m w B m w B k m η η B k m 3 B k m η μ 3 3 B δ k m 9 B k m μ B k l m λ 3 3 3 3 3 α A km 6 B δ k m 3 α A km α A km B δ k m 3 α A km 3 3 3 8 B δ k m B δ k m B δ k m 8 B kmμ B B km μ 3 3 3 3 8 B km μ9 B km μ6 B B km μ B km μ8b B k m μ 3 8 B B km μ 9 B km μ 6 B kl m λ B kl m λ 8 B kl m λ 3 3 B k l m λ B k l m λ B m w 3 B m w B m w 3 3 3 B m w B km η 3B km η B km η 3 B km η 3 3 3 3 3 3 6 B δk m 3 α A k m 6 B δ k m 6 B δ k m B δ k m 3 3 3 9 B k m μ 8 B B k m μ 9 B k m μ 6 B k l m λ 6 B k l m λ 6 B k l m λ B k l m λ 3 3 3 α A km 8 B δ k m 8 B δ k m B kmμ 6 B B km μ B km μ 3 8 B k l m λ8 B k l m λ B m w B k m η 3 α A km α A km 8 B kmμ B B km μ B km μ 3 3 3 3 B B k m μ 8 B k m μ B m w3 B m w B m w B k m η 3 3 3 B k m η B k m η 3 3 3 3 3 α A km B δ k m 8 B δ k m B km μ6 B B km μ 3 3 3 3 B km μ B kl m λ 8 B kl m λ B m w B km η 3 3 3 3 B δ k m B δ k m 8 B km μ B kl m λ B kl mλ 3 3 B δ k m 8 B k m μ B k l m λ The aboe system admits the following soltion

Y. W 99 A A 6 m( δ k l λ 3 αμ, A, k m k l k l w k 3 α k μ ( ( δ λ ( δ λ ( η B, B, ( δ k l λ B (3 This leads to the following solitary soltion 6 m( δk l λ 9 3 αμ k( m( δk λl ( δk λl ( w kη (36 isn 3 α k μ 9 ( ξ ( δ k l λ isn (37 When m, sn tanh soltions:, Eqations(36- (37 admit the following solitary wae ( δ λ ( δ λ ( w η 6( 8k k l k l k δk l λ 3 αμ 3 αk μ i coth (38 ( δ k l λ coth( i ξ (39 3. Stationary condition with respect to B Similarly by setting J / B, we obtain a set of eqations for A, A, A, B, B, B : 3 3 3 3 3 3 3 3 3 3 3 α A k m 8 B δ k m 8 B δ k m 6 B B k m μ B k m B δ k m 8 B k m μ B k l m λ B δ k m 8 B k m μ B k l m λ 3 3 3 μ λ λ η 3 3 B km 8 B kl m 8 B kl m B m w B km α A k m 8 B δ k m 8 B δ k m 6 B B k m μ B k m μ B k m 3 3 3 3 λ λ η η 3 3 3 3 3 3 3 A km A km B k m B k m A km B k m 8 B k l m 8 B k l m B m w B m w B k m B k m 3 α α δ 6 δ α 6 δ 3 3 3 3 3 6 B δ k m 9 B km μ B B km μ8 B B km μ8 B km μ 3 3 B B km μ B km μ 8 B km μ B kl m λ 6 B k l m λ 3 3 6 B k l m λ 6 B k l m λ B m w B m w B m w B m w 3 3 B k m η B k m η B k m η B k m η μ μ

9 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions 3 3 3 3 3 3 3 3 α A km 3 α A km α A km Bδ k m 6 B δ k m α A km 3 3 3 3 3 δ δ δ μ μ B k m B k m 8 B k m 9 B k m 9 B k m 8 B B k m 3 3 3 B B km μ 8 B B km μ 8 B km μ 6B B k m μ 8 B k m μ 3 3 B km μ B kl m λ 6 B kl m λ B kl m λ B kl m λ 3 3 3 3 λ 8 B k l m 3 B m w3 B m w B m w B m w3 B k m η 3 3 η η 3 B k m B k m B k m η μ 3 3 3 3 3 3 3 3 α A km α A km B δ k m 6 B δ k m α A km 6 B δ k m 3 3 3 3 6 B δ k m 9 B k m μ9 B k m μ B B k m μ8 B B k m μ 3 μ μ μ 8 B k m B B k m B k m 8 B k m 3 3 3 λ λ λ 3 3 3 μ B k l m λ 6 B k l m 6 B k l m 6 B k l m 3 B m w B m w B m w 3 B km η B km η B km η 3 3 3 3 α A km α A km B δ k m 6 B δ k m α A km 6 B δ k m 3 6 δ μ μ μ 6 B k m 9 B km B B km 8 B B km 8 B km μ 6 μ μ μ B B km B km 8 B km B kl m λ 6 B k l m 6 λ λ 6 B k l m 6 B k l m B m w B m w B m w B k m η B k m B k m η λ η 3 3 3 3 3 6 3 δ δ μ μ μ μ μ 6B B k m 8B k m 6 μ λ λ λ 6 λ η η η k m η 3 3 α A k m α A k m B δ k m B δ k m α A k m 3 3 6 3 B δ k m B δ k m B km μ B km μ B B km μ 6 B B km μ B km μ B B km μ B km μ B km μ 6 B kl m λ B kl m λ B kl m λ B kl m λ B m w B m w B m w B k m η B k m η B k m η 3a A k m 3a A k m a A k m B e k m 6B e k m a A k m B e k m B k m 8B k m 9B k m 9B k m 8B B k m B B k m 8B B k m 8B k m μ μ μ B k m B k l m 6B k l m B k l m B k l m λ 8B k l m 3B m w3b m wb m wb m w3b k m 3B k m B k m B 3 6 6 6 9 9 8 8 8 6 6 6 3 3 Soling the aboe ten eqations reslts the following cases Case

Y. W 9 A ( δ λ ( η 6km( δk l λ 6 k ( m ( δk l λ 8m ( k l wk μ 9 μ 3 α k k m δk λl 3 δk λl w kη A 9 α k μ A, ( ( ( ( B 8 ( m ( δk λl, 9 μ B, ( δ k λ l B ( Case 6 ( δ k m δ k l m λ l m λ A, 3 αμ A km( m( δ k λ l m( δ k λ l ( w kη 3 α kμ, A B, m( δ k λ l B, B ( The aboe two cases yield the following solitary soltions ( δ λ ( η 6km( δk l λ 6 k ( m ( δk l λ 8m ( k l wk μ 9 μ 3 α k k( m( δk λl 3( δk λl ( w kη isn 9 α k μ ( ξ ( 8 ( m ( δk λl ( δ k λ l isn ( ξ (3 9 μ 6 ( δ k m δ k l m λ l m λ 3 αμ km m k l m k l w k 3 α kμ ( ( δ λ ( δ λ ( η isn, ( m( δ k λ l i sn ( When m, sn tanh wae soltions respectiely:, Eqations(-( reslt in the following solitary

9 Variational Approach to Solitary Soltions sing Jacobi-elliptic Fnctions 3 3 ( ξ ( 6k( δk l λ 6( δk l λ 6( δk l λ wk η μ 9 μ 3 α k 8k( δk λl 3( δk λl ( w kη icoth 9 α k μ 6 ( δk λl ( δ k λ l coth( ξ (6 9 μ i (7 ( δ λ ( δ λ ( η 6( δk λl 8 k k l k l w k 3 αμ 3 αkμ i tanh (8 ( δ k λ l tanh( i ξ (9. CONCLUSION This paper actally coples seeral methods, i.e., the ariational theory, the semi-inerse, the exp-fnction method and elliptic fnction, in an effectie way. The soltion procedre is simple if some a mathematical software is sed. This paper sggests a new approach to solitary theory.. REFERENCES. J. H. He and X. H W, Exp-fnction method for nonlinear wae eqations, Chaos Soliton. Fract. 3, pp. 7-78, 6.. M.M. Kabir and A. Khajeh, New explicit soltions for the akhnenko and a generalized form of the nonlinear heat condction eqations ia exp-fnction method, International Jornal of Nonlinear Sciences and Nmerical Simlation, 37-38, 9. 3. A.Esen and S. Ktlay, Application of the Exp-fnction method to the two dimensional sine-gordon eqation, International Jornal of Nonlinear Sciences and Nmerical Simlation, 3-39, 9.. A. Ebaid, Generalization of He's Exp-Fnction Method and New Exact Soltions for Brgers Eqation, Zeitschrift fr Natrforschng A 6, 6-68, 9.. S. Zhang, Exp-fnction Method: Solitary, Periodic and Rational Wae Soltions of Nonlinear Eoltion Eqations, Nonlinear Science Letters A, 3-6,. 6. Z. D. Dai, C. J. Wang, S. Q. Lin, D. L. Li and G. M, The Three-wae Method for Nonlinear Eoltion Eqations, Nonlinear Science Letters A, 77-8,. 7. H.A. Zedan, New Classes of Soltions for a System of Partial Differential Eqations by sing G /G -expansion Method, Nonlinear Science Letters A, 9-38,. 8. E. M. E. Zayed and H. M. A. Rahman, The Extended Tanh-method for Finding Traeling Wae Soltions of Nonlinear Partial Differential Eqations, Nonlinear Science Letters A., 93-,.

Y. W 93 9. J. H. He, G. C. W and F. Astin, The Variational Iteration Method Which Shold Be Followed, Nonlinear Science Letters A,-3,.. N. Herian and V. Marinca, A Modified Variational Iteration Method for Strongly Nonlinear Problems, Nonlinear Science Letters A,, 83-9,.. W. J. Li, New Solitary Wae Soltion for the Bossinesq Wae Eqation Using the Semi-Inerse Method and the Exp-Fnction Method, Zeitschrift fr Natrforschng A 6, 79-7, 9.. K. W. Chow, An Exact, Periodic Soltion of the Kap-Newell Eqation, Nonlinear Science Letters A, 83-89,. 3. Y. Long and W. G. Ri, Variable Wae-form Periodic Soltions for the Dispersie Wae Eqation of a Generalized Camassa-Holm Eqation, International Jornal of Nonlinear Sciences and Nmerical Simlation., 69-7, 9.. S. Pak, Solitary Wae Soltions for the RLW Eqation by He's Semi inerse Method, International Jornal of Nonlinear Sciences and Nmerical Simlation, -8, 9.. Z. L. Tao, Solitary Soltions of the Boiti-Leon-Manna-Pempinelli Eqation Using He's Variational Method, Zeitschrift Fr Natrforschng Section A 63, 63-636, 8. 6. L. X, Variational approach to solitons of nonlinear dispersie K(m, n eqations, Chaos Solitons & Fractals. 37, 37-3, 8. 7. T. Ozis and A. Yidirim, Application of He's Semi-inerse Method to the Nonlinear Schrodinger Eqation, Compters & Mathematics With Applications, 39-, 7. 8. J. Zhang, Variational Approach to Solitary Wae Soltion of the Generalized Zakharo Eqation, Compters & Mathematics with Applications., 3-6, 7. 9. Y. W, Variational Approach to the Generalized Zakharo Eqations, International Jornal of Nonlinear Sciences and Nmerical Simlation,, -7, 9.. Y. W, Variational Approach to Higher-order Water-wae Eqations, Chaos Solitons & Fractals, 3, 9-98, 7.. A. M.Wazwaz, On Mltiple Soliton Soltions for Copled KdV-mKdV Eqation, Nonlinear Science Letters A, 89-96,.. C. Q. Dai, X. J. Pan and N. Zho, Interactions between Solitons in a (-dimensional Generalized Ablowitz-Kap-Newell-Segr System, Nonlinear Science Letters A, 6-66,. 3. J. H. He, Variational Principles for Some Nonlinear Partial Differential Eqations With Variable Coefficients, Chaos Solitons & Fractals 9, 87-8,.. M. A. Abdo and A. Elhanbaly, Constrction of Periodic and Solitary Wae Soltions by the Extended Jacobi elliptic fnction expansion method, Commnications in Nonlinear Science and Nmerical Simlation., 9-, 7.

2025 © DocPlayer.gr Πολιτική Απορρήτου | Όροι Χρήσης | Σχόλια