Non-linear electrostatic waves in pair-ion plasmas - National Centre ...

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Double layer in Pair-ion Plasmas (contd.)The mKdV equation is described asA ∂τϕ23 31+ Q ∂ξϕ1+B ∂ξϕ1+ ∂ξϕ1=0(14)where the coefficients are defined asB= ⎡ 1 1 ⎤A 2λ⎢+2 2 2 ⎥⎣(λ − μ)( λ −1)2⎦2⎡⎛ 2 1 ⎞⎛ 2 μ ⎞⎢⎜ 2 λ − ⎟⎜ 2 λ −2 22 2⎟⎢ (3λ− 1) ⎝ 3 ⎠ (3λ− 1)= −+−⎝ 3 ⎠2 52 42524⎢ 2 ( λ − 1) ( λ − 1) 2 ( λ − μ ) ( λ − μ )⎢⎣22(3λ− μ)(3λ−1)Q = −2 3 2 32( λ − μ)2( λ −1)2⎤⎥⎥⎥⎥⎦Now using the transformationη = ξ −Uτthe above mKdV equation22
Double layer in Pair-ion Plasmas (contd.)can be described in the form of energy integral equation as follows,1 ⎛ ∂ϕ1⎞⎜ ⎟2 ⎝ ∂η⎠2+ V ( ϕ ) =10(15)where the Sagdeev potential is defined asUA 2 Q 3 BV ( ϕ1)= − ϕ1+ϕ1+ϕ2 3 4For the formation of DLs, the Sagdeev potential must satisfy the following conditions,41(16)VddVϕ1)= 0, = 0 at ϕ1= 0 and ϕ = ϕm.dϕ(12V2dϕQ1=1< 0 at ϕ1= 0 and ϕ1= ϕm−32B ϕmU= −12BϕA2m(17)The above conditions are satisfied only if , coefficients ‘Q’ and ‘U’ described as,23
Page 1 and 2: Nonlinear electrostatic structures
Page 3 and 4: 1. Soliton‣ Soliton is a nonlinea
Page 5 and 6: Soliton in Plasmas (contd.)Linear w
Page 7 and 8: 2. Brief Introduction of pair-ion P
Page 9 and 10: Production of pair-ion (fullerene)
Page 11 and 12: Electrostatic Waves in pair-ion Pla
Page 13 and 14: Nonlinear Electrostatic Waves in Pa
Page 15 and 16: Nonlinear Electrostatic Waves in pa
Page 17 and 18: Nonlinear Electrostatic Waves in pa
Page 19 and 20: Nonlinear Electrostatic Waves in Pa
Page 21: Double Layer in Plasmas5. Double La
Page 25 and 26: Double layer in Pair-ion Plasmas (c
Page 27 and 28: 6. Dissipative shocks and solitons
Page 29 and 30: Dissipative shocks and solitons in
Page 31 and 32: Dissipative shocks and solitons in
Page 33 and 34: 7. Conclusion• Linear and nonline
Page 35: Thank You35

Non-linear electrostatic waves in pair-ion plasmas - National Centre ...

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