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The Existence of Traveling Wave Solution to Zakharov-Kuznetsov Type Equation

Author: HuYue
Tutor: ChenGuoWang
School: Zhengzhou University
Course: Basic mathematics
Keywords: KP equation ZK equation Traveling wave solutions Periodic traveling wave solution Anti- periodic traveling wave solutions
CLC: O175.2
Type: Master's thesis
Year: 2007
Downloads: 46
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Abstract


This paper consists of four chapters. Chapter Introduction; Chapter II study contains two parameters λ, μ Zakharov-Kuznetsov type equation; Chapters III and IV corresponding to the research results of the existing literature on the KP-type equations, respectively monotonous approach and variational method to study periodic traveling wave solutions and counter-cyclical existence of traveling wave solutions of a broad class of non-homogeneous ZK equation. The details are as follows: In the second chapter, we study the following equation: where is a given constant. When λ = 0, μ = 1, KZ type equations; When μ = 0, λ = 1 KP type equation. Equation (1) is a coupling of the ZK-type equation and the KP equation. By four Lemma last mountain pass lemma proved the existence of traveling wave solutions. The main results are as follows: Theorem 1 Let the following conditions are true: (f 1 ), f ∈ C 1 (R, R), f (0) = 0 and there are p-> , C 1 > 0, such that | f '(s) | the ≤ C 1 | s | | s | p-2 ; (f 2 ) exists v ∈ E such that F (tv) / t 2 → ∞ (t → ∞); (the f 3 ) exists γ > 2 such that (?) u ∈ R, γF (u) ≤ uf (u). Equation (1) is shaped like a W (x, y, t) = u (x-ct, y) of the traveling wave solutions. In the third chapter, we have the following existence of a class of generalized nonhomogeneous ZK-type equation countercyclical traveling wave solutions and periodic traveling wave solutions. Here f ∈ C 1 , α> 0, β ≥ 0 is a known constant, g * for the variable x, y, and T of the real-valued function. First, the problem (2) the existence of anti-periodic traveling wave solutions transformed into the following equivalent problem: assuming: (H 1 ) f: R → R is a monotone increasing continuous function; (H 2 ) G ∈ C ([0, T]; R). We have the following theorem: Theorem 2 (H 1 ) and (H 2 ) was established, and μ ≥ 0, the unique solution U (s) ∈ C < sup> 2 [0, T]. Theorem 3 Let μ ≥ 0 and f n , G n (n = 1,2, ...) satisfies (H 1 ) and (H 2 ). If f n → f (C [0, L]), 0 n , U n is the solution of the problem, U n → U 2 (in C [0, T]), where U is the problem (3) the solution. Theorem 4 Let (H 1 ) and (H 2 ) was established, if μ <0, when the the 1 (μ/4π) T 2 > 0, the problem (3) has at least one solution of equation (2) periodic traveling wave solutions of the problem of the existence of processing is completely analogous to the case of the counter-cyclical. First into equivalence problem: we have Theorem 5 Let (H 1 ) and (H 2 ) was established, and μ ≥ 0, problem (5) unique solution U (s) ∈ C 2 [0, T]. Theorem 6 Let μ ≥ 0 and f n , G n (n = 1,2, ...) satisfies (H 1 ) and (H 2 ). If f n → f (C [0, L]), 0 n , U n is the solution of the problem, then the U n U-→ (in C 2 [0, T]). Where U is the solution of the problem (5). Theorem 7 (H 1 ) and (H 2 ) was established. If μ 1/2 ) T 2 > 0, the problem (5) has at least one solution U ∈ C < sup> 2 [0, T]. In the fourth chapter, we apply the variational method of equation (2) counter-cyclical existence of traveling wave solutions and periodic traveling wave solutions. Will prove equation (2) periodic traveling wave solutions of the existence of the problem into seeking the solution of the following equivalent problem, through three lemmas main results are as follows: the theorem 8 set (i) If μ ≥ 0, the problem at least one solution U ∈ C p 2 [0, T] (?) C 2 (R). (Ii) If μ 1/2 π / μ | 1/2 , the problem (7) at least one solution U ∈ C < sub> p 2 [0, T] (?) C 2 (R). Proof of equation (2) is similar to the problem of the existence of anti-periodic traveling wave solutions comes down for the sake of the solution of the following problem: The main results are as follows: Theorem 9 set reason 8 was established, (ⅰ) μ ≥ 0, the problem (8 ) there are at least a solution U-∈ C a 2 [0, T] (?) C 2 (R); (ii) If μ <0, T <π / (2 | μ |) 1/2 , the problem (8) there is at least one solution U ∈ C a . 2 [0, T] (?) C 2 (R)

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