There are solutions of (32.2.1) such that

where

and $d$ and ${\theta }_0$ are constants.

There are also solutions of (32.2.1) such that

Next, for given initial conditions $w(0)=0$ and $w^{\prime }(0)=k$, with $k$ real, $w(x)$ has at least one pole on the real axis. There are two special values of $k$, $k_1$ and $k_2$, with the properties , , and such that:

1. (a)

   If , then $w(x)>0$ for , where $x_0$ is the first pole on the negative real axis.

2. (b)

   If , then $w(x)$ oscillates about, and is asymptotic to, $-\sqrt{\frac{1}{6}|x|}$ as $x\to -\mathrm{\infty }$.

3. (c)

   If , then $w(x)$ changes sign once, from positive to negative, as $x$ passes from $x_0$ to $0$.

For illustration see Figures 32.3.1 to 32.3.4, and for further information see Joshi and Kitaev (2005), Joshi and Kruskal (1992), Kapaev (1988), Kapaev and Kitaev (1993), and Kitaev (1994).

Consider the special case of ${\text{P}}_{\text{II}}$ with $\alpha =0$:

with boundary condition

Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\mathrm{Ai}\left(x\right)$, for some nonzero real $k$, where $\mathrm{Ai}$ denotes the Airy function (§9.2). Conversely, for any nonzero real $k$, there is a unique solution $w_k(x)$ of (32.11.4) that is asymptotic to $k\mathrm{Ai}\left(x\right)$ as $x\to +\mathrm{\infty }$.

If , then $w_k(x)$ exists for all sufficiently large $|x|$ as $x\to -\mathrm{\infty }$, and

where

and $d$ $(\ne 0)$, ${\theta }_0$ are real constants. Connection formulas for $d$ and ${\theta }_0$ are given by

where $\mathrm{\Gamma }$ is the gamma function (§5.2(i)), and the branch of the $\mathrm{ph}$ function is immaterial.

If $|k|=1$, then

If $|k|>1$, then $w_k(x)$ has a pole at a finite point $x=c_0$, dependent on $k$, and

For illustration see Figures 32.3.5 and 32.3.6, and for further information see Ablowitz and Clarkson (1991), Bassom et al. (1998), Clarkson and McLeod (1988), Deift and Zhou (1995), Segur and Ablowitz (1981), and Suleĭmanov (1987). For numerical studies see Miles (1978, 1980) and Rosales (1978).

Replacement of $w$ by $\mathrm{i}w$ in (32.11.4) gives

Any nontrivial real solution of (32.11.12) satisfies

where

with $d$ $(\ne 0)$ and $\chi$ arbitrary real constants.

In the case when

with $n\in \mathbb{Z}$, we have

where $k$ is a nonzero real constant. The connection formulas for $k$ are

In the generic case

we have

where $\sigma$, $\rho$ $(>0)$, and $\theta$ are real constants, and

The connection formulas for $\sigma$, $\rho$, and $\theta$ are

where

For ${\text{P}}_{\text{III}}$, with $\alpha =-\beta =2\nu$ $(\in ℝ)$ and $\gamma =-\delta =1$,

where $\lambda$ is an arbitrary constant such that , and

where $B$ and $\sigma$ are arbitrary constants such that $B\ne 0$ and . The connection formulas relating (32.11.25) and (32.11.26) are

See also Abdullaev (1985), Novokshënov (1985), Its and Novokshënov (1986), Kitaev (1987), Bobenko (1991), Bobenko and Its (1995), Tracy and Widom (1997), and Kitaev and Vartanian (2004).

Consider ${\text{P}}_{\text{IV}}$ with $\alpha =2\nu +1$ $(\in ℝ)$ and $\beta =0$, that is,

and with boundary condition

Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to $h{U}^2(-\nu -\frac{1}{2},\sqrt{2}x)$ as $x\to +\mathrm{\infty }$, where $h$ $(\ne 0)$ is a constant. Conversely, for any $h$ $(\ne 0)$ there is a unique solution $w_h(x)$ of (32.11.29) that is asymptotic to $h{U}^2(-\nu -\frac{1}{2},\sqrt{2}x)$ as $x\to +\mathrm{\infty }$. Here $U$ denotes the parabolic cylinder function (§12.2).

Now suppose $x\to -\mathrm{\infty }$. If , where

then $w_h(x)$ has no poles on the real axis. Furthermore, if $\nu =n=0,1,2,\mathrm{\dots }$, then

Alternatively, if $\nu$ is not zero or a positive integer, then

where

and $d$ $(>0)$ and ${\theta }_0$ are real constants. Connection formulas for $d$ and ${\theta }_0$ are given by

where

and the branch of the $\mathrm{ph}$ function is immaterial.

Next if $h=h^{\ast }$, then

and $w_{h^{\ast }}(x)$ has no poles on the real axis.

Lastly if $h>h^{\ast }$, then $w_h(x)$ has a simple pole on the real axis, whose location is dependent on $h$.

For illustration see Figures 32.3.7–32.3.10. In terms of the parameter $k$ that is used in these figures $h=2^{3/2}{k}^2$.

See also Wong and Zhang (2009a).