| This paper consists of four chapters. The first chapter is introduction. In the second chapter , we study the Zakharov-Kuznetsov type equation with two parameterλ,μ. In the third chapter and in the fourth chapter , we use the monotonic method and variational method to study the existence of antiperiodic and periodic traveling wave solution of the generalized inhomogeneous KZ equation.In the second chapter , we study the following equation:Wt + Wxxx +λDx-1Wyy +μWyyx + (f(W))x = 0, t≥0, (x, y)∈R2, (1)Where ,f∈C1(R, R),λ≥0 andμ≥0 are given constants. Ifλ=0 andμ= 1, it is called KZ type equation ; ifμ= 0,λ= 1 it is called KP type equation.Equation ( 1 ) is a coupled equation of ZK type equation and KP type equation. By establishing four lemmas and applying mountain pass theorem , we prove the following main result:Theorem 1 Suppose that the following assumptions hold :(f1) f∈C1(R, R), f(0) = 0 and there exist p > 3 and C1 > 0 such that |f'(s)|≤C1(|s| + |s|p-2;(f2) There exists v∈E such that F(tv)/t2→+∞, (t→+∞); (f3) There existsγ> 2 such that ,γF(u)≤uf(u).Then the equation (1) has a solution of the form W(x,y,t) = u(x - ct,y).In the third chapter , we study the existence of antiperiodic and periodic traveling wave solution of the following generalized inhomogeneous ZK equation :ut +αuxxx +βuyyx + (f(u))x = g*(x, y; t) t≥0, (x, y)∈R2, (2)where f∈C1(R),α> 0 andβ> 0 are given constants, g* is a real function of x,y,t.Fist of all , we deduce from the problem (2) the following equivalent problem:Assume that:(H1) f : R→R is continnuous and monotone nonincreasing;(H2) G∈([0,T]; R).Theorem 2 Let (H1) and (H2) be satisfied, andμ≥0 , then the problem (3) has a unique solution U(s)∈C2[0,T].Theorem 3 Letμ≥0 and fn, Gn (n = 1, 2,…) satisfy condtions (H1) and (H2).If Un is the solution of the following problemandfn→f in C[0, l], 0 < l < +∞,Gn→G in C[0,T]),then Un→U (in C2[0,T]) , where U is the solution of problem (3). Theorem 4 Suppose (H1) and (H2) hold ,μ< 0 and , then the problem (3) has at least one solution U∈C2[0,T].Similar to the existence of antiperiodic traveling wave solution to the equation (2).The equation (2) is equivalent to the following problem:We haveTheorem 5 Assume that (H1) and (H2) hold, andμ≥0, then the problem (5) has a unique solution U(s)∈C2[0,T].Theorem 6 Letμ≥0, fn, Gn (n = 1,2,…). satify condtions (H1) and (H2).If Un is the solution of the following problem :andthen Un→U (inC2[0, T]), where U is the solution of the problem (5).Theorem 7 Suppose (H1) and (H2) hold ,μ< 0 and , then the probem (5) has at least one solution U∈C2[0,T].In the fourth chapter , we use the variational methods to study the existence of antiperiodic and periodic traveling wave solution to the problem (2) . The existence of periodic solution to the equation (2) is equivalent to solve the following problem: The main results are the following :Theorem 8 Suppose that(i) Ifμ≥0, then the problem (8) has at least one solution U∈Cp2[0,T] (?) C2(R).(ii) Ifμ< 0 and , then the problem (8) has at least one solution U∈Cp2[0,T] (?)C2(R,).The existence of antiperiodic solution to the equation (2) is equivalent to solve the following problem:Theorem 9 Let the condition of the Theorem 8 holds .(i) Ifμ≥0, then the problem (8) has at least one solution U∈Ca2[0,T] (?) C2(R).(ii) Ifμ< 0. T <π/sqrt2|μ|, then the problem (8) has at least one solution U∈Ca2[0,T] (?) C2(R).
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